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25 \+ June 2002" }}{PARA 0 "" 0 "Top" {TEXT 23 72 "------------------------- -----------------------------------------------" }}{PARA 0 "" 0 "" {TEXT 23 72 "--------------------------------------------------------- ---------------" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 257 "" 0 "" {TEXT -1 8 " " }{HYPERLNK 17 "Procedures" 1 "" "Procedures" } {TEXT -1 0 "" }{TEXT 431 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 0 "" }{TEXT 432 8 " " }{HYPERLNK 17 "Notat ion" 1 "" "Procedure Notation" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 8 " " }{HYPERLNK 17 "Varia te Generation" 1 "" "Variate Generation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 8 " " }{HYPERLNK 17 "Continuous Distributions" 1 "" "Continuous Distributions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 8 " " }{HYPERLNK 17 "Discrete Distributions" 1 "" "Discrete Distributions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 23 72 "--------------------------------------------------------- ---------------" }}{PARA 0 "" 0 "" {TEXT 23 72 "---------------------- --------------------------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 23 72 "------------------------------------------ ------------------------------" }}{PARA 257 "" 0 "Procedure Notation" {TEXT 435 18 "Procedure Notation" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT 23 72 "--------------------------------------------------------- ---------------" }}{PARA 257 "" 0 "" {TEXT -1 503 "\n X and Y a re random variables\n Greek letters are numeric or symbolic par ameters\n x is numeric or symbolic\n n and r are positiv e integers, n >= r\n low and high are numeric\n g is a f unction\n Brackets [] denote optional parameters\n \"dou ble quotes\" denote character strings\n MATRIX is a 2 x 2 array of random variables\n A capitalized parameter indicates that i t must be\n entered as a list --> ex. Data := [1, 12.4, 34, 52. 45, 63]" }}{PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "Top" }}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 23 72 "-------------------------------------------------- ----------------------" }}{PARA 257 "" 0 "Variate Generation" {TEXT -1 0 "" }{TEXT 434 18 "Variate Generation" }}{PARA 0 "" 0 "" {TEXT 23 72 "------------------------------------------------------------------ ------" }}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 33 " ArcTanVariate(alpha, phi)" }}{PARA 257 "" 0 "" {TEXT -1 8 " " }{HYPERLNK 17 "BinomialVariate(n, p, m)" 1 "" "Binomia lVariate(n, p, m)" }{TEXT -1 35 "\n ExponentialVariate(lambda) " }}{PARA 257 "" 0 "" {TEXT -1 41 " NormalVariate(mu, sigma)\n \+ " }{HYPERLNK 17 "UniformVariate()" 1 "" "UniformVariate()" }} {PARA 257 "" 0 "" {TEXT -1 8 " " }{HYPERLNK 17 "WeibullVariate( lambda, kappa, m)" 1 "" "WeibullVariate(lambda, kappa, m)" }}{PARA 0 " " 0 "" {HYPERLNK 17 "To Top" 1 "" "Top" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 23 72 "----- -------------------------------------------------------------------" } }{PARA 0 "" 0 "Continuous Distributions" {TEXT 436 24 "Continuous Dist ributions" }}{PARA 0 "" 0 "" {TEXT 23 72 "---------------------------- --------------------------------------------" }}{PARA 0 "" 0 "" {TEXT 23 19 "# Procedure names:" }}{PARA 0 "" 0 "" {TEXT 23 22 "# \+ ArcSinRV" }}{PARA 0 "" 0 "" {TEXT 23 22 "# ArcTanRV" }} {PARA 0 "" 0 "" {TEXT 23 20 "# BetaRV" }}{PARA 0 "" 0 "" {TEXT 23 22 "# CauchyRV" }}{PARA 0 "" 0 "" {TEXT 23 19 "# \+ ChiRV" }}{PARA 0 "" 0 "" {TEXT 23 25 "# ChiSqu areRV" }}{PARA 0 "" 0 "" {TEXT 23 33 "# DoublyNoncentralFR V" }}{PARA 0 "" 0 "" {TEXT 23 33 "# DoublyNoncentralTRV" } }{PARA 0 "" 0 "" {TEXT 23 22 "# ErlangRV" }}{PARA 0 "" 0 " " {TEXT 23 21 "# ErrorRV" }}{PARA 0 "" 0 "" {TEXT 23 27 "# ExponentialRV" }}{PARA 0 "" 0 "" {TEXT 23 32 "# \+ ExponentialPowerRV" }}{PARA 0 "" 0 "" {TEXT 23 28 "# Ex tremeValueRV" }}{PARA 0 "" 0 "" {TEXT 23 17 "# FRV" }} {PARA 0 "" 0 "" {TEXT 23 21 "# GammaRV" }}{PARA 0 "" 0 "" {TEXT 23 33 "# GeneralizedParetoRV" }}{PARA 0 "" 0 "" {TEXT 23 24 "# GompertzRV" }}{PARA 0 "" 0 "" {TEXT 23 32 " # HyperbolicSecantRV" }}{PARA 0 "" 0 "" {TEXT 23 32 "# \+ HyperExponentialRV" }}{PARA 0 "" 0 "" {TEXT 23 31 "# \+ HypoExponentialRV" }}{PARA 0 "" 0 "" {TEXT 23 19 "# ID BRV" }}{PARA 0 "" 0 "" {TEXT 23 31 "# InverseGaussianRV" } }{PARA 0 "" 0 "" {TEXT 23 29 "# InvertedGammaRV" }}{PARA 0 "" 0 "" {TEXT 23 18 "# KSRV" }}{PARA 0 "" 0 "" {TEXT 23 23 "# LaPlaceRV" }}{PARA 0 "" 0 "" {TEXT 23 24 "# \+ LogGammaRV" }}{PARA 0 "" 0 "" {TEXT 23 24 "# LogisticR V" }}{PARA 0 "" 0 "" {TEXT 23 27 "# LogLogisticRV" }} {PARA 0 "" 0 "" {TEXT 23 25 "# LogNormalRV" }}{PARA 0 "" 0 "" {TEXT 23 21 "# LomaxRV" }}{PARA 0 "" 0 "" {TEXT 23 23 "# MakehamRV" }}{PARA 0 "" 0 "" {TEXT 23 20 "# \+ MuthRV" }}{PARA 0 "" 0 "" {TEXT 23 35 "# NoncentralChi SquareRV" }}{PARA 0 "" 0 "" {TEXT 23 27 "# NoncentralFRV" }}{PARA 0 "" 0 "" {TEXT 23 27 "# NoncentralTRV" }}{PARA 0 "" 0 "" {TEXT 23 22 "# NormalRV" }}{PARA 0 "" 0 "" {TEXT 23 22 "# ParetoRV" }}{PARA 0 "" 0 "" {TEXT 23 24 "# \+ RayleighRV" }}{PARA 0 "" 0 "" {TEXT 23 30 "# Standar dCauchyRV" }}{PARA 0 "" 0 "" {TEXT 23 30 "# StandardNormal RV" }}{PARA 0 "" 0 "" {TEXT 23 34 "# StandardTriangularRV " }}{PARA 0 "" 0 "" {TEXT 23 31 "# StandardUniformRV" }} {PARA 0 "" 0 "" {TEXT 23 17 "# TRV" }}{PARA 0 "" 0 "" {TEXT 23 26 "# TriangularRV" }}{PARA 0 "" 0 "" {TEXT 23 23 "# UniformRV" }}{PARA 0 "" 0 "" {TEXT 23 23 "# \+ WeibullRV" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 57 "# Other APPL Procedures Called: HypoExponentialRV calls \+ " }}{PARA 0 "" 0 "" {TEXT 23 62 "# Expo nentialRV and Convolution" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 63 "# Purpose: Def ine the common parametric continuous univariate" }}{PARA 0 "" 0 "" {TEXT 23 51 "# probability distributions shown below:" }} {PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 76 "# Distr ibution Support Parameters ParameterRestrictions " }}{PARA 0 "" 0 "" {TEXT 23 72 "# ------------ ------- --- ------- ---------------------" }}{PARA 0 "" 0 "" {TEXT 23 40 "# A rcSin 0 < x < 1 none" }}{PARA 0 "" 0 "" {TEXT 23 62 "# ArcTan x >= 0 alpha, phi alpha > 0; " }} {PARA 0 "" 0 "" {TEXT 23 67 "# \+ -inf < phi < inf" }}{PARA 0 "" 0 "" {TEXT 23 70 "# Beta \+ 0 <= x <= 1 Shape alpha, alpha > 0; beta > 0" }}{PARA 0 "" 0 "" {TEXT 23 46 "# shape beta " }}{PARA 0 "" 0 "" {TEXT 23 67 "# Cauchy -inf < x < inf L ocation a, -inf < a < inf; " }}{PARA 0 "" 0 "" {TEXT 23 61 "# \+ Scale alpha alpha > 0 " }}{PARA 0 "" 0 "" {TEXT 23 69 "# Chi x >= 0 Shape n p ositive integer n" }}{PARA 0 "" 0 "" {TEXT 23 69 "# ChiSquare \+ x >= 0 Shape n positive integer n" }}{PARA 0 "" 0 "" {TEXT 23 18 "# DblyNoncentralF" }}{PARA 0 "" 0 "" {TEXT 23 18 "# Dbl yNoncentralT" }}{PARA 0 "" 0 "" {TEXT 23 61 "# Erlang x >= 0 Scale lambda, lambda > 0" }}{PARA 0 "" 0 "" {TEXT 23 69 "# shape n positive integer n" }}{PARA 0 "" 0 "" {TEXT 23 70 "# Error -inf < x < inf mu, alpha, mu > 0; alpha >= 1;" }}{PARA 0 "" 0 "" {TEXT 23 65 "# \+ location d -inf < d < inf" }}{PARA 0 "" 0 "" {TEXT 23 61 "# Exponential x >= 0 Scale lambd a lambda > 0" }}{PARA 0 "" 0 "" {TEXT 23 72 "# Exponent Pwr x \+ >= 0 Scale lambda, lambda > 0; kappa > 0" }}{PARA 0 "" 0 "" {TEXT 23 47 "# shape kappa" }}{PARA 0 "" 0 "" {TEXT 23 70 "# ExtremeValue -inf < x < inf Scale ? alp ha, alpha > 0; beta > 0" }}{PARA 0 "" 0 "" {TEXT 23 48 "# \+ shape ? beta" }}{PARA 0 "" 0 "" {TEXT 23 71 "# \+ F x >= 0 Shape n1, positive ints n1, n2" }}{PARA 0 "" 0 "" {TEXT 23 44 "# sha pe n2" }}{PARA 0 "" 0 "" {TEXT 23 72 "# Gamma x >= 0 \+ Scale lambda, lambda > 0; kappa > 0" }}{PARA 0 "" 0 "" {TEXT 23 47 "# shape kappa" }}{PARA 0 "" 0 " " {TEXT 23 56 "# GenPareto x >= 0 gamma, delta, kapp a" }}{PARA 0 "" 0 "" {TEXT 23 71 "# Gompertz x >= 0 \+ Shape delta, delta > 0; kappa > 1" }}{PARA 0 "" 0 "" {TEXT 23 47 "# \+ shape kappa" }}{PARA 0 "" 0 "" {TEXT 23 40 "# HyperbolicSecant -inf < x < inf none" }}{PARA 0 "" 0 "" {TEXT 23 71 "# HyperExponential x >= 0 list p, list l all elts of list p: " }}{PARA 0 "" 0 "" {TEXT 23 62 "# \+ 0 <= p <= 1" }}{PARA 0 "" 0 "" {TEXT 23 70 "# all elts of \+ list l:" }}{PARA 0 "" 0 "" {TEXT 23 56 "# \+ l > 0" }}{PARA 0 "" 0 "" {TEXT 23 70 "# HypoExpo nential x >= 0 list l all elts of list l:" }}{PARA 0 "" 0 "" {TEXT 23 56 "# \+ l > 0" }}{PARA 0 "" 0 "" {TEXT 23 70 "# IDB x >= 0 \+ Shape gamma, gamma, delta >= 0, " }}{PARA 0 "" 0 "" {TEXT 23 61 "# delta, kappa kappa >= 0" }}{PARA 0 "" 0 "" {TEXT 23 69 "# InverseGaussian x > 0 Sca le lambda, lambda > 0; mu > 0" }}{PARA 0 "" 0 "" {TEXT 23 47 "# \+ location mu" }}{PARA 0 "" 0 "" {TEXT 23 70 "# InvertedGamma x >= 0 alpha, beta alpha > 0; beta > 0" }}{PARA 0 "" 0 "" {TEXT 23 5 "# KS" }}{PARA 0 "" 0 "" {TEXT 23 60 "# LaPlace -inf < x < inf omega, theta omega > 0" }} {PARA 0 "" 0 "" {TEXT 23 70 "# LogGamma -inf < x < inf alpha , beta alpha > 0; beta > 0" }}{PARA 0 "" 0 "" {TEXT 23 72 "# Logis tic -inf < x < inf Scale kappa kappa > 0; lambda > 0" }} {PARA 0 "" 0 "" {TEXT 23 52 "# shape (?) lambda" }}{PARA 0 "" 0 "" {TEXT 23 72 "# Log logistic x >= \+ 0 Scale lambda, lambda > 0; kappa > 0" }}{PARA 0 "" 0 "" {TEXT 23 47 "# shape kappa" }}{PARA 0 "" 0 "" {TEXT 23 68 "# LogNormal x >= 0 Scale mu, \+ -inf < mu < inf; " }}{PARA 0 "" 0 "" {TEXT 23 60 "# \+ shape sigma sigma > 0" }}{PARA 0 "" 0 "" {TEXT 23 72 "# Lomax x > 0 kappa, lambda kappa > 0; l ambda > 0" }}{PARA 0 "" 0 "" {TEXT 23 68 "# Makeham x >= 0 \+ Shape gamma, gamma, delta > 0;" }}{PARA 0 "" 0 "" {TEXT 23 60 "# delta, kappa kappa > 1" }} {PARA 0 "" 0 "" {TEXT 23 65 "# Muth x >= 0 Shape kappa 0 < kappa <= 1" }}{PARA 0 "" 0 "" {TEXT 23 32 "# Noncentral ChiSqre 0 <= x <= 1" }}{PARA 0 "" 0 "" {TEXT 23 32 "# NoncentralF \+ 0 <= x <= 1" }}{PARA 0 "" 0 "" {TEXT 23 32 "# NoncentralT 0 \+ <= x <= 1" }}{PARA 0 "" 0 "" {TEXT 23 67 "# Normal -inf < \+ x < inf Location mu, -inf < mu < inf;" }}{PARA 0 "" 0 "" {TEXT 23 60 "# scale sigma sigma > 0" }} {PARA 0 "" 0 "" {TEXT 23 54 "# Pareto x >= lambda Locat ion lambda, " }}{PARA 0 "" 0 "" {TEXT 23 72 "# \+ shape kappa lambda > 0; kappa > 0" }}{PARA 0 "" 0 "" {TEXT 23 61 "# Rayleigh x > 0 Scale lambda lambda > 0" }}{PARA 0 "" 0 "" {TEXT 23 40 "# StandardCauchy -inf < x < i nf none" }}{PARA 0 "" 0 "" {TEXT 23 40 "# StandardNormal -inf < x \+ < inf none" }}{PARA 0 "" 0 "" {TEXT 23 62 "# StandardTriangular0 <= x <= 1 m 0 <= m <= 1" }}{PARA 0 "" 0 "" {TEXT 23 40 "# \+ StandardUniform 0 <= x <= 1 none" }}{PARA 0 "" 0 "" {TEXT 23 69 "# T -inf < x < inf Shape n positive integer n " }}{PARA 0 "" 0 "" {TEXT 23 60 "# Triangular a <= x <= b M in a, mode m, a < m < b" }}{PARA 0 "" 0 "" {TEXT 23 41 "# \+ max b" }}{PARA 0 "" 0 "" {TEXT 23 69 "# Uniform a <= x <= b Min a, max b -inf < a < b < inf" }}{PARA 0 "" 0 "" {TEXT 23 72 "# Weibull x >= 0 Scale lambd a, lambda > 0; kappa > 0" }}{PARA 0 "" 0 "" {TEXT 23 47 "# \+ shape kappa" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 72 "# Arguments: The parameters of the di stribution of the random variable" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }} {PARA 0 "" 0 "" {TEXT 23 13 "# Algorithm:" }}{PARA 0 "" 0 "" {TEXT 23 54 "# 1. Check for the appropriate number of arguments" }} {PARA 0 "" 0 "" {TEXT 23 58 "# 2. Check parameter space when param eters are numeric" }}{PARA 0 "" 0 "" {TEXT 23 52 "# 3. Check to se e that the parameters are finite" }}{PARA 0 "" 0 "" {TEXT 23 55 "# \+ 4. Make assumptions about any symbolic parameters" }}{PARA 0 "" 0 "" {TEXT 23 56 "# 5. Assign a list of lists in the following format: " }}{PARA 0 "" 0 "" {TEXT 23 51 "# [[f(x)], [support], [\"Conti nuous\", \"XXX\"]]" }}{PARA 0 "" 0 "" {TEXT 23 71 "# where XXX \+ is one of the following: PDF, CDF, IDF, SF, HF, CHF" }}{PARA 0 "" 0 " " {TEXT 23 33 "# 6. Return the list of lists" }}{PARA 0 "" 0 "" {TEXT 23 2 "# " }}{PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "Top" }} {PARA 0 "" 0 "" {TEXT 437 78 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT 23 72 "------------------------------------- -----------------------------------" }}{PARA 0 "" 0 "Discrete Distribu tions" {TEXT -1 0 "" }{TEXT 438 22 "Discrete Distributions" }}{PARA 0 "" 0 "" {TEXT 23 72 "------------------------------------------------- -----------------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 23 30 "# Procedure names: BenfordRV" }}{PARA 0 "" 0 "" {TEXT 23 32 "# BernoulliRV" }}{PARA 0 "" 0 "" {TEXT 23 33 "# BingoCoverRV" }}{PARA 0 "" 0 "" {TEXT 23 31 "# BinomialRV" }}{PARA 0 "" 0 "" {TEXT 23 31 "# BirthdayRV" }}{PARA 0 "" 0 "" {TEXT 23 32 "# GeometricRV" }}{PARA 0 "" 0 "" {TEXT 23 39 "# \+ NegativeBinomialRV" }}{PARA 0 "" 0 "" {TEXT 23 30 "# \+ PoissonRV" }}{PARA 0 "" 0 "" {TEXT 23 38 "# \+ UniformDiscreteRV" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }} {PARA 0 "" 0 "" {TEXT 23 61 "# Purpose: Define the common parametric discrete univariate" }}{PARA 0 "" 0 "" {TEXT 23 51 "# prob ability distributions shown below:" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" } }{PARA 0 "" 0 "" {TEXT 23 48 "# NOTE: For each distribution, x is an \+ integer." }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 63 "# Distribution Support Parameters Parameter Space" } }{PARA 0 "" 0 "" {TEXT 23 63 "# ------------ ------- ------ ---- ---------------" }}{PARA 0 "" 0 "" {TEXT 23 52 "# Benford \+ x=1,2,...,9 none none" }}{PARA 0 "" 0 "" {TEXT 23 19 "# Bernoulli " }}{PARA 0 "" 0 "" {TEXT 23 44 "# BingoCover\011 x=24,25,...,75 none\011\011none" }}{PARA 0 "" 0 "" {TEXT 23 69 "# Binomial 0<=x<=n n, p n: pos int; 0 < p < 1" }}{PARA 0 "" 0 "" {TEXT 23 52 "# Birthday\011\011 x=1,2,...,365 \+ none none" }}{PARA 0 "" 0 "" {TEXT 23 57 "# Geometric \+ x >= 1 p 0 < p < 1" }}{PARA 0 "" 0 "" {TEXT 23 17 "# Hypergeometric" }}{PARA 0 "" 0 "" {TEXT 23 19 "# NegativeBinom ial" }}{PARA 0 "" 0 "" {TEXT 23 58 "# Poisson x >= 0 \+ lambda lambda > 0" }}{PARA 0 "" 0 "" {TEXT 23 18 "# UniformDis crete" }}{PARA 0 "" 0 "" {TEXT 23 3 "# " }}{PARA 0 "" 0 "" {TEXT 23 63 "# Arguments: The parameters of the distribution of the random" } }{PARA 0 "" 0 "" {TEXT 23 23 "# variable" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 13 "# Algorithm:" }} {PARA 0 "" 0 "" {TEXT 23 54 "# 1. Check for the appropriate number of arguments" }}{PARA 0 "" 0 "" {TEXT 23 58 "# 2. Check parameter space when parameters are numeric" }}{PARA 0 "" 0 "" {TEXT 23 52 "# \+ 3. Check to see that the parameters are finite" }}{PARA 0 "" 0 "" {TEXT 23 55 "# 4. Make assumptions about any symbolic parameters" }}{PARA 0 "" 0 "" {TEXT 23 56 "# 5. Assign a list of lists in the \+ following format:" }}{PARA 0 "" 0 "" {TEXT 23 49 "# [[f(x)], [s upport], [\"Discrete\", \"XXX\"]]" }}{PARA 0 "" 0 "" {TEXT 23 71 "# \+ where XXX is one of the following: PDF, CDF, IDF, SF, HF, CHF" } }{PARA 0 "" 0 "" {TEXT 23 33 "# 6. Return the list of lists" }} {PARA 0 "" 0 "" {TEXT 23 2 "# " }}{PARA 0 "" 0 "" {TEXT 23 70 "# Note : Could define a type in startup about p being between 0 and 1" }} {PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "Top" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 23 72 "--------------------------------------------------------------- ---------" }}{PARA 0 "" 0 "Procedures" {TEXT -1 0 "" }{TEXT 23 0 "" } {TEXT 264 11 "Procedures:" }}{PARA 0 "" 0 "" {TEXT 23 72 "------------ ------------------------------------------------------------" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {HYPERLNK 17 "AllCombination s(n, k)" 1 "" "AllCombinations(n,k)" }{TEXT 348 2 ", " }}{PARA 0 "" 0 "" {HYPERLNK 17 "AllPermutations(n)" 1 "" "AllPermutations(n)" }{TEXT 349 2 ", " }}{PARA 0 "" 0 "" {HYPERLNK 17 "Benford(X)" 1 "" "Benford(X )" }{TEXT 350 1 "," }}{PARA 0 "" 0 "" {HYPERLNK 17 "BootstrapRV(Data) " 1 "" "BootstrapRV(Data)" }{TEXT 351 2 ", " }}{PARA 0 "" 0 "" {HYPERLNK 17 "CDF(X, [x]))" 1 "" "CHF(X, [x])" }{TEXT 352 1 "," }} {PARA 0 "" 0 "" {HYPERLNK 17 "CHF(X, [x]))" 1 "" "CHF(X, [x])" }{TEXT 353 1 "," }}{PARA 0 "" 0 "" {HYPERLNK 17 "CleanUp(X)" 1 "" "CleanUp(X) " }{TEXT 354 1 "," }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {HYPERLNK 17 "Coef OfVar(X)" 1 "" "CoefOfVar(X)" }{TEXT 355 2 ", " }}{PARA 0 "" 0 "" {HYPERLNK 17 "Convert(X)" 1 "" "Convert(X)" }{TEXT 356 1 "," }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {HYPERLNK 17 "ConvertToNoDot(X)" 1 "" "Conve rtToNoDot(X)" }{TEXT 357 1 "," }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {HYPERLNK 17 "Convolution(X, Y)" 1 "" "Convolution(X, Y)" }{TEXT 358 2 ", " }}{PARA 0 "" 0 "" {HYPERLNK 17 "ConvolutionIID(X, n)" 1 "" "Con volutionIID(X, n)" }{TEXT 359 1 "," }}{PARA 0 "" 0 "" {HYPERLNK 17 "Cr eateMinHeap(A)" 1 "" "CriticalPoint(X, st)" }{TEXT 360 1 "," }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {HYPERLNK 17 "CriticalPoint(X, prob)" 1 "" " CriticalPoint(X, st)" }{TEXT 361 2 ", " }}{PARA 0 "" 0 "" {HYPERLNK 17 "DeleteMaxHeap(H, i, n)" 1 "" "DeleteMaxHeap(H, i, n)" }{TEXT 362 1 "," }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {HYPERLNK 17 "Determinant(MATRI X)" 1 "" "Determinant(Matrix)" }{TEXT 363 2 ", " }}{PARA 0 "" 0 "" {HYPERLNK 17 "Difference(X, Y)" 1 "" "Difference(X, Y)" }{TEXT 364 1 " ," }}{PARA 0 "" 0 "" {HYPERLNK 17 "Display(X)" 1 "" "Display(X)" } {TEXT 365 2 ", " }}{PARA 0 "" 0 "" {HYPERLNK 17 "ExpectedValue(X, [g]) " 1 "" "ExpectedValue(X, [g])" }{TEXT 366 1 "," }}{PARA 0 "" 0 "" {HYPERLNK 17 "GeometricOSMean(p)" 1 "" "GeometricOSMean(p)" }{TEXT 367 1 "," }}{PARA 0 "" 0 "" {HYPERLNK 17 "HF(X, [x]))" 1 "" "HF(X, [x] )" }{TEXT 368 1 "," }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {HYPERLNK 17 "IDF (X, [x]))" 1 "" "IDF(X, [x])" }{TEXT 369 1 "," }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {HYPERLNK 17 "IDFDiscrete(X, [x])" 1 "" "IDFDiscrete(X, [x]) " }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {HYPERLNK 17 "IncompleteBeta(p, a, b)" 1 "" "IncompleteBeta(p, a, b)" }{TEXT 370 1 "," }}{PARA 0 "" 0 " " {HYPERLNK 17 "InsertHeapConvolution()" 1 "" "InsertHeapConvolution() " }{TEXT 371 1 "," }}{PARA 0 "" 0 "" {HYPERLNK 17 "InsertHeapProduct() " 1 "" "InsertHeapProduct()" }{TEXT 372 1 "," }}{PARA 0 "" 0 "" {HYPERLNK 17 "InsertionSort(A, B)" 1 "" "InsertionSort(A, B)" }{TEXT 373 1 "," }}{PARA 0 "" 0 "" {HYPERLNK 17 "KSTest(X, Data, Parameters) " 1 "" "KSTest(X, Sample, Parameters)" }{TEXT 374 2 ", " }}{PARA 0 "" 0 "" {HYPERLNK 17 "Kurtosis(X)" 1 "" "Kurtosis(X)" }{TEXT 375 2 ", " } }{PARA 0 "" 0 "" {HYPERLNK 17 "Maximum(X, Y)" 1 "" "Maximum(X, Y)" } {TEXT 376 1 "," }}{PARA 0 "" 0 "" {HYPERLNK 17 "MaximumDiscrete(X, Y) " 1 "" "MaximumDiscrete(X, Y)" }{TEXT 377 1 "," }{TEXT -1 0 "" }} {PARA 0 "" 0 "" {HYPERLNK 17 "MaximumIID(X, n)" 1 "" "MaximumIID(X, n) " }{TEXT 378 2 ", " }}{PARA 0 "" 0 "" {HYPERLNK 17 "Mean(X)" 1 "" "Mea n(X)" }{TEXT 379 2 ", " }}{PARA 0 "" 0 "" {HYPERLNK 17 "Menu(Item)" 1 "" "Menu(Item)" }{TEXT 380 1 "," }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {HYPERLNK 17 "MGF(X)" 1 "" "MGF(X)" }{TEXT 381 2 ", " }}{PARA 0 "" 0 " " {HYPERLNK 17 "Minimum(X, Y)" 1 "" "Minimum(X, Y)" }{TEXT 382 1 "," } }{PARA 0 "" 0 "" {HYPERLNK 17 "MinimumDiscrete(X, Y)" 1 "" "MinimumDis crete(X, Y)" }{TEXT 383 1 "," }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {HYPERLNK 17 "MinimumIID(X, n)" 1 "" "MinimumIID(X, n)" }{TEXT 384 2 " , " }}{PARA 0 "" 0 "" {HYPERLNK 17 "Mixture(MixParameters, MixRVs)" 1 "" "Mixture(MixParameters, MixRVs)" }{TEXT 385 1 "," }}{PARA 0 "" 0 " " {HYPERLNK 17 "MLE(X, Data, Parameters, [Rightcensor])" 1 "" "MLE(X, \+ Sample, Parameters, [Censor])" }{TEXT 386 1 "," }}{PARA 0 "" 0 "" {HYPERLNK 17 "MLENHPP(X, Data, Parameters, obstime)," 1 "" "MLENHPP" } }{PARA 0 "" 0 "" {HYPERLNK 17 "MLEWeibull(Data, [Rightcensor])" 1 "" " MLEWeibull(Sample, Rightcensor)" }{TEXT 387 2 ", " }}{PARA 0 "" 0 "" {HYPERLNK 17 "MOM(X, Data, Parameters)" 1 "" "MOM(X, Sample, Parameter s)" }{TEXT 388 1 "," }}{PARA 0 "" 0 "" {HYPERLNK 17 "MOMErlang(Sample) " 1 "" "MOMErlang(Sample)" }{TEXT 389 1 "," }}{PARA 0 "" 0 "" {HYPERLNK 17 "MovingHeapConvolutionMethod(X, Y)" 1 "" "MovingHeapConvo lutionMethod(X, Y)" }{TEXT 390 1 "," }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {HYPERLNK 17 "MovingHeapProductMethod(X, Y)" 1 "" "MovingHeapProductMe thod(X, Y)" }{TEXT 391 1 "," }}{PARA 0 "" 0 "" {HYPERLNK 17 "MovingHea pProductQ1Q3(X, Y)" 1 "" "MovingHeapProductQ1Q3(X, Y)" }{TEXT 392 1 ", " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {HYPERLNK 17 "MovingHeapProductQ2Q4 (X, Y)" 1 "" "MovingHeapProductQ2Q4(X, Y)" }{TEXT 393 1 "," }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {HYPERLNK 17 "NextCombination(Previous, size)" 1 "" "NextCombination(Previous, N)" }{TEXT 394 2 ", " }}{PARA 0 "" 0 " " {HYPERLNK 17 "NextPermutation(Previous)" 1 "" "NextPermutation(Previ ous)" }{TEXT 395 1 "," }}{PARA 0 "" 0 "" {HYPERLNK 17 "OrderStat(X, n, r, [\"wo\"])" 1 "" "OrderStat(X, n, r, [\"wo\"])" }{TEXT 396 2 ", " } }{PARA 0 "" 0 "" {HYPERLNK 17 "PDF(X, [x]))" 1 "" "PDF(X, [x])" } {TEXT 397 1 "," }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {HYPERLNK 17 "PlotDis t(X, [low], [high])," 1 "" "PlotDist(X, low, high)" }}{PARA 0 "" 0 "" {HYPERLNK 17 "PlotEmpCDF(Data, [low], [high])" 1 "" "PlotEmpCDF(Sample , [low], [high])" }{TEXT 398 1 "," }}{PARA 0 "" 0 "" {HYPERLNK 17 "Plo tEmpCIF(Data, [low], [high])" 1 "" "PlotEmpCIF(Sample, lo, hi)" } {TEXT 399 2 ", " }}{PARA 0 "" 0 "" {HYPERLNK 17 "PlotEmpSF(Data, Censo r)" 1 "" "PlotEmpSF(Sample, Censor)" }{TEXT 400 1 "," }}{PARA 0 "" 0 " " {HYPERLNK 17 "PlotEmpVsFittedCDF(X, Data, Parameters, [low], [high]) " 1 "" "PlotEmpVsFittedCDF(X, Sample, Parameters, [low], [high])" } {TEXT 401 1 "," }}{PARA 0 "" 0 "" {HYPERLNK 17 "PlotEmpVsFittedCIF(X, \+ Data, Parameters, [low], [high])" 1 "" "PlotEmpVsFittedCIF(X, Sample, \+ Parameters, low, high)" }{TEXT 402 1 "," }}{PARA 0 "" 0 "" {HYPERLNK 17 "PlotEmpVsFittedSF(X, Data, Parameters, Censor, low, high)" 1 "" "P lotEmpVsFittedSF(X, Sample, Parameters, Censor)" }{TEXT 403 1 "," }} {PARA 0 "" 0 "" {HYPERLNK 17 "PPPlot(X, Data, Parameters)" 1 "" "PPPlo t(X, Sample, Parameters)" }{TEXT 404 2 ", " }}{PARA 0 "" 0 "" {HYPERLNK 17 "Product(X, Y)" 1 "" "Product(X, Y)" }{TEXT 405 1 "," }} {PARA 0 "" 0 "" {HYPERLNK 17 "ProductContinuous(X, Y)" 1 "" "ProductCo ntinuous(X, Y)" }{TEXT 406 1 "," }}{PARA 0 "" 0 "" {HYPERLNK 17 "Produ ctDiscrete(X, Y)" 1 "" "ProductDiscrete(X, Y)" }{TEXT 407 2 ", " }} {PARA 0 "" 0 "" {HYPERLNK 17 "ProductIID(X, n)" 1 "" "ProductIID(X, n) " }{TEXT 408 1 "," }}{PARA 0 "" 0 "" {HYPERLNK 17 "QQPlot(X, Data, Par ameters)" 1 "" "QQPlot(X, Sample, Parameters)" }{TEXT 409 2 ", " }} {PARA 0 "" 0 "" {HYPERLNK 17 "RangeStat(X, n, [\"wo\"])" 1 "" "RangeSt at(X, n, [\"wo\"])" }{TEXT 410 1 "," }}{PARA 0 "" 0 "" {HYPERLNK 17 "S F(X, [x]))" 1 "" "SF(X, [x])" }{TEXT 411 1 "," }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {HYPERLNK 17 "Stealth" 1 "" "Stealth" }{TEXT 412 1 "," } {TEXT -1 0 "" }}{PARA 0 "" 0 "" {HYPERLNK 17 "Skewness(X)" 1 "" "Skewn ess(X)" }{TEXT 413 2 ", " }}{PARA 0 "" 0 "" {HYPERLNK 17 "TransformDis crete(X, g)" 1 "" "TransformDiscrete(X, gX)" }{TEXT 414 2 ", " }} {PARA 0 "" 0 "" {HYPERLNK 17 "Truncate(X, low, high)" 1 "" "Truncate(X , a, b)" }{TEXT 415 1 "," }}{PARA 0 "" 0 "" {HYPERLNK 17 "UniformVaria te()" 1 "" "UniformVariate()" }{TEXT 416 1 "," }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {HYPERLNK 17 "Valid(X)" 1 "" "Valid(X)" }{TEXT 417 1 "," } {TEXT -1 0 "" }}{PARA 0 "" 0 "" {HYPERLNK 17 "Variance(X)" 1 "" "Varia nce(X)" }{TEXT 418 2 ", " }}{PARA 0 "" 0 "" {HYPERLNK 17 "VerifyPDF(X) " 1 "" "VerifyPDF(X)" }{TEXT 419 1 "," }}{PARA 0 "" 0 "" {HYPERLNK 17 "WeibullVariate(lambda, kappa, m)" 1 "" "WeibullVariate(lambda, kappa, m)" }{TEXT 420 1 "," }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {HYPERLNK 17 "Continuous Distributions" 1 "" "Continu ous Distributions" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 256 " " 0 "AllCombinations(n,k)" {TEXT 23 20 "AllCombinations(n,k)" }}{PARA 0 "" 0 "" {TEXT 23 72 "----------------------------------------------- -------------------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 23 48 "# Check for the appropriate number of arguments" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 36 "# Che ck that k is no greater than n" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }} {PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "Top" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "AllPermutations(n)" {TEXT 256 18 "AllPer mutations(n)" }}{PARA 0 "" 0 "" {TEXT 23 72 "------------------------- -----------------------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 23 48 "# Check for the appropri ate number of arguments" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 " " 0 "" {HYPERLNK 17 "To Top" 1 "" "Top" }}{PARA 0 "" 0 "" {TEXT 23 0 " " }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 256 "" 0 "Benford(X)" {TEXT 23 10 "Benford(X)" }{TEXT 257 0 "" }}{PARA 0 "" 0 "" {TEXT 23 72 "---- --------------------------------------------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 23 65 "# Othe r APPL Procedures Called: PDF, CDF, TruncateRV, Transform," }}{PARA 0 "" 0 "" {TEXT 23 42 "# MixtureRV" }} {PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 72 "# Purpo se: Given a continuous random variable X, this procedure returns" }} {PARA 0 "" 0 "" {TEXT 23 68 "# the PDF of Z, as defined in L eemis, Schmeiser, and Evans" }}{PARA 0 "" 0 "" {TEXT 23 18 "# \+ (2000)" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 46 "# Arguments: X: A continuous random variable." }}{PARA 0 "" 0 "" {TEXT 23 2 "# " }}{PARA 0 "" 0 "" {TEXT 23 13 "# Algorithm:" }}{PARA 0 "" 0 "" {TEXT 23 39 "# 1. Check for 1 argument, the RV X" }} {PARA 0 "" 0 "" {TEXT 23 61 "# 2. Check that the RV X is in the li st of 3 lists format" }}{PARA 0 "" 0 "" {TEXT 23 43 "# 3. Check th at the RV X is continuous " }}{PARA 0 "" 0 "" {TEXT 23 37 "# 4. Co nvert the RV X to PDF form" }}{PARA 0 "" 0 "" {TEXT 23 58 "# 5. Ch eck that the RV supports are numeric and finite" }}{PARA 0 "" 0 "" {TEXT 23 53 "# 6. Compute the lower and upper summation limits" }} {PARA 0 "" 0 "" {TEXT 23 64 "# 7. Create lists Weight, TruncX, and Xprime that will hold " }}{PARA 0 "" 0 "" {TEXT 23 31 "# (Hi - Lo + 1) elements" }}{PARA 0 "" 0 "" {TEXT 23 67 "# 8. Compute eac h \"weight\" and \"truncated distribution\" for the" }}{PARA 0 "" 0 " " {TEXT 23 72 "# intervals [Lo, Lo + 1], [Lo + 1, Lo + 2], ..., [d, d + 1], ...," }}{PARA 0 "" 0 "" {TEXT 23 40 "# [Hi - 2, Hi - 1], [Hi - 1, Hi] " }}{PARA 0 "" 0 "" {TEXT 23 65 "# 9. For each value d, where d is a value s.t. Lo <= d <= Hi," }}{PARA 0 "" 0 "" {TEXT 23 72 "# transform the truncated distribution by the func tion x -> x - d" }}{PARA 0 "" 0 "" {TEXT 23 66 "# 10. Send the list of weights along with its list of truncated" }}{PARA 0 "" 0 "" {TEXT 23 72 "# distributions to the procedure Mixture to compute the \+ PDF of Z," }}{PARA 0 "" 0 "" {TEXT 23 58 "# as defined in Leemi s, Schmeiser, and Evans (2000)" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }} {PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "Top" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "Binomial Variate(n, p, m)" {TEXT 258 24 "BinomialVariate(n, p, m)" }}{PARA 0 " " 0 "" {TEXT 23 72 "-------------------------------------------------- ----------------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 23 47 "# Other APPL Procedures Called: UniformVariate" }} {PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 71 "# Purpo se: Given the parameters n and p for the binomial distribution," }} {PARA 0 "" 0 "" {TEXT 23 82 "# BinomialVariate returns m bin omial variates, and the mean and variance" }}{PARA 0 "" 0 "" {TEXT 23 39 "# of the m binomial variates " }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 45 "# NOTE: only works for n = 16 at this moment" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "Top" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }} {PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "BootstrapRV(Data)" {TEXT 259 17 "BootstrapRV(Data)" }}{PARA 0 "" 0 "" {TEXT 23 72 "------ ------------------------------------------------------------------" }} {PARA 0 "" 0 "" {TEXT 23 32 "# Arguments: Data: list of data" }} {PARA 0 "" 0 "" {TEXT 23 2 "# " }}{PARA 0 "" 0 "" {TEXT 23 0 "" }} {PARA 0 "" 0 "" {TEXT 23 32 "# BruteForceConvolutionMethod()" }} {PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 41 "# Other APPL Procedures Called: HeapSort" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }} {PARA 0 "" 0 "" {TEXT 23 47 "# Arguments: X, Y: Continuous random var iables" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 44 "# \+ Procedure Name: BruteForceProductMethod()" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 41 "# Other APPL Procedures Called: \+ HeapSort" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 47 "# Arguments: X, Y: Continuous random variables" }}{PARA 0 "" 0 " " {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 26 "# BruteForceProductQ1Q3()" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 41 "# Other APPL Procedures Called: He apSort" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 47 "# Arguments: X, Y: Continuous random variables" }}{PARA 0 "" 0 " " {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 26 "# BruteForceProductQ2Q4()" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 41 "# Other APPL Procedures Called: He apSort" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 47 "# Arguments: X, Y: Continuous random variables" }}{PARA 0 "" 0 " " {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "Top" }} {PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "CDF(X, [x])" {TEXT 260 11 "CDF(X, [x])" }}{PARA 0 "" 0 "" {TEXT 23 72 "--------------------------------------------------------- ---------------" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 45 "# Other APPL Procedures Called: PDF, Convert" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 36 "# Purpose: CDF \+ is a procedure that:" }}{PARA 0 "" 0 "" {TEXT 23 65 "# (1) Re turns the cumulative distribution function of a " }}{PARA 0 "" 0 "" {TEXT 23 70 "# random variable X in the APPL list of 3 li sts format if" }}{PARA 0 "" 0 "" {TEXT 23 47 "# the only \+ argument given is X, or" }}{PARA 0 "" 0 "" {TEXT 23 71 "# (2) Returns the value Pr(X <= x) if it is given the optional" }}{PARA 0 " " 0 "" {TEXT 23 49 "# argument x in addition to the RV X " }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 59 "# A rguments: X: A continuous or discrete random variable; " }}{PARA 0 "" 0 "" {TEXT 23 65 "# x (optional argument): A numeric value entered when" }}{PARA 0 "" 0 "" {TEXT 23 47 "# trying \+ to determine Pr(X <= x)" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 " " 0 "" {TEXT 23 65 "# Algorithm: 1. Perform error checking on user en tered arguments" }}{PARA 0 "" 0 "" {TEXT 23 65 "# 2. The C DF of X is determined by sending it to its " }}{PARA 0 "" 0 "" {TEXT 23 65 "# appropriate category, which is based on whethe r:" }}{PARA 0 "" 0 "" {TEXT 23 47 "# A. X is continuous or discrete" }}{PARA 0 "" 0 "" {TEXT 23 69 "# B. X is \+ entered in its PDF, CDF, SF, HF, CHF, or IDF" }}{PARA 0 "" 0 "" {TEXT 23 27 "# format " }}{PARA 0 "" 0 "" {TEXT 23 70 "# \+ 3. If only 1 argument (the random variable X) is entered" } }{PARA 0 "" 0 "" {TEXT 23 71 "# by the user, return the cumulative density function of" }}{PARA 0 "" 0 "" {TEXT 23 54 "# \+ X in the APPL list of 3 lists format " }}{PARA 0 "" 0 "" {TEXT 23 63 "# 4. If 2 arguments are provided, return the \+ value " }}{PARA 0 "" 0 "" {TEXT 23 27 "# Pr(X <= x)" }} {PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {HYPERLNK 17 "To Top " 1 "" "Top" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "CHF(X, [x])" {TEXT 261 11 "CHF(X, [x])" }} {PARA 0 "" 0 "" {TEXT 23 72 "----------------------------------------- -------------------------------" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }} {PARA 0 "" 0 "" {TEXT 23 48 "# Other APPL Procedures Called: HF, Conv ert, SF" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 36 "# Purpose: CHF is a procedure that:" }}{PARA 0 "" 0 "" {TEXT 23 71 "# (1) Returns the cumulative hazard function of a distrib ution" }}{PARA 0 "" 0 "" {TEXT 23 72 "# in the APPL list \+ of 3 sublists form if given only the one" }}{PARA 0 "" 0 "" {TEXT 23 29 "# argument X, or" }}{PARA 0 "" 0 "" {TEXT 23 72 "# \+ (2) Returns the value -log(1 - Pr(X <= x)) if it is given the" }}{PARA 0 "" 0 "" {TEXT 23 58 "# optional argument x in a ddition to the RV X" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 " " {TEXT 23 59 "# Arguments: X: A continuous or discrete random variab le; " }}{PARA 0 "" 0 "" {TEXT 23 65 "# x (optional argumen t): A numeric value entered when" }}{PARA 0 "" 0 "" {TEXT 23 57 "# \+ trying to determine -log(1 - Pr(X <= x))" }}{PARA 0 "" 0 " " {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 65 "# Algorithm: 1. Perfo rm error checking on user entered arguments" }}{PARA 0 "" 0 "" {TEXT 23 65 "# 2. The CHF of X is determined by sending it to it s " }}{PARA 0 "" 0 "" {TEXT 23 65 "# appropriate catego ry, which is based on whether:" }}{PARA 0 "" 0 "" {TEXT 23 47 "# \+ A. X is continuous or discrete" }}{PARA 0 "" 0 "" {TEXT 23 69 "# B. X is entered in its PDF, CDF, SF, HF, CHF, or \+ IDF" }}{PARA 0 "" 0 "" {TEXT 23 27 "# format " }} {PARA 0 "" 0 "" {TEXT 23 70 "# 3. If only 1 argument (the \+ random variable X) is entered" }}{PARA 0 "" 0 "" {TEXT 23 71 "# \+ by the user, return the cumulative hazard function of " }} {PARA 0 "" 0 "" {TEXT 23 54 "# X in the APPL list of 3 \+ lists format " }}{PARA 0 "" 0 "" {TEXT 23 63 "# 4. If 2 ar guments are provided, return the value " }}{PARA 0 "" 0 "" {TEXT 23 37 "# -log(1 - Pr(X <= x))" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "Top" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "Cle anUp(X)" {TEXT 262 10 "CleanUp(X)" }}{PARA 0 "" 0 "" {TEXT 23 72 "---- --------------------------------------------------------------------" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 56 "# Purp ose: Clean up the list of sublists data structure" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 32 "# Arguments: X: Random \+ variable" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 13 "# Algorithm:" }}{PARA 0 "" 0 "" {TEXT 23 54 "# 1. Check for t he appropriate number of arguments" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" } }{PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "Top" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "CoefOfVa r(X)" {TEXT 263 12 "CoefOfVar(X)" }}{PARA 0 "" 0 "" {TEXT 23 72 "----- -------------------------------------------------------------------" } }{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 24 "# Purpo se: Returns the " }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 32 "# Arguments: X: Random Variable" }}{PARA 0 "" 0 "" {TEXT 23 2 "# " }}{PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "Top" }} {PARA 0 "" 0 "" {TEXT 23 0 "" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "Convert(X)" {TEXT 346 10 "Convert(X)" }}{PARA 0 "" 0 "" {TEXT 23 73 "----------------------------------------------- ------------------------- " }}{PARA 0 "" 0 "" {TEXT 23 26 "# NOTES: \+ Discrete Cases:" }}{PARA 0 "" 0 "" {TEXT 23 23 "# \011 (I) dot c ases" }}{PARA 0 "" 0 "" {TEXT 23 29 "# \011(a) incremented by 1 " }}{PARA 0 "" 0 "" {TEXT 23 32 "# \011(b) transformed by g(x)" }}{PARA 0 "" 0 "" {TEXT 23 35 "# \011\011(i) incremented by k" }}{PARA 0 "" 0 "" {TEXT 23 35 "# \011\011(ii) incremente d by 1" }}{PARA 0 "" 0 "" {TEXT 23 29 "# \011(c) incremented by \+ k" }}{PARA 0 "" 0 "" {TEXT 23 25 "# \011 (II) no dot cases" }} {PARA 0 "" 0 "" {TEXT 23 23 "# \011(a) arrow case" }}{PARA 0 "" 0 "" {TEXT 23 26 "# \011(b) no arrow case" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 34 "# standard \"dot\" supp ort format: " }}{PARA 0 "" 0 "" {TEXT 23 63 "# [value .. value, \+ incremented by k, transformed by g(x)]" }}{PARA 0 "" 0 "" {TEXT 23 3 " # " }}{PARA 0 "" 0 "" {TEXT 23 44 "# standard \"no dot\" pdf and sup port format:" }}{PARA 0 "" 0 "" {TEXT 23 66 "# [frac1, frac2, fr ac3, ... , fracN], [a1, a2, a3, ... , aN]" }}{PARA 0 "" 0 "" {TEXT 23 55 "# Example: [0.1, 0.3, 0.2, 0.4], [1, 4, 11, 34/3]" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 59 "# Other APPL \+ Procedures Called: PDF, Convert, ConvertNoDot" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 66 "# Purpose: (1) Converts discrete RVs with a \"dot\" support type to" }}{PARA 0 "" 0 "" {TEXT 23 56 "# the standard APPL \"dot\" support format: " }} {PARA 0 "" 0 "" {TEXT 23 71 "# [value .. value, incremen ted by k, transformed by g(x)]" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }} {PARA 0 "" 0 "" {TEXT 23 67 "# (2) Converts discrete RV's wi th a \"no dot\" support type" }}{PARA 0 "" 0 "" {TEXT 23 64 "# \+ to the standard \"no dot\" pdf and support format:" }}{PARA 0 " " 0 "" {TEXT 23 71 "# [frac1, frac2, frac3, ... , fracN], [ a1, a2, a3, ... , aN]" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 71 "# Arguments: A discrete random variable X in the li st-of-lists format:" }}{PARA 0 "" 0 "" {TEXT 23 68 "# [[f( x)], [support], [\"Discrete\", \"XXX\"]], where XXX is" }}{PARA 0 "" 0 "" {TEXT 23 44 "# PDF, CDF, IDF, SF, HF, or CHF " }} {PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 12 "# Algor ithm" }}{PARA 0 "" 0 "" {TEXT 23 54 "# 1. Check for the appropriat e number of arguments" }}{PARA 0 "" 0 "" {TEXT 23 61 "# 2. Check t hat the RV X is in the list of 3 lists format" }}{PARA 0 "" 0 "" {TEXT 23 46 "# 3. Check that the given RV X is discrete" }}{PARA 0 "" 0 "" {TEXT 23 59 "# 4. Convert X to PDF form if not already i n that form " }}{PARA 0 "" 0 "" {TEXT 23 69 "# 5. Check to see whe ther the RV X is in the discrete \"dot\" case." }}{PARA 0 "" 0 "" {TEXT 23 71 "# In startup.map, \"dot\" is a defined structure t ype for discrete" }}{PARA 0 "" 0 "" {TEXT 23 55 "# RVs. The str ucture `type/dot is defined to be:" }}{PARA 0 "" 0 "" {TEXT 23 65 "# \+ `type/dot` := \{constant .. constant, symbol .. constant," }} {PARA 0 "" 0 "" {TEXT 23 52 "# constant .. symbol, symbol \+ .. symbol\}:" }}{PARA 0 "" 0 "" {TEXT 23 65 "# 6. If X is in the \+ \"dot\" case with support incremented by 1," }}{PARA 0 "" 0 "" {TEXT 23 60 "# [value .. value], then convert its support list to:" } }{PARA 0 "" 0 "" {TEXT 23 36 "# [value .. value, 1, x -> x]" }} {PARA 0 "" 0 "" {TEXT 23 68 "# 7. If X is in the \"dot\" case with support incremented by 1 and" }}{PARA 0 "" 0 "" {TEXT 23 62 "# \+ transformed by g(x): [value .. value, transformed by " }}{PARA 0 "" 0 "" {TEXT 23 50 "# g(x)], then convert its support list to: " }}{PARA 0 "" 0 "" {TEXT 23 34 "# [value .. value, 1, g(x)]" }} {PARA 0 "" 0 "" {TEXT 23 65 "# 8. If X is in the \"dot\" case with support incremented by k," }}{PARA 0 "" 0 "" {TEXT 23 69 "# [v alue .. value, incremented by k], then convert its support" }}{PARA 0 "" 0 "" {TEXT 23 18 "# list to: " }}{PARA 0 "" 0 "" {TEXT 23 36 "# [value .. value, k, x -> x]" }}{PARA 0 "" 0 "" {TEXT 23 68 "# 9. If the discrete RV X is not in the \"dot\" case, it is in the" }}{PARA 0 "" 0 "" {TEXT 23 65 "# \"no dot\" case, which h as a support list that looks like:" }}{PARA 0 "" 0 "" {TEXT 23 57 "# \+ [1, 2, 5], or [15], or [2, 8, 13, 43, 100] " }}{PARA 0 "" 0 "" {TEXT 23 65 "# 10. If X is in the \"no dot\" case, then determ ine if the PDF " }}{PARA 0 "" 0 "" {TEXT 23 69 "# sublist conta ins a Maple procedure. If so, then X is in the" }}{PARA 0 "" 0 "" {TEXT 23 68 "# \"arrow\" (->) case and needs to be converted to the standard" }}{PARA 0 "" 0 "" {TEXT 23 33 "# \"no dot\" form , which is:" }}{PARA 0 "" 0 "" {TEXT 23 62 "# [[fraction1, frac tion2, fraction3, ... , fractionN], " }}{PARA 0 "" 0 "" {TEXT 23 54 "# [a1, a2, a3, ... , aN], [\"Discrete\", \"PDF\"]]" }}{PARA 0 " " 0 "" {TEXT 23 69 "# 11. Check that symbols or infinity are not su pport values in the" }}{PARA 0 "" 0 "" {TEXT 23 23 "# \"no dot \" cases" }}{PARA 0 "" 0 "" {TEXT 23 48 "# 12. Convert X to the sta ndard \"no dot\" form" }}{PARA 0 "" 0 "" {TEXT 23 69 "# 13. Return \+ the converted discrete random variable to the user in" }}{PARA 0 "" 0 "" {TEXT 23 21 "# its new form" }}{PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "Top" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }{TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "ConvertToNoDot(X)" {TEXT 265 17 "ConvertToNoDot(X)" }}{PARA 0 "" 0 "" {TEXT 23 72 "------ ------------------------------------------------------------------" }} {PARA 0 "" 0 "" {TEXT 23 1 " " }}{PARA 0 "" 0 "" {TEXT 23 34 "# stand ard \"dot\" support format: " }}{PARA 0 "" 0 "" {TEXT 23 63 "# [ value .. value, incremented by k, transformed by g(x)]" }}{PARA 0 "" 0 "" {TEXT 23 3 "# " }}{PARA 0 "" 0 "" {TEXT 23 43 "# standard \"nod ot\" pdf and support format:" }}{PARA 0 "" 0 "" {TEXT 23 66 "# [ frac1, frac2, frac3, ... , fracN], [a1, a2, a3, ... , aN]" }}{PARA 0 " " 0 "" {TEXT 23 55 "# Example: [0.1, 0.3, 0.2, 0.4], [1, 4, 11, \+ 34/3]" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 36 "# Other APPL Procedures Called: PDF" }}{PARA 0 "" 0 "" {TEXT 23 1 "# " }}{PARA 0 "" 0 "" {TEXT 23 68 "# Purpose: Converts discrete RVs wit h a standard APPL \"dot\" support" }}{PARA 0 "" 0 "" {TEXT 23 65 "# \+ format to the standard APPL \"no dot\" support format: " }} {PARA 0 "" 0 "" {TEXT 23 70 "# [frac1, frac2, frac3, ... , f racN], [a1, a2, a3, ... , aN]" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }} {PARA 0 "" 0 "" {TEXT 23 71 "# Arguments: A discrete random variable \+ X in the list-of-lists format:" }}{PARA 0 "" 0 "" {TEXT 23 68 "# \+ [[f(x)], [support], [\"Discrete\", \"XXX\"]], where XXX is" }} {PARA 0 "" 0 "" {TEXT 23 44 "# PDF, CDF, IDF, SF, HF, or C HF " }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 12 "# Algorithm" }}{PARA 0 "" 0 "" {TEXT 23 54 "# 1. Check for the app ropriate number of arguments" }}{PARA 0 "" 0 "" {TEXT 23 61 "# 2. \+ Check that the RV X is in the list of 3 lists format" }}{PARA 0 "" 0 " " {TEXT 23 46 "# 3. Check that the given RV X is discrete" }} {PARA 0 "" 0 "" {TEXT 23 59 "# 4. Convert X to PDF form if not alr eady in that form " }}{PARA 0 "" 0 "" {TEXT 23 71 "# 5. Check to s ee that the RV X has the standard APPL \"dot\" format." }}{PARA 0 "" 0 "" {TEXT 23 71 "# In startup.map, \"dot\" is a defined struct ure type for discrete" }}{PARA 0 "" 0 "" {TEXT 23 55 "# RVs. Th e structure `type/dot is defined to be:" }}{PARA 0 "" 0 "" {TEXT 23 65 "# `type/dot` := \{constant .. constant, symbol .. constant, " }}{PARA 0 "" 0 "" {TEXT 23 52 "# constant .. symbol, sym bol .. symbol\}:" }}{PARA 0 "" 0 "" {TEXT 23 51 "# The standard APPL \"dot\" support format is:" }}{PARA 0 "" 0 "" {TEXT 23 64 "# \+ [value .. value, incremented by k, transformed by g(x)]" }}{PARA 0 "" 0 "" {TEXT 23 69 "# 6. Convert to the PDF and support of X to the standard APPL \"no" }}{PARA 0 "" 0 "" {TEXT 23 22 "# dot\" formats." }}{PARA 0 "" 0 "" {TEXT 23 69 "# 7. Return the converte d discrete random variable to the user in" }}{PARA 0 "" 0 "" {TEXT 23 21 "# its new form" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "Top" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "Convolution(X, Y)" {TEXT 266 17 "Convolution(X, Y)" }}{PARA 0 "" 0 "" {TEXT 23 72 "------ ------------------------------------------------------------------" }} {PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 67 "# Other \+ APPL Procedures Called: PDF, Transform, Product, Convert, " }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 18 "# Arguments: X, Y" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {HYPERLNK 17 "T o Top" 1 "" "Top" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "ConvolutionIID(X, n)" {TEXT 267 20 "Conv olutionIID(X, n)" }}{PARA 0 "" 0 "" {TEXT 23 72 "--------------------- ---------------------------------------------------" }}{PARA 0 "" 0 " " {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 49 "# Other APPL Procedure s Called: PDF, Convolution" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 43 "# Arguments: X: Continuous random variable" }} {PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {HYPERLNK 17 "To Top " 1 "" "Top" }}{PARA 0 "" 0 "" {TEXT 23 1 "\n" }}{PARA 0 "" 0 "CreateH eap(A, B)" {TEXT 268 16 "CreateHeap(A, B)" }}{PARA 0 "" 0 "" {TEXT 23 72 "------------------------------------------------------------------ ------" }}{PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "Top" }}{PARA 0 " " 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "C reateMinHeap(A)" {TEXT 269 16 "CreateMinHeap(A)" }}{PARA 0 "" 0 "" {TEXT 23 72 "--------------------------------------------------------- ---------------" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 51 "# Purpose: Create minimum heap for just 2 elements" }} {PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {HYPERLNK 17 "To Top " 1 "" "Top" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "CriticalPoint(X, st)" {TEXT 271 20 "CriticalPo int(X, st)" }{TEXT 270 1 " " }}{PARA 0 "" 0 "" {TEXT 23 72 "---------- --------------------------------------------------------------" }} {PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 36 "# Other \+ APPL Procedures Called: CDF" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 82 "# Purpose: CriticalPoint is a procedure that r eturns the critical point of a List" }}{PARA 0 "" 0 "" {TEXT 23 35 "# \+ of Lists specificed pdf" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }} {PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "Top" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "DeleteMa xHeap(H, i, n)" {TEXT 272 22 "DeleteMaxHeap(H, i, n)" }}{PARA 0 "" 0 " " {TEXT 23 72 "------------------------------------------------------- -----------------" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 33 "# Other APPL Procedures Called: " }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "Top" }} {PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "Determinant(Matrix)" {TEXT 274 19 "Determinant(Matrix)" } {TEXT 273 1 " " }}{PARA 0 "" 0 "" {TEXT 23 72 "----------------------- -------------------------------------------------" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 56 "# Other APPL Procedures \+ Called: Product, Convolution " }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }} {PARA 0 "" 0 "" {TEXT 23 108 "# Purpose: Returns the distribution of \+ the determinant of a 2x2 matrix # with random variables as elements" }}{PARA 0 "" 0 "" {TEXT 23 3 "# " }}{PARA 0 "" 0 "" {TEXT 23 53 "# A rguments: Matrix: A 2x2 array of random variables" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 13 "# Algorithm:" }}{PARA 0 "" 0 "" {TEXT 23 59 "# 1. Check to see if the matrix is a 2x2 sq uare matrix " }}{PARA 0 "" 0 "" {TEXT 23 53 "# 2. Compute the dist ribution of the determinant " }}{PARA 0 "" 0 "" {TEXT 23 70 "# \+ to contain all 1's to indicate the data values are uncensored" }} {PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {HYPERLNK 17 "To Top " 1 "" "Top" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "Difference(X, Y)" {TEXT 276 15 "Difference(X , Y" }{TEXT 275 1 ")" }}{PARA 0 "" 0 "" {TEXT 23 72 "----------------- -------------------------------------------------------" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 37 "# Other APPL Proced ures Called: PDF " }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 32 "# Arguments: X: Random variable" }}{PARA 0 "" 0 "" {TEXT 23 32 "# Y: Random variable" }}{PARA 0 "" 0 "" {TEXT 23 2 "# " }}{PARA 0 "" 0 "" {TEXT 23 13 "# Algorithm:" }}{PARA 0 "" 0 "" {TEXT 23 53 "# 1. Check for the appropriate number of arg uments" }}{PARA 0 "" 0 "" {TEXT 23 60 "# 2. Check that the RV X is \+ in the list of 3 lists format" }}{PARA 0 "" 0 "" {TEXT 23 57 "# 3. \+ Convert X to PDF form if not already in that form" }}{PARA 0 "" 0 "" {TEXT 23 56 "# 4. Check whether the RV X is Continuous or Discrete " }}{PARA 0 "" 0 "" {TEXT 23 66 "# 5. If the RV X is continuous, th en compute its expected value" }}{PARA 0 "" 0 "" {TEXT 23 70 "# 6. \+ If the RV X is discrete, then determine if the support of X is" }} {PARA 0 "" 0 "" {TEXT 23 37 "# in the \"dot\" or \"no dot\" form " }}{PARA 0 "" 0 "" {TEXT 23 69 "# 7. If the random variable X is i n the discrete \"dot\" case, first" }}{PARA 0 "" 0 "" {TEXT 23 63 "# \+ call procedure Convert(X) to convert its support to the" }}{PARA 0 "" 0 "" {TEXT 23 39 "# *standard dot* support format: " }} {PARA 0 "" 0 "" {TEXT 23 69 "# [anything .. anything, incremente d by k, transformed by h(x)]" }}{PARA 0 "" 0 "" {TEXT 23 51 "# 8. T ransform the discrete dot pdf f(x) by h(x)" }}{PARA 0 "" 0 "" {TEXT 23 54 "# 9. Find the expected value of the transformed pdf" }} {PARA 0 "" 0 "" {TEXT 23 71 "# 10. Similarly, if the random variable X is in the discrete \"no dot\"" }}{PARA 0 "" 0 "" {TEXT 23 71 "# \+ case, first call procedure Convert(X) to convert its support to" }} {PARA 0 "" 0 "" {TEXT 23 57 "# the *standard no dot* pdf/support format: " }}{PARA 0 "" 0 "" {TEXT 23 66 "# [frac1, frac2 , frac3, ... , fracN], [a1, a2, a3, ... , aN]" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 30 "# Procedure names: BenfordRV" }}{PARA 0 "" 0 "" {TEXT 23 32 "# BernoulliRV" }}{PARA 0 "" 0 "" {TEXT 23 33 "# BingoCoverRV" }}{PARA 0 "" 0 "" {TEXT 23 31 " # BinomialRV" }}{PARA 0 "" 0 "" {TEXT 23 31 "# \+ BirthdayRV" }}{PARA 0 "" 0 "" {TEXT 23 32 "# \+ GeometricRV" }}{PARA 0 "" 0 "" {TEXT 23 39 "# \+ NegativeBinomialRV" }}{PARA 0 "" 0 "" {TEXT 23 30 "# \+ PoissonRV" }}{PARA 0 "" 0 "" {TEXT 23 38 "# U niformDiscreteRV" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 61 "# Purpose: Define the common parametric discrete univar iate" }}{PARA 0 "" 0 "" {TEXT 23 51 "# probability distribu tions shown below:" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 " " {TEXT 23 48 "# NOTE: For each distribution, x is an integer." }} {PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 63 "# Distr ibution Support Parameters Parameter Space" }}{PARA 0 "" 0 "" {TEXT 23 63 "# ------------ ------- ---------- ---- -----------" }}{PARA 0 "" 0 "" {TEXT 23 52 "# Benford x=1,2,. ..,9 none none" }}{PARA 0 "" 0 "" {TEXT 23 19 "# Bernoull i " }}{PARA 0 "" 0 "" {TEXT 23 52 "# BingoCover\011 x=24,25 ,...,75 none\011\011 none" }}{PARA 0 "" 0 "" {TEXT 23 69 "# Bi nomial 0<=x<=n n, p n: pos int; 0 < p < 1" }} {PARA 0 "" 0 "" {TEXT 23 52 "# Birthday\011\011 x=1,2,...,365 n one none" }}{PARA 0 "" 0 "" {TEXT 23 57 "# Geometric x >= 1 p 0 < p < 1" }}{PARA 0 "" 0 "" {TEXT 23 17 " # Hypergeometric" }}{PARA 0 "" 0 "" {TEXT 23 19 "# NegativeBinomial " }}{PARA 0 "" 0 "" {TEXT 23 58 "# Poisson x >= 0 lam bda lambda > 0" }}{PARA 0 "" 0 "" {TEXT 23 18 "# UniformDiscre te" }}{PARA 0 "" 0 "" {TEXT 23 3 "# " }}{PARA 0 "" 0 "" {TEXT 23 63 " # Arguments: The parameters of the distribution of the random" }} {PARA 0 "" 0 "" {TEXT 23 23 "# variable" }}{PARA 0 "" 0 " " {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 13 "# Algorithm:" }} {PARA 0 "" 0 "" {TEXT 23 54 "# 1. Check for the appropriate number of arguments" }}{PARA 0 "" 0 "" {TEXT 23 58 "# 2. Check parameter space when parameters are numeric" }}{PARA 0 "" 0 "" {TEXT 23 52 "# \+ 3. Check to see that the parameters are finite" }}{PARA 0 "" 0 "" {TEXT 23 55 "# 4. Make assumptions about any symbolic parameters" }}{PARA 0 "" 0 "" {TEXT 23 56 "# 5. Assign a list of lists in the \+ following format:" }}{PARA 0 "" 0 "" {TEXT 23 49 "# [[f(x)], [s upport], [\"Discrete\", \"XXX\"]]" }}{PARA 0 "" 0 "" {TEXT 23 71 "# \+ where XXX is one of the following: PDF, CDF, IDF, SF, HF, CHF" } }{PARA 0 "" 0 "" {TEXT 23 33 "# 6. Return the list of lists" }} {PARA 0 "" 0 "" {TEXT 23 2 "# " }}{PARA 0 "" 0 "" {TEXT 23 70 "# Note : Could define a type in startup about p being between 0 and 1" }} {PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {HYPERLNK 17 "To Top " 1 "" "Top" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 2 " " }}{PARA 0 "" 0 "Display(X)" {TEXT 277 10 "Display(X)" }} {PARA 0 "" 0 "" {TEXT 23 72 "----------------------------------------- -------------------------------" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }} {PARA 0 "" 0 "" {TEXT 23 80 "# Purpose: Prints the List of List rand om variable X in a more readable format" }}{PARA 0 "" 0 "" {TEXT 23 49 "# given a List of Lists specificed PDF" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 45 "# Arguments: X: A co ntinuous random variable" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 " " 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 2 "# " }}{PARA 0 "" 0 "" {TEXT 23 24 "# D1, D2, D3, ... , D10" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "Top" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "Exp ectedValue(X, [g])" {TEXT 278 21 "ExpectedValue(X, [g])" }}{PARA 0 "" 0 "" {TEXT 23 72 "---------------------------------------------------- --------------------" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 " " {TEXT 23 36 "# Other APPL Procedures Called: PDF" }}{PARA 0 "" 0 " " {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 46 "# Purpose: Returns th e expected value of g(X)" }}{PARA 0 "" 0 "" {TEXT 23 71 "# E x: Suppose X ~ ExponentialRV(lambda); then E(X^2) would be" }}{PARA 0 "" 0 "" {TEXT 23 69 "# entered as ExpectedValue(X, g), where g is the procedure " }}{PARA 0 "" 0 "" {TEXT 23 25 "# g := \+ x -> x^2" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 32 "# Arguments: X: Random variable" }}{PARA 0 "" 0 "" {TEXT 23 28 "# g: A procedure" }}{PARA 0 "" 0 "" {TEXT 23 2 "# " }} {PARA 0 "" 0 "" {TEXT 23 13 "# Algorithm:" }}{PARA 0 "" 0 "" {TEXT 23 53 "# 1. Check for the appropriate number of arguments" }}{PARA 0 "" 0 "" {TEXT 23 60 "# 2. Check that the RV X is in the list of 3 lists format" }}{PARA 0 "" 0 "" {TEXT 23 57 "# 3. Convert X to PDF form if not already in that form" }}{PARA 0 "" 0 "" {TEXT 23 56 "# \+ 4. Check whether the RV X is Continuous or Discrete" }}{PARA 0 "" 0 " " {TEXT 23 66 "# 5. If the RV X is continuous, then compute its exp ected value" }}{PARA 0 "" 0 "" {TEXT 23 70 "# 6. If the RV X is dis crete, then determine if the support of X is" }}{PARA 0 "" 0 "" {TEXT 23 37 "# in the \"dot\" or \"no dot\" form" }}{PARA 0 "" 0 "" {TEXT 23 69 "# 7. If the random variable X is in the discrete \"dot \" case, first" }}{PARA 0 "" 0 "" {TEXT 23 63 "# call procedure \+ Convert(X) to convert its support to the" }}{PARA 0 "" 0 "" {TEXT 23 39 "# *standard dot* support format: " }}{PARA 0 "" 0 "" {TEXT 23 69 "# [anything .. anything, incremented by k, transformed by h(x)]" }}{PARA 0 "" 0 "" {TEXT 23 51 "# 8. Transform the discrete \+ dot pdf f(x) by h(x)" }}{PARA 0 "" 0 "" {TEXT 23 54 "# 9. Find the \+ expected value of the transformed pdf" }}{PARA 0 "" 0 "" {TEXT 23 71 " # 10. Similarly, if the random variable X is in the discrete \"no do t\"" }}{PARA 0 "" 0 "" {TEXT 23 71 "# case, first call procedure Convert(X) to convert its support to" }}{PARA 0 "" 0 "" {TEXT 23 57 " # the *standard no dot* pdf/support format: " }}{PARA 0 " " 0 "" {TEXT 23 66 "# [frac1, frac2, frac3, ... , fracN], [a1, a 2, a3, ... , aN]" }}{PARA 0 "" 0 "" {TEXT 23 42 "# 11. Find the expe cted value of the pdf" }}{PARA 0 "" 0 "" {TEXT 23 41 "# 12. Return t he expected value of g(X)" }}{PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 " " "Top" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 0 " " }}{PARA 0 "" 0 "ExponentialVariate(lambda)" {TEXT 279 26 "Exponentia lVariate(lambda)" }}{PARA 0 "" 0 "" {TEXT 23 72 "--------------------- ---------------------------------------------------" }}{PARA 0 "" 0 " " {HYPERLNK 17 "To Top" 1 "" "Top" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }} {PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "GeometricOSMean(p)" {TEXT 281 18 "GeometricOSMean(p)" }{TEXT 280 1 " " }}{PARA 0 "" 0 "" {TEXT 23 72 "----------------------------- -------------------------------------------" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 48 "# Other APPL Procedures Called : OrderStat, Mean" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 69 "# Purpose: Reproduce appendix from Margolin, Winokur, \" Exact Moments" }}{PARA 0 "" 0 "" {TEXT 23 66 "# of the Order Statisti cs of the Geometric Distribution and their" }}{PARA 0 "" 0 "" {TEXT 23 69 "# Relation to Inverse Sampling and Reliability of Redundant Sy stems\"" }}{PARA 0 "" 0 "" {TEXT 23 56 "# in the American Statistical Association Journal, 1967" }}{PARA 0 "" 0 "" {TEXT 23 3 "# " }} {PARA 0 "" 0 "" {TEXT 23 41 "# Arguments: p: A probability, 0 < p < 1 " }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {HYPERLNK 17 "To \+ Top" 1 "" "Top" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 1 " " }}{PARA 0 "" 0 "GeometricOSVariance(p)" {TEXT 283 22 "G eometricOSVariance(p)" }{TEXT 282 1 " " }}{PARA 0 "" 0 "" {TEXT 23 72 "--------------------------------------------------------------------- ---" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 52 "# \+ Other APPL Procedures Called: OrderStat, Variance" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 69 "# Purpose: Reproduce ap pendix from Margolin, Winokur, \"Exact Moments" }}{PARA 0 "" 0 "" {TEXT 23 66 "# of the Order Statistics of the Geometric Distribution \+ and their" }}{PARA 0 "" 0 "" {TEXT 23 69 "# Relation to Inverse Sampl ing and Reliability of Redundant Systems\"" }}{PARA 0 "" 0 "" {TEXT 23 56 "# in the American Statistical Association Journal, 1967" }} {PARA 0 "" 0 "" {TEXT 23 3 "# " }}{PARA 0 "" 0 "" {TEXT 23 41 "# Arg uments: p: A probability, 0 < p < 1" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "Top" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "Procedur e Name: HeapSort(A, B)" {TEXT 284 30 "Procedure Name: HeapSort(A, B)" }}{PARA 0 "" 0 "" {TEXT 23 72 "--------------------------------------- ---------------------------------" }}{PARA 0 "" 0 "" {HYPERLNK 17 "To \+ Top" 1 "" "Top" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 2 " " }}{PARA 0 "" 0 "HF(X, \+ [x])" {TEXT 285 10 "HF(X, [x])" }}{PARA 0 "" 0 "" {TEXT 23 72 "------- -----------------------------------------------------------------" }} {PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 49 "# Other \+ APPL Procedures Called: SF, PDF, Convert" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 35 "# Purpose: HF is a procedure tha t:" }}{PARA 0 "" 0 "" {TEXT 23 72 "# (1) Returns the hazard f unction of a random variable X in the" }}{PARA 0 "" 0 "" {TEXT 23 65 " # APPL list of 3 sublists form if given only the one" }} {PARA 0 "" 0 "" {TEXT 23 29 "# argument X, or" }}{PARA 0 "" 0 "" {TEXT 23 70 "# (2) Returns the value Pr(X = x) / Pr(X >= x) if it is given" }}{PARA 0 "" 0 "" {TEXT 23 62 "# t he optional argument x in addition to the RV X" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 59 "# Arguments: X: A conti nuous or discrete random variable; " }}{PARA 0 "" 0 "" {TEXT 23 65 "# \+ x (optional argument): A numeric value entered when" }} {PARA 0 "" 0 "" {TEXT 23 60 "# trying to determine Pr(X = x) / Pr(X >= x) " }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 " " {TEXT 23 65 "# Algorithm: 1. Perform error checking on user entered arguments" }}{PARA 0 "" 0 "" {TEXT 23 64 "# 2. The HF of \+ X is determined by sending it to its " }}{PARA 0 "" 0 "" {TEXT 23 65 " # appropriate category, which is based on whether:" }} {PARA 0 "" 0 "" {TEXT 23 47 "# A. X is continuous or di screte" }}{PARA 0 "" 0 "" {TEXT 23 69 "# B. X is entere d in its PDF, CDF, SF, HF, CHF, or IDF" }}{PARA 0 "" 0 "" {TEXT 23 27 "# format " }}{PARA 0 "" 0 "" {TEXT 23 70 "# \+ 3. If only 1 argument (the random variable X) is entered" }} {PARA 0 "" 0 "" {TEXT 23 68 "# by the user, return the \+ hazard function of X in the" }}{PARA 0 "" 0 "" {TEXT 23 45 "# \+ APPL list of 3 lists format " }}{PARA 0 "" 0 "" {TEXT 23 63 "# \+ 4. If 2 arguments are provided, return the value " }} {PARA 0 "" 0 "" {TEXT 23 39 "# Pr(X = x) / Pr(X >= x)" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {HYPERLNK 17 "To To p" 1 "" "Top" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 2 " " }}{PARA 0 "" 0 "IDF(X, [x])" {TEXT 286 11 "IDF(X, [x])" }} {PARA 0 "" 0 "" {TEXT 23 72 "----------------------------------------- -------------------------------" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }} {PARA 0 "" 0 "" {TEXT 23 49 "# Other APPL Procedures Called: CDF, IDF Discrete" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 36 "# Purpose: IDF is a procedure that:" }}{PARA 0 "" 0 "" {TEXT 23 62 "# (1) returns the inverse distribution function of a " }} {PARA 0 "" 0 "" {TEXT 23 71 "# random variable X in the A PPL list of sublists format if" }}{PARA 0 "" 0 "" {TEXT 23 47 "# \+ the only argument given is X, or" }}{PARA 0 "" 0 "" {TEXT 23 68 "# (2) Given the optional value x, returns the value alpha , " }}{PARA 0 "" 0 "" {TEXT 23 39 "# where that F(x) = al pha." }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 59 " # Arguments: X: A continuous or discrete random variable; " }}{PARA 0 "" 0 "" {TEXT 23 65 "# x (optional argument): A numeric \+ value entered when" }}{PARA 0 "" 0 "" {TEXT 23 48 "# al pha, where alpha = Pr(X <= x)" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }} {PARA 0 "" 0 "" {TEXT 23 65 "# Algorithm: 1. Perform error checking o n user entered arguments" }}{PARA 0 "" 0 "" {TEXT 23 65 "# \+ 2. The IDF of X is determined by sending it to its " }}{PARA 0 "" 0 " " {TEXT 23 65 "# appropriate category, which is based o n whether:" }}{PARA 0 "" 0 "" {TEXT 23 47 "# A. X is co ntinuous or discrete" }}{PARA 0 "" 0 "" {TEXT 23 69 "# \+ B. X is entered in its PDF, CDF, SF, HF, CHF, or IDF" }}{PARA 0 "" 0 " " {TEXT 23 27 "# format " }}{PARA 0 "" 0 "" {TEXT 23 70 "# 3. If only 1 argument (the random variable X) is \+ entered" }}{PARA 0 "" 0 "" {TEXT 23 70 "# by the user, \+ return the inverse distribution function" }}{PARA 0 "" 0 "" {TEXT 23 57 "# of X in the APPL list of 3 lists format " }} {PARA 0 "" 0 "" {TEXT 23 70 "# 4. If 2 arguments are provi ded, return the value alpha, " }}{PARA 0 "" 0 "" {TEXT 23 41 "# \+ where alpha = Pr(X <= x)" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }} {PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "Top" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "IDFDiscrete(X, [x])" {TEXT 23 2 " " }} {PARA 0 "" 0 "" {TEXT 287 19 "IDFDiscrete(X, [x])" }}{PARA 0 "" 0 "" {TEXT 23 72 "--------------------------------------------------------- ---------------" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 45 "# Other APPL Procedures Called: Convert, CDF" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 70 "# Defn: For a d istribution F, mapping a quantile x into a probability" }}{PARA 0 "" 0 "" {TEXT 23 65 "# alpha, the inverse distribution function G \+ performs the " }}{PARA 0 "" 0 "" {TEXT 23 66 "# corresponding i nverse mapping from alpha into x. Thus, if" }}{PARA 0 "" 0 "" {TEXT 23 54 "# Pr(X <= x) = F(x) = alpha, then G(alpha) = x." }} {PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 36 "# Purpo se: IDF is a procedure that:" }}{PARA 0 "" 0 "" {TEXT 23 62 "# \+ (1) returns the inverse distribution function of a " }}{PARA 0 "" 0 "" {TEXT 23 71 "# random variable X in the APPL list of s ublists format if" }}{PARA 0 "" 0 "" {TEXT 23 47 "# the o nly argument given is X, or" }}{PARA 0 "" 0 "" {TEXT 23 68 "# \+ (2) Given the optional value x, returns the value alpha, " }}{PARA 0 "" 0 "" {TEXT 23 39 "# where that F(x) = alpha." }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 45 "# Arguments: \+ X: A discrete random variable; " }}{PARA 0 "" 0 "" {TEXT 23 66 "# \+ x (optional argument): A numeric value entered when" }}{PARA 0 "" 0 "" {TEXT 23 69 "# trying to determine alpha, whe re alpha = Pr(X <= x) " }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 65 "# Algorithm: 1. Perform error checking on user ente red arguments" }}{PARA 0 "" 0 "" {TEXT 23 65 "# 2. The IDF of X is determined by sending it to its " }}{PARA 0 "" 0 "" {TEXT 23 65 "# appropriate category, which is based on whether: " }}{PARA 0 "" 0 "" {TEXT 23 69 "# A. X is entered in i ts PDF, CDF, SF, HF, CHF, or IDF" }}{PARA 0 "" 0 "" {TEXT 23 27 "# \+ format " }}{PARA 0 "" 0 "" {TEXT 23 70 "# 3 . If only 1 argument (the random variable X) is entered" }}{PARA 0 "" 0 "" {TEXT 23 70 "# by the user, return the inverse dis tribution function" }}{PARA 0 "" 0 "" {TEXT 23 57 "# of X in the APPL list of 3 lists format " }}{PARA 0 "" 0 "" {TEXT 23 69 "# 4. If 2 arguments are provided, return the value alpha, " }}{PARA 0 "" 0 "" {TEXT 23 41 "# where alpha = Pr(X < = x)" }}{PARA 0 "" 0 "" {TEXT 23 2 "# " }}{PARA 0 "" 0 "" {TEXT 23 55 "# CAUTION: Made x local to find the inverse of the CDF" }}{PARA 0 " " 0 "" {TEXT 23 2 "# " }}{PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "T op" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "IncompleteBeta(p, \+ a, b)" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 288 23 "IncompleteBeta(p, a, b)" }}{PARA 0 "" 0 "" {TEXT 23 72 "------------------------------- -----------------------------------------" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 49 "# Purpose: Computes the Incomplet e Beta Function" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "Top" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }} {PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "InsertHeapConvolution() " {TEXT 289 23 "InsertHeapConvolution()" }}{PARA 0 "" 0 "" {TEXT 23 72 "------------------------------------------------------------------ ------" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 45 "# Purpose: making fX and fY global variables" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "Top" }} {PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "InsertHeapProduct()" {TEXT 290 19 "InsertHeapProduct()" }} {PARA 0 "" 0 "" {TEXT 23 72 "----------------------------------------- -------------------------------" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }} {PARA 0 "" 0 "" {TEXT 23 45 "# Purpose: making fX and fY global varia bles" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "Top" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "InsertHeapProductQ1()" {TEXT 291 21 "Ins ertHeapProductQ1()" }}{PARA 0 "" 0 "" {TEXT 23 72 "------------------- -----------------------------------------------------" }}{PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "Top" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }} {PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "InsertHeapProductQ2()" {TEXT 292 21 "InsertHeapProductQ2()" }}{PARA 0 "" 0 "" {TEXT 23 72 "-- ---------------------------------------------------------------------- " }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 45 "# Pu rpose: making fX and fY global variables" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "Top" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "InsertHeapProductQ3()" {TEXT 23 0 " " }}{PARA 0 "" 0 "" {TEXT 293 21 "InsertHeapProductQ3()" }}{PARA 0 "" 0 "" {TEXT 23 72 "---------------------------------------------------- --------------------" }}{PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "To p" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }} {PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "InsertHeapProductQ4()" {TEXT 294 21 "InsertHeapProductQ4()" }}{PARA 0 "" 0 "" {TEXT 23 72 "-- ---------------------------------------------------------------------- " }}{PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "Top" }}{PARA 0 "" 0 " " {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "InsertionSort(A, B)" {TEXT 295 19 "Inser tionSort(A, B)" }}{PARA 0 "" 0 "" {TEXT 23 72 "----------------------- -------------------------------------------------" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 33 "# Other APPL Procedures \+ Called: " }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 26 "# Date: September 8, 2000" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }} {PARA 0 "" 0 "" {TEXT 23 69 "# Purpose: Returns two sorted lists; the procedure sorts list A and " }}{PARA 0 "" 0 "" {TEXT 23 50 "# \+ uses list A to sort list B accordingly" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "Top" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "KSRV(n)" {TEXT 23 2 " " }}{PARA 0 "" 0 "" {TEXT 296 7 "KSRV(n)" }}{PARA 0 "" 0 "" {TEXT 23 72 "--------- ---------------------------------------------------------------" }} {PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 69 "# Purpos e: KS returns the PDF of the KS statistic for sample size n " }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "Top" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "KSTest(X, Sample , Parameters)" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 297 29 "KSTest(X, Sample, Parameters)" }}{PARA 0 "" 0 "" {TEXT 23 72 "----------------- -------------------------------------------------------" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 69 "# Purpose: Calculat es the Kolmogorov-Smirnoff test statistic for the" }}{PARA 0 "" 0 "" {TEXT 23 71 "# empirical CDF of the sample data versus the C DF of a fitted" }}{PARA 0 "" 0 "" {TEXT 23 47 "# distributio n with random variable X" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 " " 0 "" {TEXT 23 34 "# Arguments: X: Random variable, " }}{PARA 0 "" 0 "" {TEXT 23 49 "# Sample: List of sample data points," } }{PARA 0 "" 0 "" {TEXT 23 73 "# Parameters: List of parame ters set equal to their estimated" }}{PARA 0 "" 0 "" {TEXT 23 58 "# \+ values to be substituted into the CDF of X" }}{PARA 0 "" 0 "" {TEXT 23 3 "# " }}{PARA 0 "" 0 "" {TEXT 23 13 "# Algorithm:" }} {PARA 0 "" 0 "" {TEXT 23 54 "# 1. Check for the appropriate number of arguments" }}{PARA 0 "" 0 "" {TEXT 23 57 "# 2. Check that the \+ RV X is in a list-of-lists format" }}{PARA 0 "" 0 "" {TEXT 23 68 "# \+ 3. Check that X is in its PDF, CDF, SF, HF, CHF, or IDF format" }} {PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {HYPERLNK 17 "To Top " 1 "" "Top" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "Kurtosis(X)" {TEXT 298 11 "Kurtosis(X)" }} {PARA 0 "" 0 "" {TEXT 23 72 "----------------------------------------- -------------------------------" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }} {PARA 0 "" 0 "" {TEXT 23 32 "# Purpose: Returns the kurtosis" }} {PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 32 "# Argum ents: X: Random Variable" }}{PARA 0 "" 0 "" {TEXT 23 2 "# " }}{PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "Top" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "Maximum(X, Y)" {TEXT 299 13 "Maximum(X, Y)" }}{PARA 0 "" 0 "" {TEXT 23 72 "---------- --------------------------------------------------------------" }} {PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 56 "# Other \+ APPL Procedures Called: PDF, Transform, Minimum" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 71 "# Purpose: Maximum is \+ a procedure that returns the PDF of the maximum" }}{PARA 0 "" 0 "" {TEXT 23 36 "# of two independent RVs " }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 59 "# Arguments: X, Y: Inde pendent continuous random variables" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "Top" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "MaximumDiscrete(X, Y)" {TEXT 23 0 "" }} {PARA 0 "" 0 "" {TEXT 300 21 "MaximumDiscrete(X, Y)" }}{PARA 0 "" 0 " " {TEXT 23 72 "------------------------------------------------------- -----------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 23 71 "# Purpose: MaximumDiscrete is a procedure that returns \+ the PDF of the" }}{PARA 0 "" 0 "" {TEXT 23 53 "# maximum of two independent discrete RVs " }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }} {PARA 0 "" 0 "" {TEXT 23 57 "# Arguments: X, Y: Independent discrete \+ random variables" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "Top" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }} {PARA 0 "" 0 "MaximumIID(X, n)" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 301 16 "MaximumIID(X, n)" }}{PARA 0 "" 0 "" {TEXT 23 72 "------------- -----------------------------------------------------------" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 33 "# Other APPL P rocedures Called: " }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 " " {TEXT 23 23 "# Date: June 12, 2001" }}{PARA 0 "" 0 "" {TEXT 23 1 " #" }}{PARA 0 "" 0 "" {TEXT 23 66 "# Purpose: MaximumIID is a procedu re that returns the PDF of the" }}{PARA 0 "" 0 "" {TEXT 23 47 "# \+ maximum of n iid random variables " }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 62 "# Arguments: X: Discrete random \+ variable, n: positive integer" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }} {PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "Top" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "Mean(X) " {TEXT 302 7 "Mean(X)" }}{PARA 0 "" 0 "" {TEXT 23 72 "--------------- ---------------------------------------------------------" }}{PARA 0 " " 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 45 "# Other APPL Proc edures Called: PDF, Convert" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 48 "# Purpose: Returns the mean of a distribution. " }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 32 "# \+ Arguments: X: Random Variable" }}{PARA 0 "" 0 "" {TEXT 23 2 "# " }} {PARA 0 "" 0 "" {TEXT 23 13 "# Algorithm:" }}{PARA 0 "" 0 "" {TEXT 23 53 "# 1. Check for the appropriate number of arguments" }}{PARA 0 "" 0 "" {TEXT 23 60 "# 2. Check that the RV X is in the list of 3 lists format" }}{PARA 0 "" 0 "" {TEXT 23 57 "# 3. Convert X to PDF form if not already in that form" }}{PARA 0 "" 0 "" {TEXT 23 71 "# \+ 4. Call the procedure ExpectedValue with the RV X and g := x -> x " } }{PARA 0 "" 0 "" {TEXT 23 35 "# 5. Return the mean of the RV X" }} {PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {HYPERLNK 17 "To Top " 1 "" "Top" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "Menu(Item)" {TEXT 304 10 "Menu(Item)" }{TEXT 303 52 ", where Item is either All, Continuous, Discrete, or" }}{PARA 0 "" 0 "" {TEXT 23 72 "----------------------------------------------- -------------------------" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 " " 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 24 "# Purpose: APPL \+ Menu " }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 24 "# Arguments: (optional)" }}{PARA 0 "" 0 "" {TEXT 23 77 "# \+ Menu or Menu(All): Returns the list of continuous and discrete " }}{PARA 0 "" 0 "" {TEXT 23 61 "# distributions and the l ist of APPL procedures" }}{PARA 0 "" 0 "" {TEXT 23 81 "# M enu(Continuous): Returns the list of continuous distributions only" }} {PARA 0 "" 0 "" {TEXT 23 77 "# Menu(Discrete): Returns the list of discrete distributions only" }}{PARA 0 "" 0 "" {TEXT 23 72 "# Menu(Procedures): Returns the list of APPL procedures onl y" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "Top" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "MGF(X)" {TEXT 305 6 "MGF(X)" }}{PARA 0 " " 0 "" {TEXT 23 72 "-------------------------------------------------- ----------------------" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 46 "# Other APPL Procedures Called: ExpectedValue" }} {PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 70 "# Purpo se: Returns the moment-generating function of a distribution. " }} {PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 32 "# Argum ents: X: Random Variable" }}{PARA 0 "" 0 "" {TEXT 23 2 "# " }}{PARA 0 "" 0 "" {TEXT 23 13 "# Algorithm:" }}{PARA 0 "" 0 "" {TEXT 23 53 "# \+ 1. Check for the appropriate number of arguments" }}{PARA 0 "" 0 "" {TEXT 23 60 "# 2. Check that the RV X is in the list of 3 lists for mat" }}{PARA 0 "" 0 "" {TEXT 23 64 "# 3. Check that X is given as a PDF, CDF, SF, HF, CHF, or IDF" }}{PARA 0 "" 0 "" {TEXT 23 58 "# 4. Call the procedure ExpectedValue with the RV X and" }}{PARA 0 "" 0 " " {TEXT 23 29 "# g := x -> exp(t * x) " }}{PARA 0 "" 0 "" {TEXT 23 34 "# 5. Return the MGF of the RV X" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "Top" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 2 " " }}{PARA 0 "" 0 "M inimum(X, Y)" {TEXT 306 13 "Minimum(X, Y)" }}{PARA 0 "" 0 "" {TEXT 23 72 "------------------------------------------------------------------ ------" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 53 "# Other APPL Procedures Called: PDF, ReduceList, CDF" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 71 "# Purpose: Minimu m is a procedure that returns the PDF of the minimum" }}{PARA 0 "" 0 " " {TEXT 23 36 "# of two independent RVs " }}{PARA 0 "" 0 " " {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 59 "# Arguments: X, Y: In dependent continuous random variables" }}{PARA 0 "" 0 "" {TEXT 23 1 "# " }}{PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "Top" }}{PARA 0 "" 0 " " {TEXT 23 0 "" }}{PARA 0 "" 0 "MinimumDiscrete(X, Y)" {TEXT 23 0 "" } }{PARA 0 "" 0 "" {TEXT 307 21 "MinimumDiscrete(X, Y)" }}{PARA 0 "" 0 " " {TEXT 23 72 "------------------------------------------------------- -----------------" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 71 "# Purpose: MinimumDiscrete is a procedure that returns \+ the PDF of the" }}{PARA 0 "" 0 "" {TEXT 23 53 "# minimum of two independent discrete RVs " }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }} {PARA 0 "" 0 "" {TEXT 23 57 "# Arguments: X, Y: Independent discrete \+ random variables" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "Top" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }} {PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "MinimumIID(X, n)" {TEXT 308 16 "MinimumIID(X, n)" }}{PARA 0 "" 0 "" {TEXT 23 72 "------------- -----------------------------------------------------------" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 33 "# Other APPL P rocedures Called: " }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 " " {TEXT 23 23 "# Date: June 12, 2001" }}{PARA 0 "" 0 "" {TEXT 23 1 " #" }}{PARA 0 "" 0 "" {TEXT 23 66 "# Purpose: MinimumIID is a procedu re that returns the PDF of the" }}{PARA 0 "" 0 "" {TEXT 23 47 "# \+ maximum of n iid random variables " }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 62 "# Arguments: X: Discrete random \+ variable, n: positive integer" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }} {PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "Top" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "Mixture( MixParameters, MixRVs)" {TEXT 309 30 "Mixture(MixParameters, MixRVs)" }}{PARA 0 "" 0 "" {TEXT 23 72 "--------------------------------------- ---------------------------------" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }} {PARA 0 "" 0 "" {TEXT 23 37 "# Other APPL Procedures Called: PDF " }} {PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 83 "# Purpo se: \"Mixes\" random variables X1, X2, ... Xn by taking weighted sums \+ of the " }}{PARA 0 "" 0 "" {TEXT 23 16 "# distributions" }}{PARA 0 " " 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 77 "# Arguments: Mix Parameters: A list of probabilities p1, p2, ... pn, where p#" }}{PARA 0 "" 0 "" {TEXT 23 80 "# determines what proportion of the mixture de nsity comes from distribution X#," }}{PARA 0 "" 0 "" {TEXT 23 31 "# a lso p1 + p2 + ... + pn = 1;" }}{PARA 0 "" 0 "" {TEXT 23 81 "# MixRVs: X1, X2, ... Xn are the random variables of the component distribution s" }}{PARA 0 "" 0 "" {TEXT 23 40 "# that make up the mixture distribu tion" }}{PARA 0 "" 0 "" {TEXT 23 2 "# " }}{PARA 0 "" 0 "" {TEXT 23 13 "# Algorithm:" }}{PARA 0 "" 0 "" {TEXT 23 54 "# 1. Check for the \+ appropriate number of arguments" }}{PARA 0 "" 0 "" {TEXT 23 48 "# 2 . Check that the first argument is a list" }}{PARA 0 "" 0 "" {TEXT 23 49 "# 3. Check that the second argument is a list" }}{PARA 0 " " 0 "" {TEXT 23 64 "# 4. Check that the length of MixParameters an d MixRVs match" }}{PARA 0 "" 0 "" {TEXT 23 46 "# 5. Check for symb olic mixture parameters" }}{PARA 0 "" 0 "" {TEXT 23 74 "# 6. Check each element of MixParameters to make sure it is in [0, 1] " }}{PARA 0 "" 0 "" {TEXT 23 78 "# 7. Check that the sum of the elements of \+ the MixParameters argument is 1" }}{PARA 0 "" 0 "" {TEXT 23 84 "# 8 . Check that each element of MixRVs represents a continuous random va riable " }}{PARA 0 "" 0 "" {TEXT 23 63 "# 9. Convert MixRVs to PD F form if not already in that form" }}{PARA 0 "" 0 "" {TEXT 23 82 "# \+ 10. Compute the support of the mixture as the union of the supports \+ of MixRVs" }}{PARA 0 "" 0 "" {TEXT 23 71 "# 11. Compute and return \+ the mixed PDF in the list-of-3-lists format" }}{PARA 0 "" 0 "" {TEXT 23 77 "# (NOTE: A conditional expression was placed in the mixe d PDF loop to " }}{PARA 0 "" 0 "" {TEXT 23 75 "# check for PDF 's that are NOT of the form [x -> !]; the Maple \"op\"" }}{PARA 0 "" 0 "" {TEXT 23 76 "# command reads these PDF's differently sinc e the operation x is not" }}{PARA 0 "" 0 "" {TEXT 23 34 "# in \+ the PDF's expression)" }}{PARA 0 "" 0 "" {TEXT 23 2 "# " }}{PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "Top" }}{PARA 0 "" 0 "" {TEXT 421 1 " \+ " }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }} {PARA 256 "" 0 "PMLE(X, Sample, Parameters, [Rightcensor])" {TEXT 23 43 "PMLE(X, Sample, Parameters, [Rightcensor]) " }}{PARA 0 "" 0 "" {TEXT 23 72 "--------------------------------------------------------- ---------------" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 40 "# Other APPL Procedures Called: CHF, HF" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 66 "# Purpose: Returns t he Maximum Likelihood Estimates (MLE) of a RV" }}{PARA 0 "" 0 "" {TEXT 23 69 "# given a sample data set drawn from the RV's d istribution." }}{PARA 0 "" 0 "" {TEXT 23 70 "# The optional \+ argument, Rightcensor, allows for data values" }}{PARA 0 "" 0 "" {TEXT 23 33 "# to be right-censored." }}{PARA 0 "" 0 "" {TEXT 23 3 "# " }}{PARA 0 "" 0 "" {TEXT 23 33 "# Arguments: X: Rando m Variable " }}{PARA 0 "" 0 "" {TEXT 23 48 "# Sample: List of sample data points" }}{PARA 0 "" 0 "" {TEXT 23 60 "# P arameters: List of parameters to be estimated" }}{PARA 0 "" 0 "" {TEXT 23 71 "# Rightcensor (optional): Right-censoring vec tor of 1's and" }}{PARA 0 "" 0 "" {TEXT 23 70 "# 0's, wher e 1 indicates an observed value and 0 indicates" }}{PARA 0 "" 0 "" {TEXT 23 36 "# a right-censored value" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 13 "# Algorithm:" }}{PARA 0 "" 0 "" {TEXT 23 62 "# 1. Check for the appropriate number of ar guments: 3 or 4" }}{PARA 0 "" 0 "" {TEXT 23 46 "# 4. Make sure the re are no piecewise RV's" }}{PARA 0 "" 0 "" {TEXT 23 65 "# 3. Chec k that the RV X is given in a list of 3 lists format" }}{PARA 0 "" 0 " " {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 66 "# 2. If number of a rguments is equal to 3, use *standard* (?) " }}{PARA 0 "" 0 "" {TEXT 23 31 "# loglikelihood function" }}{PARA 0 "" 0 "" {TEXT 23 0 " " }}{PARA 0 "" 0 "" {TEXT 23 69 "# . If number of arguments is eq ual to 4 (which indicated right-" }}{PARA 0 "" 0 "" {TEXT 23 63 "# \+ censoring), use *censoring* (?) loglikelihood function" }}{PARA 0 "" 0 "" {TEXT 23 38 "# 5. Compute HF and CHF forms of X" }}{PARA 0 "" 0 "" {TEXT 23 62 "# 6. Split up the sample data into two list s, censored and" }}{PARA 0 "" 0 "" {TEXT 23 19 "# uncensored" } }{PARA 0 "" 0 "" {TEXT 23 57 "# 7. Compute and simplify the log li kelihood function" }}{PARA 0 "" 0 "" {TEXT 23 70 "# 8. Differentia te the log likelihood function wrt each parameter " }}{PARA 0 "" 0 "" {TEXT 23 24 "# and equate to 0" }}{PARA 0 "" 0 "" {TEXT 23 59 " # 9. Solve the system of log likehood equations from #8" }}{PARA 0 "" 0 "" {TEXT 23 25 "# 10. Return the MLE's" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 70 "# CHANGE: if only 3 arg s, use Andy's formula from mle1.map, sometimes" }}{PARA 0 "" 0 "" {TEXT 23 68 "# likeloglihood function with CHF and HF has pro blems -- so" }}{PARA 0 "" 0 "" {TEXT 23 60 "# don't use this \+ formula unless there is censoring!" }}{PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "Top" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "MLE(X , Sample, Parameters, [Censor])" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 23 36 "MLE(X, Sample, Parameters, [Censor])" }}{PARA 0 "" 0 "" {TEXT 23 72 "--------------------------------------------------------- ---------------" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 75 "# Purpose: Returns the Maximum Likelihood Estimates (MLE ) of a RV given a " }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 " " {HYPERLNK 17 "To Top" 1 "" "Top" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "MLENHPP" {TEXT 310 7 "ML ENHPP" }}{PARA 0 "" 0 "" {TEXT 23 72 "-------------------------------- ----------------------------------------" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 42 "# Purpose: Estimate the paramet ers for " }}{PARA 0 "" 0 "" {TEXT 23 67 "# via maximum lik elihood with arbitrary right censoring" }}{PARA 0 "" 0 "" {TEXT 23 1 " #" }}{PARA 0 "" 0 "" {TEXT 23 18 "# Arguments: 1. " }}{PARA 0 "" 0 " " {TEXT 23 18 "# 2. " }}{PARA 0 "" 0 "" {TEXT 23 58 "# \+ 4. A list of the parameters to be estimated" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "Top " }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }} {PARA 0 "" 0 "MLEWeibull(Sample, Rightcensor)" {TEXT 311 31 "MLEWeibul l(Sample, Rightcensor)" }}{PARA 0 "" 0 "" {TEXT 23 72 "--------------- ---------------------------------------------------------" }}{PARA 0 " " 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 66 "# Purpose: Esti mate the parameters for a Weibull distribution " }}{PARA 0 "" 0 "" {TEXT 23 67 "# via maximum likelihood with arbitrary right censoring" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 69 "# Arguments: Sample: The failure/right censoring times in list form;" }}{PARA 0 "" 0 "" {TEXT 23 71 "# Delta: An indicat or list (1 observed, 0 right censored). " }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 70 "# Algorithm: From Braxton and Le emis (1998), \"A fixed-point algorithm" }}{PARA 0 "" 0 "" {TEXT 23 72 "# for computing Weibull MLEs for a right-censored data se t.\"" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "Top" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "MO M(X, Sample, Parameters)" {TEXT 23 2 " " }}{PARA 0 "" 0 "" {TEXT 312 27 "MOM(X, Sample, Parameters) " }}{PARA 0 "" 0 "" {TEXT 23 72 "------ ------------------------------------------------------------------" }} {PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 51 "# Other \+ APPL Procedures Called: PDF, ExpectedValue" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 72 "# Purpose: Returns the Method of Moments (MOM) Estimates for a complete" }}{PARA 0 "" 0 "" {TEXT 23 27 "# sample data set" }}{PARA 0 "" 0 "" {TEXT 23 3 "# \+ " }}{PARA 0 "" 0 "" {TEXT 23 33 "# Arguments: X: Random variable," }} {PARA 0 "" 0 "" {TEXT 23 53 "# Sample: List of sample data points, and" }}{PARA 0 "" 0 "" {TEXT 23 60 "# Parameters: List of parameters to be estimated" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 14 "# Algorithm: " }}{PARA 0 "" 0 "" {TEXT 23 54 "# 1. Check for the appropriate number of arguments" } }{PARA 0 "" 0 "" {TEXT 23 65 "# 2. Check that the RV X is given in a list of 3 lists format" }}{PARA 0 "" 0 "" {TEXT 23 58 "# 3. Con vert X to PDF form if not already in that form" }}{PARA 0 "" 0 "" {TEXT 23 70 "# 4. Check that parameters to be estimated are the sa me parameters" }}{PARA 0 "" 0 "" {TEXT 23 71 "# that have been \+ assigned to the random variable X; that is, if " }}{PARA 0 "" 0 "" {TEXT 23 72 "# X := GammaRV(a, b), then the parameters to be es timated must be" }}{PARA 0 "" 0 "" {TEXT 23 72 "# [a, b]. Retu rn an error message if this is not the case. Also," }}{PARA 0 "" 0 "" {TEXT 23 70 "# return an error message if a parameter variable \+ name has been" }}{PARA 0 "" 0 "" {TEXT 23 67 "# used in the cur rent Maple session without being unassigned" }}{PARA 0 "" 0 "" {TEXT 23 62 "# before defining the random variable X and running MOM " }}{PARA 0 "" 0 "" {TEXT 23 69 "# 5. Convert the values in the li st Sample to rational numbers so" }}{PARA 0 "" 0 "" {TEXT 23 66 "# \+ that Maple commands such as sum and solve will give exact" }} {PARA 0 "" 0 "" {TEXT 23 59 "# solutions instead of floating po int approximations" }}{PARA 0 "" 0 "" {TEXT 23 65 "# 6. Compute an d simplify the sample moments and distribution" }}{PARA 0 "" 0 "" {TEXT 23 16 "# moments" }}{PARA 0 "" 0 "" {TEXT 23 72 "# 7. \+ Equate the distribution moments with their corresponding sample" }} {PARA 0 "" 0 "" {TEXT 23 16 "# moments" }}{PARA 0 "" 0 "" {TEXT 23 69 "# 8. Solve the system of equations; if Maple cannot d etermine the" }}{PARA 0 "" 0 "" {TEXT 23 65 "# exact solution(s ) with \"solve,\" then use Maple's numeric" }}{PARA 0 "" 0 "" {TEXT 23 25 "# solver, \"fsolve\"" }}{PARA 0 "" 0 "" {TEXT 23 34 "# \+ 9. Return the MOMs as a list" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }} {PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "Top" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "MOMErlan g(Sample)" {TEXT 313 17 "MOMErlang(Sample)" }}{PARA 0 "" 0 "" {TEXT 23 72 "--------------------------------------------------------------- ---------" }}{PARA 0 "" 0 "" {TEXT 23 1 " " }}{PARA 0 "" 0 "" {TEXT 23 45 "# Other APPL Procedures Called: ErlangRV, " }}{PARA 0 "" 0 " " {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 67 "# Purpose: Estimate the parameters for a Erlang distribution via" }}{PARA 0 "" 0 "" {TEXT 23 42 "# method of moments estimation" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 47 "# Arguments: Sampl e: The data set in list form" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }} {PARA 0 "" 0 "" {TEXT 23 37 "# Algorithm: 1. Check for 1 argument" }} {PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {HYPERLNK 17 "To Top " 1 "" "Top" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "MovingHeapConvolutionMethod(X, Y)" {TEXT 314 33 "MovingHeapConvolutionMethod(X, Y)" }}{PARA 0 "" 0 "" {TEXT 23 72 " ---------------------------------------------------------------------- --" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 47 "# \+ Arguments: X, Y: Continuous random variables" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "Top" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "MovingHeapProductMethod(X, Y)" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 315 29 "MovingHeapProductMethod( X, Y)" }}{PARA 0 "" 0 "" {TEXT 23 72 "-------------------------------- ----------------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 23 47 "# Arguments: X, Y: Continuous ran dom variables" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "Top" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }} {PARA 0 "" 0 "MovingHeapProductQ1Q3(X, Y)" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 316 27 "MovingHeapProductQ1Q3(X, Y)" }}{PARA 0 "" 0 "" {TEXT 23 72 "--------------------------------------------------------- ---------------" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 45 "# Arguments: X, Y: Discrete random variables" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "T op" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "MovingHeapProductQ 2Q4(X, Y)" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 317 27 "MovingHeapPro ductQ2Q4(X, Y)" }}{PARA 0 "" 0 "" {TEXT 23 72 "----------------------- -------------------------------------------------" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 45 "# Arguments: X, Y: Discr ete random variables" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "Top" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }} {PARA 0 "" 0 "NextCombination(Previous, N)" {TEXT 23 0 "" }}{PARA 0 " " 0 "" {TEXT 318 28 "NextCombination(Previous, N)" }}{PARA 0 "" 0 "" {TEXT 23 72 "--------------------------------------------------------- ---------------" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 70 "# Purpose: The code generates the next lexicographical ( \"alphabetical" }}{PARA 0 "" 0 "" {TEXT 23 67 "# order\") co mbination of size n (the size of the previous" }}{PARA 0 "" 0 "" {TEXT 23 71 "# combination) of the integers \{1, 2, ..., N\} . It was created" }}{PARA 0 "" 0 "" {TEXT 23 42 "# for use i n the OrderStat code." }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 67 "# Arguments: Previous: A set of integers (written i n a Maple list)" }}{PARA 0 "" 0 "" {TEXT 23 70 "# which is the previous combination. WLOG the code assumes" }}{PARA 0 "" 0 "" {TEXT 23 56 "# that the underlying set is \{1, 2, ..., N\} ." }}{PARA 0 "" 0 "" {TEXT 23 71 "# N: The size of the und erlying set of integers from which " }}{PARA 0 "" 0 "" {TEXT 23 46 "# \+ the combination is to be formed." }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 14 "# Algorithm: " }}{PARA 0 "" 0 "" {TEXT 23 55 "# 1. Error check for the correct number of argume nts" }}{PARA 0 "" 0 "" {TEXT 23 67 "# 2. If the final position in t he combination is not the maximum" }}{PARA 0 "" 0 "" {TEXT 23 71 "# \+ value it can attain, N, then increment it by 1. For example, if" } }{PARA 0 "" 0 "" {TEXT 23 65 "# the previous combination is [1, \+ 2, 4, 6] and N = 10, then" }}{PARA 0 "" 0 "" {TEXT 23 61 "# incr ement the final position by 1 and return the next" }}{PARA 0 "" 0 "" {TEXT 23 36 "# combination as [1, 2, 4, 7]." }}{PARA 0 "" 0 "" {TEXT 23 71 "# 3. If the final position in the combination list is \+ already at its" }}{PARA 0 "" 0 "" {TEXT 23 68 "# maximum value, \+ N, then move left through the combination and" }}{PARA 0 "" 0 "" {TEXT 23 70 "# find the next possible value that can be incremen ted. In other" }}{PARA 0 "" 0 "" {TEXT 23 67 "# words, scan the \+ previous combination from right to left to " }}{PARA 0 "" 0 "" {TEXT 23 66 "# locate the rightmost element that has not yet attained \+ its" }}{PARA 0 "" 0 "" {TEXT 23 71 "# maximum value. If n is the size of the combination, then index " }}{PARA 0 "" 0 "" {TEXT 23 64 " # position i's maximum attainable value is N + i - n. For " }} {PARA 0 "" 0 "" {TEXT 23 66 "# example, if the previous combinat ion is [1, 4, 9, 10] and " }}{PARA 0 "" 0 "" {TEXT 23 63 "# N = \+ 10, then increment position 2 by 1: [1, 5, 9, 10]. " }}{PARA 0 "" 0 " " {TEXT 23 68 "# 4. Upon incrementing the rightmost element in posi tion i, reset" }}{PARA 0 "" 0 "" {TEXT 23 69 "# each value in th e jth position (j = 1, 2, ... , n - i) to the" }}{PARA 0 "" 0 "" {TEXT 23 65 "# right of the ith position to 1 more than the valu e in the" }}{PARA 0 "" 0 "" {TEXT 23 66 "# preceding position. I n the example above from step 3, the " }}{PARA 0 "" 0 "" {TEXT 23 62 " # values in positions 3 and 4 need to be reset. The next" }} {PARA 0 "" 0 "" {TEXT 23 36 "# combination is [1, 5, 6, 7]." }} {PARA 0 "" 0 "" {TEXT 23 36 "# 5. Return the next combination." }} {PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {HYPERLNK 17 "To Top " 1 "" "Top" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "NextPermu tation(Previous)" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 319 25 "NextPe rmutation(Previous)" }}{PARA 0 "" 0 "" {TEXT 23 72 "------------------ ------------------------------------------------------" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 70 "# Purpose: The code generates the next lexicographical (\"alphabetical" }}{PARA 0 "" 0 " " {TEXT 23 70 "# order\") permutation. WLOG the code assumes the elements of" }}{PARA 0 "" 0 "" {TEXT 23 68 "# the set t o be permuted are \{1, 2, ..., k\}. This code was" }}{PARA 0 "" 0 "" {TEXT 23 50 "# created for use in the OrderStat code." }} {PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 67 "# Argum ents: Previous: A set of integers (written in a Maple list)" }}{PARA 0 "" 0 "" {TEXT 23 49 "# which is the previous permutation . " }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 14 "# \+ Algorithm: " }}{PARA 0 "" 0 "" {TEXT 23 56 "# 1. Error check for t he correct number of arguments." }}{PARA 0 "" 0 "" {TEXT 23 71 "# 2 . Find the largest index value i for which Next[i] < Next[i + 1]." }} {PARA 0 "" 0 "" {TEXT 23 71 "# 3. Find the smallest value Next[j] f or which Next[i] < Next[j] and" }}{PARA 0 "" 0 "" {TEXT 23 14 "# \+ i < j." }}{PARA 0 "" 0 "" {TEXT 23 51 "# 4. Interchange the values Next[i] and Next[j]." }}{PARA 0 "" 0 "" {TEXT 23 68 "# 5. Reverse \+ the order of the values to the right of the leftmost" }}{PARA 0 "" 0 " " {TEXT 23 44 "# swapped value, which is now Next[j]." }}{PARA 0 "" 0 "" {TEXT 23 36 "# 6. Return the next permutation." }}{PARA 0 "" 0 "" {TEXT 23 3 "# " }}{PARA 0 "" 0 "" {TEXT 23 61 "# Example : Let A := [1, 4, 3, 2]. Find NextPermutation(A)." }}{PARA 0 "" 0 "" {TEXT 23 35 "# 1. Error check ok: 1 argument." }}{PARA 0 "" 0 "" {TEXT 23 71 "# 2. Assign Next := [1, 4, 3, 2]. The largest index va lue i in which" }}{PARA 0 "" 0 "" {TEXT 23 39 "# Next[i] < Next[ i + 1] is i = 1." }}{PARA 0 "" 0 "" {TEXT 23 65 "# 3. The smallest \+ value Next[j] s.t. Next[i] < Next[j] is 2 = " }}{PARA 0 "" 0 "" {TEXT 23 16 "# Next[4]." }}{PARA 0 "" 0 "" {TEXT 23 65 "# 4. Swap t he values in position i = 1 and j = 4. Next becomes" }}{PARA 0 "" 0 " " {TEXT 23 21 "# [2, 4, 3, 1]." }}{PARA 0 "" 0 "" {TEXT 23 71 "# 5. Reverse the order of the values to the right of 2. Next becomes " }}{PARA 0 "" 0 "" {TEXT 23 21 "# [2, 1, 3, 4]." }}{PARA 0 "" 0 "" {TEXT 23 52 "# 6. Return [2, 1, 3, 4] as the next permutation. " }}{PARA 0 "" 0 "" {TEXT 23 2 "# " }}{PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "Top" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "Norma lVariate(mu, sigma)" {TEXT 23 2 " " }}{PARA 0 "" 0 "" {TEXT 320 24 "N ormalVariate(mu, sigma)" }}{PARA 0 "" 0 "" {TEXT 23 72 "-------------- ----------------------------------------------------------" }}{PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "Top" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "OrderStat(X, n, r, \+ [\"wo\"])" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 321 26 "OrderStat(X, \+ n, r, [\"wo\"])" }}{PARA 0 "" 0 "" {TEXT 23 72 "---------------------- --------------------------------------------------" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 55 "# Other APPL Procedures \+ Called: PDF, CDF, Convert, SF," }}{PARA 0 "" 0 "" {TEXT 23 49 "# \+ ConvertToNumeric" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 69 "# Purpose: OrderStat is a pro cedure that returns the PDF of the rth " }}{PARA 0 "" 0 "" {TEXT 23 64 "# order statistic of a random variable when ordering n" }}{PARA 0 "" 0 "" {TEXT 23 59 "# observations from the popul ation with PDF fX(x)" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 35 "# Arguments: X: A random variable," }}{PARA 0 "" 0 " " {TEXT 23 71 "# n: The number of items drawn randomly fro m the population" }}{PARA 0 "" 0 "" {TEXT 23 33 "# with PDF fX(x), " }}{PARA 0 "" 0 "" {TEXT 23 58 "# r: The inde x of the desired order statistic," }}{PARA 0 "" 0 "" {TEXT 23 70 "# \+ \"wo\" (optional): A variable that indicates if the items " }}{PARA 0 "" 0 "" {TEXT 23 67 "# drawn from the populat ion are done so without (wo)" }}{PARA 0 "" 0 "" {TEXT 23 62 "# \+ replacement (for discrete distributions only)" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 13 "# Algorithm:" }} {PARA 0 "" 0 "" {TEXT 23 53 "# 1. Check for the appropriate number \+ of arguments" }}{PARA 0 "" 0 "" {TEXT 23 68 "# 2. Check that the or der statistic index, r, is not greater than" }}{PARA 0 "" 0 "" {TEXT 23 63 "# the number of random items drawn from the population, n " }}{PARA 0 "" 0 "" {TEXT 23 51 "# 3. Check that the RV X is a list of 3 sublists" }}{PARA 0 "" 0 "" {TEXT 23 57 "# 4. Convert X to PD F form if not already in that form" }}{PARA 0 "" 0 "" {TEXT 23 64 "# \+ 5. If the RV X is continuous, find the PDF of the rth order" }} {PARA 0 "" 0 "" {TEXT 23 68 "# statistic. The RV X must have a t ractable CDF for this to be" }}{PARA 0 "" 0 "" {TEXT 23 17 "# po ssible." }}{PARA 0 "" 0 "" {TEXT 23 70 "# 6. If the RV X is discret e, then call the procedure Convert(X) to" }}{PARA 0 "" 0 "" {TEXT 23 68 "# convert the discrete random variable's support to a standa rd" }}{PARA 0 "" 0 "" {TEXT 23 54 "# APPL discrete random variab le support format: " }}{PARA 0 "" 0 "" {TEXT 23 42 "# standa rd \"dot\" support format:" }}{PARA 0 "" 0 "" {TEXT 23 69 "# [an ything .. anything, incremented by k, transformed by g(x)]" }}{PARA 0 "" 0 "" {TEXT 23 53 "# standard \"no dot\" support format: \+ " }}{PARA 0 "" 0 "" {TEXT 23 30 "# [a1, a2, a3, ... , aN] " }}{PARA 0 "" 0 "" {TEXT 23 69 "# 7. Determine if the random varia ble X has a \"dot\" support format" }}{PARA 0 "" 0 "" {TEXT 23 68 "# \+ (which indicates a symbolic PDF) or a \"nodot\" support format" } }{PARA 0 "" 0 "" {TEXT 23 67 "# (which indicates a numeric suppo rt format). Return an error" }}{PARA 0 "" 0 "" {TEXT 23 69 "# me ssage for symbolic support values in the \"dot\" format, such" }} {PARA 0 "" 0 "" {TEXT 23 60 "# as X := [[x -> x/6], [1 .. n], [ \"Discrete\", \"PDF\"]];" }}{PARA 0 "" 0 "" {TEXT 23 67 "# 8. If on ly 3 arguments are given, by default items are sampled" }}{PARA 0 "" 0 "" {TEXT 23 26 "# with replacement " }}{PARA 0 "" 0 "" {TEXT 23 71 "# 9. If X has a numeric PDF, then calculate the PDF of the r th order" }}{PARA 0 "" 0 "" {TEXT 23 70 "# statistic by looping \+ through the formula given in the code for" }}{PARA 0 "" 0 "" {TEXT 23 67 "# each value of x from min(support of X) to max(support of X )" }}{PARA 0 "" 0 "" {TEXT 23 72 "# 10. If X has a symbolic PDF, the n calculate the PDF of the rth order" }}{PARA 0 "" 0 "" {TEXT 23 72 "# statistic by obtaining a general expression for the PDF with th e" }}{PARA 0 "" 0 "" {TEXT 23 70 "# same formula used in the num eric case. Since the first term of" }}{PARA 0 "" 0 "" {TEXT 23 68 "# \+ the PDF is calculated with FX = 1, calculate the first term " }} {PARA 0 "" 0 "" {TEXT 23 72 "# separate from the other general t erms of the PDF. If the general" }}{PARA 0 "" 0 "" {TEXT 23 70 "# \+ expression for the order stat PDF evaluated at x = min(support" }} {PARA 0 "" 0 "" {TEXT 23 71 "# of X) is equal to the same value \+ as the first term of the order" }}{PARA 0 "" 0 "" {TEXT 23 71 "# \+ stat PDF, then combine them as one expression. Otherwise, leave" }} {PARA 0 "" 0 "" {TEXT 23 64 "# the order stat PDF as a piecewise function with 2 terms." }}{PARA 0 "" 0 "" {TEXT 23 68 "# 11. If 4 a rguments are provided, check if the fourth argument is" }}{PARA 0 "" 0 "" {TEXT 23 69 "# the string \"wo.\" If it is, then items are \+ drawn from the RV's" }}{PARA 0 "" 0 "" {TEXT 23 39 "# population without replacement " }}{PARA 0 "" 0 "" {TEXT 23 68 "# 12. Determin e if the random variable X has a symbolic PDF. If it" }}{PARA 0 "" 0 " " {TEXT 23 65 "# does and also has finite support, then call the procedure" }}{PARA 0 "" 0 "" {TEXT 23 71 "# ConvertToNumeric to convert its PDF and support to the standard" }}{PARA 0 "" 0 "" {TEXT 23 70 "# APPL numeric list format. The conversion is necessary i n order" }}{PARA 0 "" 0 "" {TEXT 23 66 "# to have list inputs fo r the procedures NextPermutation and" }}{PARA 0 "" 0 "" {TEXT 23 24 "# NextCombination." }}{PARA 0 "" 0 "" {TEXT 23 71 "# 13. Check \+ that the sample size n is not greater than the population" }}{PARA 0 " " 0 "" {TEXT 23 50 "# size N. If it is, return an error message. " }}{PARA 0 "" 0 "" {TEXT 23 71 "# 14. If X has fijite support, chec k to see if it has equally likely " }}{PARA 0 "" 0 "" {TEXT 23 72 "# \+ probabilities. If so, compute the PDF of the rth order statistic " }}{PARA 0 "" 0 "" {TEXT 23 53 "# by a combinatorial formula gi ven in the code." }}{PARA 0 "" 0 "" {TEXT 23 70 "# 15. If X has fini te support, but not equally likely probabilities," }}{PARA 0 "" 0 "" {TEXT 23 70 "# the PDF of the rth order statistic is computed di fferently if " }}{PARA 0 "" 0 "" {TEXT 23 70 "# n = 1, n = N, or n = 2, 3, ... N - 1. If n = 1, the PDF of the" }}{PARA 0 "" 0 "" {TEXT 23 73 "# rth order statistic is the same as the population PDF. If n = N, " }}{PARA 0 "" 0 "" {TEXT 23 72 "# the PDF of th e rth order statistic 1 for x = r, and 0 otherwise." }}{PARA 0 "" 0 " " {TEXT 23 70 "# 16. If n = 2, 3, ... N - 1, create an \"n by N\" ar ray, ProbStorage," }}{PARA 0 "" 0 "" {TEXT 23 70 "# to store the rth order statistic PDF values for r from 1 to n," }}{PARA 0 "" 0 "" {TEXT 23 71 "# and x from 1 to N. Initialize ProbStorage to cont ain all zeros." }}{PARA 0 "" 0 "" {TEXT 23 65 "# 17. Create the firs t lexicographical combination of n items. " }}{PARA 0 "" 0 "" {TEXT 23 69 "# 18. Assign perm as the current combination. For each combin ation," }}{PARA 0 "" 0 "" {TEXT 23 62 "# the algorithm finds eve ry possible permutation of that" }}{PARA 0 "" 0 "" {TEXT 23 20 "# \+ combination." }}{PARA 0 "" 0 "" {TEXT 23 71 "# 19. For each permut ation, compute the probability of obtaining that" }}{PARA 0 "" 0 "" {TEXT 23 26 "# given permutation." }}{PARA 0 "" 0 "" {TEXT 23 72 "# 20. Order each permutation and determine which value sits in t he rth" }}{PARA 0 "" 0 "" {TEXT 23 68 "# ordered position. Store the permutation's probability in the" }}{PARA 0 "" 0 "" {TEXT 23 55 " # appropriate positions in the array ProbStorage." }}{PARA 0 "" 0 "" {TEXT 23 72 "# 21. Find the next lexicographical combination, a nd repeat steps 18 -" }}{PARA 0 "" 0 "" {TEXT 23 11 "# 20." }} {PARA 0 "" 0 "" {TEXT 23 40 "# 22. If X has infinite support, then: " }}{PARA 0 "" 0 "" {TEXT 23 70 "# (a) if n = 1, the PDF of th e rth order statistic is the same" }}{PARA 0 "" 0 "" {TEXT 23 36 "# \+ as the population PDF," }}{PARA 0 "" 0 "" {TEXT 23 71 "# \+ (b) if n = 2 and r = 1, a general formula for the PDF of the " }} {PARA 0 "" 0 "" {TEXT 23 46 "# rth order statistic is comp uted," }}{PARA 0 "" 0 "" {TEXT 23 72 "# (c) at present, OrderS tat is unable to handle infinite support" }}{PARA 0 "" 0 "" {TEXT 23 69 "# without replacement cases for n > 2, r > 1. This is \+ an " }}{PARA 0 "" 0 "" {TEXT 23 69 "# open area for future work. Return a message to the user" }}{PARA 0 "" 0 "" {TEXT 23 27 "# \+ stating this." }}{PARA 0 "" 0 "" {TEXT 23 65 "# 23. If 4 arguments are provided and the fourth one is not the" }}{PARA 0 "" 0 "" {TEXT 23 49 "# string \"wo\", then return an error message" } }{PARA 0 "" 0 "" {TEXT 23 70 "# 24. If the RV X is neither continuou s or discrete, return an error" }}{PARA 0 "" 0 "" {TEXT 23 15 "# \+ message" }}{PARA 0 "" 0 "" {TEXT 23 69 "# 25. Return the rth order \+ statistic in the list of sublists format" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "Top" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "PDF(X, [x])" {TEXT 23 2 " " }} {PARA 0 "" 0 "" {TEXT 322 11 "PDF(X, [x])" }}{PARA 0 "" 0 "" {TEXT 23 72 "------------------------------------------------------------------ ------" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 58 "# Other APPL Procedures Called: IDF, Convert, HF, SF, CDF" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 36 "# Purpose: PD F is a procedure that:" }}{PARA 0 "" 0 "" {TEXT 23 68 "# (1) \+ Returns the probability density function (continuous)" }}{PARA 0 "" 0 "" {TEXT 23 64 "# or the probability mass function (discr ete) of a " }}{PARA 0 "" 0 "" {TEXT 23 70 "# random varia ble X in the APPL list of 3 lists format if" }}{PARA 0 "" 0 "" {TEXT 23 47 "# the only argument given is X, or" }}{PARA 0 "" 0 "" {TEXT 23 70 "# (2) Returns the value Pr(X = x) if it is \+ given the optional" }}{PARA 0 "" 0 "" {TEXT 23 49 "# argu ment x in addition to the RV X" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }} {PARA 0 "" 0 "" {TEXT 23 59 "# Arguments: X: A continuous or discrete random variable; " }}{PARA 0 "" 0 "" {TEXT 23 65 "# x (op tional argument): A numeric value entered when" }}{PARA 0 "" 0 "" {TEXT 23 46 "# trying to determine Pr(X = x)" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 65 "# Algorithm: \+ 1. Perform error checking on user entered arguments" }}{PARA 0 "" 0 " " {TEXT 23 65 "# 2. The PDF of X is determined by sending \+ it to its " }}{PARA 0 "" 0 "" {TEXT 23 65 "# appropriat e category, which is based on whether:" }}{PARA 0 "" 0 "" {TEXT 23 47 "# A. X is continuous or discrete" }}{PARA 0 "" 0 "" {TEXT 23 69 "# B. X is entered in its PDF, CDF, SF, HF, CHF, or IDF" }}{PARA 0 "" 0 "" {TEXT 23 27 "# forma t " }}{PARA 0 "" 0 "" {TEXT 23 70 "# 3. If only 1 argument (the random variable X) is entered" }}{PARA 0 "" 0 "" {TEXT 23 72 "# \+ by the user, return the probability density function of " }}{PARA 0 "" 0 "" {TEXT 23 54 "# X in the APPL list o f 3 lists format " }}{PARA 0 "" 0 "" {TEXT 23 72 "# 4. If \+ 2 arguments are provided, return the value Pr(X = x)" }}{PARA 0 "" 0 " " {TEXT 23 2 "# " }}{PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "Top" } }{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "PercolateDown(H, B, i, \+ n)" {TEXT 23 0 "" }}{PARA 256 "" 0 "" {TEXT 23 25 "PercolateDown(H, B, i, n)" }}{PARA 0 "" 0 "" {TEXT 23 72 "------------------------------- -----------------------------------------" }}{PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "Top" }}{PARA 256 "" 0 "" {TEXT 23 0 "" }} {PARA 256 "" 0 "" {TEXT 23 0 "" }}{PARA 256 "" 0 "PercolateDownHeap(H, i, n)" {TEXT 23 0 "" }}{PARA 256 "" 0 "" {TEXT 23 26 "PercolateDownHe ap(H, i, n)" }}{PARA 0 "" 0 "" {TEXT 23 72 "-------------------------- ----------------------------------------------" }}{PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "Top" }}{PARA 256 "" 0 "" {TEXT 23 0 "" }} {PARA 256 "" 0 "" {TEXT 23 0 "" }}{PARA 256 "" 0 "PercolateDownHeapQ1( H, i, n)" {TEXT 23 0 "" }}{PARA 256 "" 0 "" {TEXT 23 28 "PercolateDown HeapQ1(H, i, n)" }}{PARA 0 "" 0 "" {TEXT 23 72 "---------------------- --------------------------------------------------" }}{PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "Top" }}{PARA 256 "" 0 "" {TEXT 23 0 "" }} {PARA 256 "" 0 "" {TEXT 23 0 "" }}{PARA 256 "" 0 "PercolateDownHeapQ2( H, i, n)" {TEXT 23 0 "" }}{PARA 256 "" 0 "" {TEXT 23 28 "PercolateDown HeapQ2(H, i, n)" }}{PARA 0 "" 0 "" {TEXT 23 72 "---------------------- --------------------------------------------------" }}{PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "Top" }}{PARA 256 "" 0 "" {TEXT 23 0 "" }} {PARA 256 "" 0 "" {TEXT 23 0 "" }}{PARA 256 "" 0 "PercolateDownHeapQ3( i, n)" {TEXT 23 0 "" }}{PARA 256 "" 0 "" {TEXT 23 25 "PercolateDownHea pQ3(i, n)" }}{PARA 0 "" 0 "" {TEXT 23 72 "---------------------------- --------------------------------------------" }}{PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "Top" }}{PARA 256 "" 0 "" {TEXT 23 0 "" }} {PARA 256 "" 0 "" {TEXT 23 0 "" }}{PARA 256 "" 0 "Procedure Name: Perc olateDownHeapQ4(i, n)" {TEXT 23 0 "" }}{PARA 256 "" 0 "PercolateDownHe apQ4(i, n)" {TEXT 23 25 "PercolateDownHeapQ4(i, n)" }}{PARA 0 "" 0 "" {TEXT 23 72 "--------------------------------------------------------- ---------------" }}{PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "Top" }} {PARA 256 "" 0 "" {TEXT 23 0 "" }}{PARA 256 "" 0 "" {TEXT 23 0 "" }} {PARA 256 "" 0 "PlotDist(X, low, high)" {TEXT 23 0 "" }}{PARA 256 "" 0 "" {TEXT 23 22 "PlotDist(X, low, high)" }}{PARA 0 "" 0 "" {TEXT 23 72 "------------------------------------------------------------------ ------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 70 "# Purpose: For the random variable X, plot the functi on indicated in " }}{PARA 0 "" 0 "" {TEXT 23 68 "# the random variab le X's list-of-sublists (PDF, CDF, SF, HF, CHF," }}{PARA 0 "" 0 "" {TEXT 23 11 "# or IDF) " }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 " " 0 "" {TEXT 23 34 "# Arguments: X: Random variable; " }}{PARA 0 "" 0 "" {TEXT 23 70 "# low, high (optional values): values pr ovided by the user" }}{PARA 0 "" 0 "" {TEXT 23 71 "# to in dicate over what range the function is to be plotted" }}{PARA 0 "" 0 " " {TEXT 23 3 "# " }}{PARA 0 "" 0 "" {TEXT 23 13 "# Algorithm:" }} {PARA 0 "" 0 "" {TEXT 23 54 "# 1. Check for the appropriate number of arguments" }}{PARA 0 "" 0 "" {TEXT 23 57 "# 2. Check that the \+ RV X is in a list-of-lists format" }}{PARA 0 "" 0 "" {TEXT 23 68 "# \+ 3. Check that X is in its PDF, CDF, SF, HF, CHF, or IDF format" }} {PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 65 "# NOTE \+ (6/24/00): plotdist3.map having trouble with KSRV(n) when" }}{PARA 0 " " 0 "" {TEXT 23 63 "# lo and hi are given -- see plotdist3.mws; Fixed in plotdist4" }}{PARA 0 "" 0 "" {TEXT 23 72 "# Problem -- if range g iven by user, the new max range value for X does" }}{PARA 0 "" 0 "" {TEXT 23 70 "# not automatically become that high value -- need to fi gure out what" }}{PARA 0 "" 0 "" {TEXT 23 68 "# segment that high val ue falls in in a piecewise function ... same" }}{PARA 0 "" 0 "" {TEXT 23 11 "# with low" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 " " {TEXT 23 48 "# NOTE: plotdist6.map has discrete capabilities" }} {PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 67 "# NOTE: plotdist7.map defines lo and hi for infinite support range" }}{PARA 0 "" 0 "" {TEXT 23 34 "# as IDF(X, 0.025), IDF(X, 0.975)" }}{PARA 0 " " 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 70 "# NOTE: plotdist .map only works for 1 argument in discrete case right" }}{PARA 0 "" 0 "" {TEXT 23 71 "# now -- user cannot supply his own range ... APPL de termines the best" }}{PARA 0 "" 0 "" {TEXT 23 29 "# support range for graphing" }}{PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "Top" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "PlotEmpCDF(Sample, [low], [hig h])" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 323 33 "PlotEmpCDF(Sample, \+ [low], [high])" }}{PARA 0 "" 0 "" {TEXT 23 72 "----------------------- -------------------------------------------------" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 36 "# Other APPL Procedures \+ Called: PDF" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 68 "# Purpose: Plot the empirical cumulative distribution function of a" }}{PARA 0 "" 0 "" {TEXT 23 25 "# random sample" }} {PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 39 "# Argum ents: Sample: A random sample; " }}{PARA 0 "" 0 "" {TEXT 23 71 "# \+ low, high (optional values): Values provided by the user " }} {PARA 0 "" 0 "" {TEXT 23 66 "# indicating the left and rig ht end values of the plot" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 13 "# Algorithm:" }}{PARA 0 "" 0 "" {TEXT 23 54 "# \+ 1. Check for the appropriate number of arguments" }}{PARA 0 "" 0 " " {TEXT 23 29 "# 2. Sort the sample list" }}{PARA 0 "" 0 "" {TEXT 23 58 "# Example: Sample := [1, 10, 5, 10, 5, 7, 3, 2.5];" }} {PARA 0 "" 0 "" {TEXT 23 63 "# SortedSample := [1, 2. 5, 3, 5, 5, 7, 10, 10]" }}{PARA 0 "" 0 "" {TEXT 23 71 "# 3. Constr uct the empirical CDF as a set of lists, where each list" }}{PARA 0 " " 0 "" {TEXT 23 25 "# contains either:" }}{PARA 0 "" 0 "" {TEXT 23 70 "# (a) The endpoints of a given \"step\" of the em pirical CDF, or" }}{PARA 0 "" 0 "" {TEXT 23 38 "# Example: [[1, 1/8], [2, 1/8]]" }}{PARA 0 "" 0 "" {TEXT 23 75 "# (b) The poi nts needed to connect the \"steps\" of the empirical CDF" }}{PARA 0 " " 0 "" {TEXT 23 36 "# Example: [[1, 0], [1, 1/8]]" }}{PARA 0 " " 0 "" {TEXT 23 70 "# The counter is used to determine the leng th of the line which" }}{PARA 0 "" 0 "" {TEXT 23 34 "# connects the steps in (b)" }}{PARA 0 "" 0 "" {TEXT 23 72 "# 4. The empiric al CDF is plotted on the range low to high. If the " }}{PARA 0 "" 0 " " {TEXT 23 70 "# values low and high are not provided by the us er, the default" }}{PARA 0 "" 0 "" {TEXT 23 73 "# value for low is the smallest sample point and the default value" }}{PARA 0 "" 0 " " {TEXT 23 45 "# for high is the largest sample point" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "Top" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "PlotEmpCIF(Sampl e, lo, hi)" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 324 26 "PlotEmpCIF(S ample, lo, hi)" }}{PARA 0 "" 0 "" {TEXT 23 72 "----------------------- -------------------------------------------------" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 60 "# Purpose: Plot the empi rical cumulative intensity function" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 39 "# Arguments: Sample: A random sample; \+ " }}{PARA 0 "" 0 "" {TEXT 23 71 "# low, high (optional val ues): Values provided by the user " }}{PARA 0 "" 0 "" {TEXT 23 66 "# \+ indicating the left and right end values of the plot" }} {PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 13 "# Algor ithm:" }}{PARA 0 "" 0 "" {TEXT 23 54 "# 1. Check for the appropria te number of arguments" }}{PARA 0 "" 0 "" {TEXT 23 29 "# 2. Sort t he sample list" }}{PARA 0 "" 0 "" {TEXT 23 58 "# Example: Samp le := [1, 10, 5, 10, 5, 7, 3, 2.5];" }}{PARA 0 "" 0 "" {TEXT 23 63 "# \+ SortedSample := [1, 2.5, 3, 5, 5, 7, 10, 10]" }} {PARA 0 "" 0 "" {TEXT 23 71 "# 3. Construct the empirical CDF as a set of lists, where each list" }}{PARA 0 "" 0 "" {TEXT 23 25 "# \+ contains either:" }}{PARA 0 "" 0 "" {TEXT 23 70 "# (a) The e ndpoints of a given \"step\" of the empirical CDF, or" }}{PARA 0 "" 0 "" {TEXT 23 38 "# Example: [[1, 1/8], [2, 1/8]]" }}{PARA 0 "" 0 "" {TEXT 23 75 "# (b) The points needed to connect the \"ste ps\" of the empirical CDF" }}{PARA 0 "" 0 "" {TEXT 23 36 "# Exa mple: [[1, 0], [1, 1/8]]" }}{PARA 0 "" 0 "" {TEXT 23 70 "# The \+ counter is used to determine the length of the line which" }}{PARA 0 " " 0 "" {TEXT 23 34 "# connects the steps in (b)" }}{PARA 0 "" 0 "" {TEXT 23 72 "# 4. The empirical CDF is plotted on the range l ow to high. If the " }}{PARA 0 "" 0 "" {TEXT 23 70 "# values l ow and high are not provided by the user, the default" }}{PARA 0 "" 0 "" {TEXT 23 73 "# value for low is the smallest sample point an d the default value" }}{PARA 0 "" 0 "" {TEXT 23 45 "# for high \+ is the largest sample point" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "Top" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "PlotEmpSF(Sample, Censor)" {TEXT 325 25 "PlotEmpSF(Sample, Censor)" }}{PARA 0 "" 0 "" {TEXT 23 72 "--------------------------------------------------------- ---------------" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 67 "# Purpose: Plot the empirical survivor function of a ran dom sample" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 39 "# Arguments: Sample: A random sample; " }}{PARA 0 "" 0 "" {TEXT 23 44 "# Censor: Right-censoring vector" }}{PARA 0 " " 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 13 "# Algorithm:" }} {PARA 0 "" 0 "" {TEXT 23 54 "# 1. Check for the appropriate number of arguments" }}{PARA 0 "" 0 "" {TEXT 23 29 "# 2. Sort the sample list" }}{PARA 0 "" 0 "" {TEXT 23 58 "# Example: Sample := [1, 10, 5, 10, 5, 7, 3, 2.5];" }}{PARA 0 "" 0 "" {TEXT 23 63 "# \+ SortedSample := [1, 2.5, 3, 5, 5, 7, 10, 10]" }}{PARA 0 "" 0 " " {TEXT 23 70 "# 3. Construct the empirical SF as a set of lists, \+ where each list" }}{PARA 0 "" 0 "" {TEXT 23 25 "# contains eith er:" }}{PARA 0 "" 0 "" {TEXT 23 70 "# (a) The endpoints of a g iven \"step\" of the empirical CDF, or" }}{PARA 0 "" 0 "" {TEXT 23 38 "# Example: [[1, 1/8], [2, 1/8]]" }}{PARA 0 "" 0 "" {TEXT 23 71 "# (b) The points needed to connect the \"steps\" of the em pirical" }}{PARA 0 "" 0 "" {TEXT 23 18 "# CDF" }}{PARA 0 "" 0 "" {TEXT 23 36 "# Example: [[1, 0], [1, 1/8]]" }}{PARA 0 " " 0 "" {TEXT 23 70 "# The counter is used to determine the leng th of the line which" }}{PARA 0 "" 0 "" {TEXT 23 34 "# connects the steps in (b)" }}{PARA 0 "" 0 "" {TEXT 23 71 "# 4. The empiric al SF is plotted on the range low to high. If the " }}{PARA 0 "" 0 " " {TEXT 23 70 "# values low and high are not provided by the us er, the default" }}{PARA 0 "" 0 "" {TEXT 23 67 "# value for low is the smallest sample point and the default" }}{PARA 0 "" 0 "" {TEXT 23 14 "# value" }}{PARA 0 "" 0 "" {TEXT 23 45 "# f or high is the largest sample point" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "Top" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "PlotEmpVsFittedCDF(X, Sample, Parameters , [low], [high])" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 326 56 "PlotEm pVsFittedCDF(X, Sample, Parameters, [low], [high])" }}{PARA 0 "" 0 "" {TEXT 23 72 "--------------------------------------------------------- ---------------" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 58 "# Other APPL Procedures Called: CDF, PlotDist, PlotEmpCD F" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 69 "# \+ Purpose: Plot the random variable X's empirical CDF and fitted CDF" }} {PARA 0 "" 0 "" {TEXT 23 20 "# together" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 34 "# Arguments: X: Random \+ variable; " }}{PARA 0 "" 0 "" {TEXT 23 48 "# Sample: List \+ of sample data points" }}{PARA 0 "" 0 "" {TEXT 23 74 "# Pa rameters: List of parameters set equatl to their estimated" }}{PARA 0 "" 0 "" {TEXT 23 58 "# values to be substituted into the CDF of X" }}{PARA 0 "" 0 "" {TEXT 23 3 "# " }}{PARA 0 "" 0 "" {TEXT 23 13 "# Algorithm:" }}{PARA 0 "" 0 "" {TEXT 23 54 "# 1. Check fo r the appropriate number of arguments" }}{PARA 0 "" 0 "" {TEXT 23 57 " # 2. Check that the RV X is in a list-of-lists format" }}{PARA 0 " " 0 "" {TEXT 23 68 "# 3. Check that X is in its PDF, CDF, SF, HF, \+ CHF, or IDF format" }}{PARA 0 "" 0 "" {TEXT 23 59 "# 4. Define Emp CDFPlot as the plot of the empirical CDF" }}{PARA 0 "" 0 "" {TEXT 23 59 "# 5. Define FittedCDFPlot as the plot of the fitted CDF" }} {PARA 0 "" 0 "" {TEXT 23 55 "# 6. Plot the empirical CDF and fitte d CDF together" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "Top" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }} {PARA 0 "" 0 "PlotEmpVsFittedCIF(X, Sample, Parameters, low, high)" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 327 52 "PlotEmpVsFittedCIF(X, Sa mple, Parameters, low, high)" }}{PARA 0 "" 0 "" {TEXT 23 72 "--------- ---------------------------------------------------------------" }} {PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 55 "# NEED T O WORK ON COMMENTS AT A LATER TIME (7/18/2000)" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 75 "# Purpose: Plot the emp irical cumulative distribution function of a random" }}{PARA 0 "" 0 " " {TEXT 23 18 "# sample" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }} {PARA 0 "" 0 "" {TEXT 23 39 "# Arguments: Sample: A random sample; " }}{PARA 0 "" 0 "" {TEXT 23 71 "# low, high (optional value s): Values provided by the user " }}{PARA 0 "" 0 "" {TEXT 23 66 "# \+ indicating the left and right end values of the plot" }} {PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 13 "# Algor ithm:" }}{PARA 0 "" 0 "" {TEXT 23 54 "# 1. Check for the appropria te number of arguments" }}{PARA 0 "" 0 "" {TEXT 23 29 "# 2. Sort t he sample list" }}{PARA 0 "" 0 "" {TEXT 23 58 "# Example: Samp le := [1, 10, 5, 10, 5, 7, 3, 2.5];" }}{PARA 0 "" 0 "" {TEXT 23 63 "# \+ SortedSample := [1, 2.5, 3, 5, 5, 7, 10, 10]" }} {PARA 0 "" 0 "" {TEXT 23 71 "# 3. Construct the empirical CDF as a set of lists, where each list" }}{PARA 0 "" 0 "" {TEXT 23 25 "# \+ contains either:" }}{PARA 0 "" 0 "" {TEXT 23 70 "# (a) The e ndpoints of a given \"step\" of the empirical CDF, or" }}{PARA 0 "" 0 "" {TEXT 23 38 "# Example: [[1, 1/8], [2, 1/8]]" }}{PARA 0 "" 0 "" {TEXT 23 75 "# (b) The points needed to connect the \"ste ps\" of the empirical CDF" }}{PARA 0 "" 0 "" {TEXT 23 36 "# Exa mple: [[1, 0], [1, 1/8]]" }}{PARA 0 "" 0 "" {TEXT 23 70 "# The \+ counter is used to determine the length of the line which" }}{PARA 0 " " 0 "" {TEXT 23 34 "# connects the steps in (b)" }}{PARA 0 "" 0 "" {TEXT 23 72 "# 4. The empirical CDF is plotted on the range l ow to high. If the " }}{PARA 0 "" 0 "" {TEXT 23 70 "# values l ow and high are not provided by the user, the default" }}{PARA 0 "" 0 "" {TEXT 23 73 "# value for low is the smallest sample point an d the default value" }}{PARA 0 "" 0 "" {TEXT 23 45 "# for high \+ is the largest sample point" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "Top" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "PlotEmpVsFittedSF(X, Sample, Parameters, Censor) " {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 328 48 "PlotEmpVsFittedSF(X, S ample, Parameters, Censor)" }}{PARA 0 "" 0 "" {TEXT 23 72 "----------- -------------------------------------------------------------" }} {PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 60 "# Purpos e: Plot the empirical survivor function of a random" }}{PARA 0 "" 0 " " {TEXT 23 18 "# sample" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }} {PARA 0 "" 0 "" {TEXT 23 39 "# Arguments: Sample: A random sample; " }}{PARA 0 "" 0 "" {TEXT 23 71 "# low, high (optional value s): Values provided by the user " }}{PARA 0 "" 0 "" {TEXT 23 66 "# \+ indicating the left and right end values of the plot" }} {PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 13 "# Algor ithm:" }}{PARA 0 "" 0 "" {TEXT 23 54 "# 1. Check for the appropria te number of arguments" }}{PARA 0 "" 0 "" {TEXT 23 29 "# 2. Sort t he sample list" }}{PARA 0 "" 0 "" {TEXT 23 58 "# Example: Samp le := [1, 10, 5, 10, 5, 7, 3, 2.5];" }}{PARA 0 "" 0 "" {TEXT 23 63 "# \+ SortedSample := [1, 2.5, 3, 5, 5, 7, 10, 10]" }} {PARA 0 "" 0 "" {TEXT 23 71 "# 3. Construct the empirical CDF as a set of lists, where each list" }}{PARA 0 "" 0 "" {TEXT 23 25 "# \+ contains either:" }}{PARA 0 "" 0 "" {TEXT 23 70 "# (a) The e ndpoints of a given \"step\" of the empirical CDF, or" }}{PARA 0 "" 0 "" {TEXT 23 38 "# Example: [[1, 1/8], [2, 1/8]]" }}{PARA 0 "" 0 "" {TEXT 23 71 "# (b) The points needed to connect the \"ste ps\" of the empirical" }}{PARA 0 "" 0 "" {TEXT 23 7 "# \011 CDF" }} {PARA 0 "" 0 "" {TEXT 23 36 "# Example: [[1, 0], [1, 1/8]]" }} {PARA 0 "" 0 "" {TEXT 23 70 "# The counter is used to determine the length of the line which" }}{PARA 0 "" 0 "" {TEXT 23 34 "# \+ connects the steps in (b)" }}{PARA 0 "" 0 "" {TEXT 23 72 "# 4. Th e empirical CDF is plotted on the range low to high. If the " }} {PARA 0 "" 0 "" {TEXT 23 70 "# values low and high are not prov ided by the user, the default" }}{PARA 0 "" 0 "" {TEXT 23 67 "# \+ value for low is the smallest sample point and the default" }}{PARA 0 "" 0 "" {TEXT 23 9 "# \011 value" }}{PARA 0 "" 0 "" {TEXT 23 45 "# \+ for high is the largest sample point" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "Top" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "PPPlot(X, Sample, Parameters)" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 329 29 "PPPlot(X, Sample, Parame ters)" }}{PARA 0 "" 0 "" {TEXT 23 72 "-------------------------------- ----------------------------------------" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 67 "# Purpose: Plots the model probab ility F^hat(X) versus the sample " }}{PARA 0 "" 0 "" {TEXT 23 24 "# \+ probability " }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 32 "# Arguments: X: Random variable" }}{PARA 0 "" 0 "" {TEXT 23 48 "# Sample: List of sample data points" }} {PARA 0 "" 0 "" {TEXT 23 24 "# Parameters" }}{PARA 0 "" 0 "" {TEXT 23 3 "# " }}{PARA 0 "" 0 "" {TEXT 23 13 "# Algorithm:" }} {PARA 0 "" 0 "" {TEXT 23 54 "# 1. Check for the appropriate number of arguments" }}{PARA 0 "" 0 "" {TEXT 23 57 "# 2. Check that the \+ RV X is in a list-of-lists format" }}{PARA 0 "" 0 "" {TEXT 23 68 "# \+ 3. Check that X is in its PDF, CDF, SF, HF, CHF, or IDF format" }} {PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }} {PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 33 "# Proce dure names: BirthdayProb" }}{PARA 0 "" 0 "" {TEXT 23 30 "# \+ CrapsProb" }}{PARA 0 "" 0 "" {TEXT 23 37 "# \+ HatCheckGirlProb" }}{PARA 0 "" 0 "" {TEXT 23 31 "# \+ PointsProb" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 63 "# Arguments: The parameters of the distribution of the \+ random" }}{PARA 0 "" 0 "" {TEXT 23 23 "# variable" }} {PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 46 "# Algor ithm: (NEED TO REDO FOR PROBABILITIES)" }}{PARA 0 "" 0 "" {TEXT 23 54 "# 1. Check for the appropriate number of arguments" }}{PARA 0 "" 0 "" {TEXT 23 58 "# 2. Check parameter space when parameters are n umeric" }}{PARA 0 "" 0 "" {TEXT 23 52 "# 3. Check to see that the \+ parameters are finite" }}{PARA 0 "" 0 "" {TEXT 23 55 "# 4. Make as sumptions about any symbolic parameters" }}{PARA 0 "" 0 "" {TEXT 23 33 "# 6. Return the list of lists" }}{PARA 0 "" 0 "" {TEXT 23 2 "# " }}{PARA 0 "" 0 "" {TEXT 23 70 "# Note: Could define a type in sta rtup about p being between 0 and 1" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" } }{PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "Top" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "ProductIID(X, n)" {TEXT 23 1 " " }} {PARA 0 "" 0 "" {TEXT 330 16 "ProductIID(X, n)" }}{PARA 0 "" 0 "" {TEXT 23 72 "--------------------------------------------------------- ---------------" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 47 "# Other APPL Procedures Called: PDFRV, Product" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 43 "# Arguments: \+ X: Continuous random variable" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }} {PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "Top" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "Product(X, Y)" {TEXT 23 0 "" }}{PARA 0 " " 0 "" {TEXT 331 13 "Product(X, Y)" }}{PARA 0 "" 0 "" {TEXT 23 72 "--- --------------------------------------------------------------------- " }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 67 "# Ot her APPL Procedures Called: ProductContinuous, ProductDiscrete" }} {PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 18 "# Argum ents: X, Y" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "Top" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }} {PARA 0 "" 0 "ProductContinuous(X, Y)" {TEXT 23 2 " " }}{PARA 0 "" 0 "" {TEXT 332 23 "ProductContinuous(X, Y)" }}{PARA 0 "" 0 "" {TEXT 23 72 "------------------------------------------------------------------ ------" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 57 "# Other APPL Procedures Called: PDF, ReduceList, CleanUp" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 47 "# Arguments: X, Y: Continuous random variables" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }} {PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "Top" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "ProductDiscrete(X, Y)" {TEXT 23 0 "" }} {PARA 0 "" 0 "" {TEXT 333 21 "ProductDiscrete(X, Y)" }}{PARA 0 "" 0 " " {TEXT 23 72 "------------------------------------------------------- -----------------" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 18 "# Arguments: X, Y" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }} {PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "Top" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "QQPlot(X, Sample, Parameters)" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 334 29 "QQPlot(X, Sample, Parameters)" }} {PARA 0 "" 0 "" {TEXT 23 72 "----------------------------------------- -------------------------------" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }} {PARA 0 "" 0 "" {TEXT 23 36 "# Other APPL Procedures Called: IDF" }} {PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 71 "# Purpo se: Plots the q_i quantile of a fitted distribution function X " }} {PARA 0 "" 0 "" {TEXT 23 62 "# versus the q_i quantile of th e Sample distribution" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 32 "# Arguments: X: Random variable" }}{PARA 0 "" 0 "" {TEXT 23 48 "# Sample: List of sample data points" }} {PARA 0 "" 0 "" {TEXT 23 24 "# Parameters" }}{PARA 0 "" 0 "" {TEXT 23 3 "# " }}{PARA 0 "" 0 "" {TEXT 23 13 "# Algorithm:" }} {PARA 0 "" 0 "" {TEXT 23 54 "# 1. Check for the appropriate number of arguments" }}{PARA 0 "" 0 "" {TEXT 23 57 "# 2. Check that the \+ RV X is in a list-of-lists format" }}{PARA 0 "" 0 "" {TEXT 23 68 "# \+ 3. Check that X is in its PDF, CDF, SF, HF, CHF, or IDF format" }} {PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {HYPERLNK 17 "To Top " 1 "" "Top" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "RangeStat (X, n, [\"wo\"])" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 335 23 "RangeS tat(X, n, [\"wo\"])" }}{PARA 0 "" 0 "" {TEXT 23 72 "------------------ ------------------------------------------------------" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 40 "# Other APPL Proced ures Called: Convert" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 70 "# Purpose: RangeStat is a procedure that returns the \+ PDF of the range" }}{PARA 0 "" 0 "" {TEXT 23 68 "# X_(n) - X _(1), of a population when n items are sampled " }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 35 "# Arguments: X: A rando m variable," }}{PARA 0 "" 0 "" {TEXT 23 71 "# n: The numbe r of items drawn randomly from the population" }}{PARA 0 "" 0 "" {TEXT 23 33 "# with PDF fX(x), " }}{PARA 0 "" 0 "" {TEXT 23 70 "# \"wo\" (optional): A variable that indicate s if the items " }}{PARA 0 "" 0 "" {TEXT 23 67 "# drawn from the population are done so without (wo)" }}{PARA 0 "" 0 "" {TEXT 23 62 "# replacement (for discrete distributions \+ only)" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 13 "# Algorithm:" }}{PARA 0 "" 0 "" {TEXT 23 53 "# 1. Check for the a ppropriate number of arguments" }}{PARA 0 "" 0 "" {TEXT 23 51 "# 2. Check that the RV X is a list of 3 sublists" }}{PARA 0 "" 0 "" {TEXT 23 70 "# 3. Check that n >= 2, if not return an error message stati ng that" }}{PARA 0 "" 0 "" {TEXT 23 63 "# there is no range sinc e only 1 item is sampled from the" }}{PARA 0 "" 0 "" {TEXT 23 18 "# \+ population" }}{PARA 0 "" 0 "" {TEXT 23 57 "# 4. Convert X to PD F form if not already in that form" }}{PARA 0 "" 0 "" {TEXT 23 68 "# \+ 5. Call the procedure Convert(X) to convert the discrete random" }} {PARA 0 "" 0 "" {TEXT 23 70 "# variable's support to a standard \+ APPL discrete random variable" }}{PARA 0 "" 0 "" {TEXT 23 24 "# \+ support format: " }}{PARA 0 "" 0 "" {TEXT 23 42 "# standard \+ \"dot\" support format:" }}{PARA 0 "" 0 "" {TEXT 23 69 "# [anyth ing .. anything, incremented by k, transformed by g(x)]" }}{PARA 0 "" 0 "" {TEXT 23 53 "# standard \"no dot\" support format: \+ " }}{PARA 0 "" 0 "" {TEXT 23 30 "# [x1, x2, x3, ... , xN]" }} {PARA 0 "" 0 "" {TEXT 23 69 "# 6. Determine if the random variable \+ X has a \"dot\" support format" }}{PARA 0 "" 0 "" {TEXT 23 69 "# \+ (which indicates a formulaic PDF) or a \"nodot\" support format" }} {PARA 0 "" 0 "" {TEXT 23 68 "# (which indicates a numeric PDF). \+ Return an error message for" }}{PARA 0 "" 0 "" {TEXT 23 64 "# fo rmulaic PDFs with non-numeric support values, such as:" }}{PARA 0 "" 0 "" {TEXT 23 57 "# X := [[x -> x/6], [1 .. n], [\"Discrete\", \+ \"PDF\"]];" }}{PARA 0 "" 0 "" {TEXT 23 71 "# 7. Check that N >= 2, \+ if not return an error message stating that " }}{PARA 0 "" 0 "" {TEXT 23 69 "# the population consists of only 1 element and so thhere is no" }}{PARA 0 "" 0 "" {TEXT 23 13 "# range" }}{PARA 0 "" 0 " " {TEXT 23 67 "# 8. If only 2 arguments are given, by default items are sampled" }}{PARA 0 "" 0 "" {TEXT 23 26 "# with replacement \+ " }}{PARA 0 "" 0 "" {TEXT 23 70 "# 9. If X has a numeric PDF, then calculate the PDF of the range by" }}{PARA 0 "" 0 "" {TEXT 23 37 "# \+ formula from Paul Stockmeyer." }}{PARA 0 "" 0 "" {TEXT 23 72 "# \+ 10. If X has a formulaic PDF, then calculate the PDF of the range .. " }}{PARA 0 "" 0 "" {TEXT 23 62 "# by obtaining a general expres sion for the PDF with the" }}{PARA 0 "" 0 "" {TEXT 23 47 "# same formula used in the numeric case. " }}{PARA 0 "" 0 "" {TEXT 23 67 "# \+ 11. If 3 arguments are provided, check if the third argument is" }} {PARA 0 "" 0 "" {TEXT 23 69 "# the string \"wo.\" If it is, then items are drawn from the RV's" }}{PARA 0 "" 0 "" {TEXT 23 39 "# \+ population without replacement " }}{PARA 0 "" 0 "" {TEXT 23 69 "# 1 2. Determine if the random variable X has a formulaic PDF. If it" }} {PARA 0 "" 0 "" {TEXT 23 65 "# does and also has finite support, then call the procedure" }}{PARA 0 "" 0 "" {TEXT 23 71 "# Conve rtToNumeric to convert its PDF and support to the standard" }}{PARA 0 "" 0 "" {TEXT 23 70 "# APPL numeric list format. The conversion \+ is necessary in order" }}{PARA 0 "" 0 "" {TEXT 23 66 "# to have \+ list inputs for the procedures NextPermutation and" }}{PARA 0 "" 0 "" {TEXT 23 24 "# NextCombination." }}{PARA 0 "" 0 "" {TEXT 23 71 " # 15. If X has finite support and if n = N, then the PDF of the rang e" }}{PARA 0 "" 0 "" {TEXT 23 40 "# is 1 for x = n, and 0 otherw ise." }}{PARA 0 "" 0 "" {TEXT 23 70 "# 16. If n = 2, 3, ... N - 1, c reate an array of size N - n to store" }}{PARA 0 "" 0 "" {TEXT 23 71 " # the range values x from 1 to N - n + 1. Initialize the array t o" }}{PARA 0 "" 0 "" {TEXT 23 26 "# contain all zeros." }}{PARA 0 "" 0 "" {TEXT 23 65 "# 17. Create the first lexicographical combin ation of n items. " }}{PARA 0 "" 0 "" {TEXT 23 69 "# 18. Assign perm as the current combination. For each combination," }}{PARA 0 "" 0 "" {TEXT 23 62 "# the algorithm finds every possible permutation of that" }}{PARA 0 "" 0 "" {TEXT 23 20 "# combination." }}{PARA 0 "" 0 "" {TEXT 23 71 "# 19. For each permutation, compute the probabi lity of obtaining that" }}{PARA 0 "" 0 "" {TEXT 23 26 "# given p ermutation." }}{PARA 0 "" 0 "" {TEXT 23 68 "# 20. Find the maximum a nd minimum elements of the permutation and" }}{PARA 0 "" 0 "" {TEXT 23 61 "# then determine their diference, the range. Store the " }}{PARA 0 "" 0 "" {TEXT 23 68 "# permutation's probability in th e appropriate position in the" }}{PARA 0 "" 0 "" {TEXT 23 14 "# \+ array." }}{PARA 0 "" 0 "" {TEXT 23 71 "# 21. Find the next lexicogra phical combination, and repeat steps 18 " }}{PARA 0 "" 0 "" {TEXT 23 13 "# - 20." }}{PARA 0 "" 0 "" {TEXT 23 71 "# 22. If X has inf inite support, then at present, RangeStat is unable" }}{PARA 0 "" 0 " " {TEXT 23 69 "# to compute the range for infinite support witho ut replacement" }}{PARA 0 "" 0 "" {TEXT 23 53 "# cases. This is \+ an open area for future work. " }}{PARA 0 "" 0 "" {TEXT 23 64 "# 23. If 3 arguments are provided and the third one is not the" }}{PARA 0 " " 0 "" {TEXT 23 49 "# string \"wo\", then return an error messag e" }}{PARA 0 "" 0 "" {TEXT 23 66 "# 24. Return the PDF of the range \+ in the list of sublists format" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }} {PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "Top" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "RebuildHeap()" {TEXT 23 2 " " }}{PARA 256 "" 0 "" {TEXT 23 13 "RebuildHeap()" }}{PARA 0 "" 0 "" {TEXT 23 72 "--------------------------------------------------------------------- ---" }}{PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "Top" }}{PARA 256 " " 0 "" {TEXT 23 0 "" }}{PARA 256 "" 0 "" {TEXT 23 0 "" }}{PARA 256 "" 0 "Procedure Name: RebuildHeapQ1()" {TEXT 23 2 " " }}{PARA 256 "" 0 " " {TEXT 23 15 "RebuildHeapQ1()" }}{PARA 0 "" 0 "" {TEXT 23 72 "------- -----------------------------------------------------------------" }} {PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "Top" }}{PARA 256 "" 0 "" {TEXT 23 0 "" }}{PARA 256 "" 0 "" {TEXT 23 0 "" }}{PARA 256 "" 0 "Proc edure Name: RebuildHeapQ2()" {TEXT 23 2 " " }}{PARA 256 "" 0 "Rebuild HeapQ2()" {TEXT 23 15 "RebuildHeapQ2()" }}{PARA 0 "" 0 "" {TEXT 23 72 "--------------------------------------------------------------------- ---" }}{PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "Top" }}{PARA 256 " " 0 "" {TEXT 23 0 "" }}{PARA 256 "" 0 "" {TEXT 23 0 "" }}{PARA 256 "" 0 "" {TEXT 23 2 " " }}{PARA 256 "" 0 "RebuildHeapQ3()" {TEXT 23 15 "R ebuildHeapQ3()" }}{PARA 0 "" 0 "" {TEXT 23 72 "----------------------- -------------------------------------------------" }}{PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "Top" }}{PARA 256 "" 0 "" {TEXT 23 0 "" }} {PARA 256 "" 0 "" {TEXT 23 0 "" }}{PARA 256 "" 0 "" {TEXT 23 2 " " }} {PARA 256 "" 0 "RebuildHeapQ4()" {TEXT 23 15 "RebuildHeapQ4()" }} {PARA 0 "" 0 "" {TEXT 23 72 "----------------------------------------- -------------------------------" }}{PARA 0 "" 0 "" {HYPERLNK 17 "To To p" 1 "" "Top" }}{PARA 256 "" 0 "" {TEXT 23 0 "" }}{PARA 256 "" 0 "" {TEXT 23 0 "" }}{PARA 256 "" 0 "" {TEXT 23 0 "" }}{PARA 256 "" 0 "SF(X , [x])" {TEXT 23 10 "SF(X, [x])" }}{PARA 0 "" 0 "" {TEXT 23 72 "------ ------------------------------------------------------------------" }} {PARA 256 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 59 "# Othe r APPL Procedures Called: CDF, CHF, Convert, PDF, HF" }}{PARA 0 "" 0 " " {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 36 "# Purpose: SF is a pr ocedure that: " }}{PARA 0 "" 0 "" {TEXT 23 71 "# (1) Returns the survivor function of a random variable X in" }}{PARA 0 "" 0 "" {TEXT 23 69 "# the APPL list of 3 lists format if the on ly argument " }}{PARA 0 "" 0 "" {TEXT 23 30 "# given is \+ X, or" }}{PARA 0 "" 0 "" {TEXT 23 71 "# (2) Returns the valu e Pr(X > x) if it is given the optional" }}{PARA 0 "" 0 "" {TEXT 23 50 "# argument x in addition to the RV X" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 59 "# Arguments: X: A \+ continuous or discrete random variable; " }}{PARA 0 "" 0 "" {TEXT 23 65 "# x (optional argument): A numeric value entered when " }}{PARA 0 "" 0 "" {TEXT 23 46 "# trying to determine \+ Pr(X > x)" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 65 "# Algorithm: 1. Perform error checking on user entered argumen ts" }}{PARA 0 "" 0 "" {TEXT 23 64 "# 2. The SF of X is det ermined by sending it to its " }}{PARA 0 "" 0 "" {TEXT 23 65 "# \+ appropriate category, which is based on whether:" }}{PARA 0 " " 0 "" {TEXT 23 47 "# A. X is continuous or discrete" } }{PARA 0 "" 0 "" {TEXT 23 69 "# B. X is entered in its \+ PDF, CDF, SF, HF, CHF, or IDF" }}{PARA 0 "" 0 "" {TEXT 23 27 "# \+ format " }}{PARA 0 "" 0 "" {TEXT 23 70 "# 3. I f only 1 argument (the random variable X) is entered" }}{PARA 0 "" 0 " " {TEXT 23 66 "# by the user, return the survivor funct ion of X in" }}{PARA 0 "" 0 "" {TEXT 23 49 "# the APPL \+ list of 3 lists format " }}{PARA 0 "" 0 "" {TEXT 23 72 "# \+ 4. If 2 arguments are provided, return the value Pr(X > x)" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 71 "# CAUTION: Made \"x\" local because assumption made on x in the discrete" }}{PARA 0 " " 0 "" {TEXT 23 14 "# dot HF case" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" } }{PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "Top" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "Skewness (X)" {TEXT 336 11 "Skewness(X)" }}{PARA 0 "" 0 "" {TEXT 23 72 "------- -----------------------------------------------------------------" }} {PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 32 "# Purpos e: Returns the skewness" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 " " 0 "" {TEXT 23 32 "# Arguments: X: Random Variable" }}{PARA 0 "" 0 " " {TEXT 23 2 "# " }}{PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "Top" } }{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "Stealth" {TEXT 23 2 " \+ " }}{PARA 0 "" 0 "" {TEXT 337 7 "Stealth" }}{PARA 0 "" 0 "" {TEXT 23 72 "------------------------------------------------------------------ ------" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 78 "# Other APPL Procedures Called: ExpectedValue, NormalRV, UniformRV, \+ Variance " }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 60 "# Purpose: Stealth for E(X, g), N(mu, sigma), U(a, b), V(X)" } }{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {HYPERLNK 17 "To Top " 1 "" "Top" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "TransformDiscrete(X, gX)" {TEXT 338 24 "Transf ormDiscrete(X, gX)" }}{PARA 0 "" 0 "" {TEXT 23 72 "------------------- -----------------------------------------------------" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 70 "# Purpose: TransformD iscrete finds the PDF of Y = g(X), where X and Y" }}{PARA 0 "" 0 "" {TEXT 23 67 "# are discrete random variables. The arguments are X, a " }}{PARA 0 "" 0 "" {TEXT 23 67 "# discrete random variable, and g(X), the transformation." }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 19 "# Arguments: X, gX" }}{PARA 0 " " 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "To p" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "Truncate(X, a, b)" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 339 17 "Truncate(X, a, b)" }} {PARA 0 "" 0 "" {TEXT 23 72 "----------------------------------------- -------------------------------" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }} {PARA 0 "" 0 "" {TEXT 23 41 "# Other APPL Procedures Called: PDF, CDF " }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 72 "# P urpose: Truncates a random variable X on the left by a and on the " }}{PARA 0 "" 0 "" {TEXT 23 24 "# right by b. " }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 33 "# Arguments: X: Ra ndom Variable " }}{PARA 0 "" 0 "" {TEXT 23 37 "# a, b: Tru ncation Points" }}{PARA 0 "" 0 "" {TEXT 23 2 "# " }}{PARA 0 "" 0 "" {TEXT 23 13 "# Algorithm:" }}{PARA 0 "" 0 "" {TEXT 23 54 "# 1. Ch eck for the appropriate number of arguments" }}{PARA 0 "" 0 "" {TEXT 23 61 "# 2. Check that the RV X is in the list of 3 lists format" }}{PARA 0 "" 0 "" {TEXT 23 58 "# 3. Convert X to PDF form if not a lready in that form" }}{PARA 0 "" 0 "" {TEXT 23 50 "# 4. Check tha t the truncation pts are numeric" }}{PARA 0 "" 0 "" {TEXT 23 47 "# \+ 5. Check that the RV supports are numeric" }}{PARA 0 "" 0 "" {TEXT 23 80 "# 6. Check that lower truncation pt 'a' is less than upper \+ truncation pt 'b'" }}{PARA 0 "" 0 "" {TEXT 23 78 "# 7. Check that \+ lower truncation pt 'a' is less than upper support of RV X" }}{PARA 0 "" 0 "" {TEXT 23 81 "# 8. If 'a' is less than the min value of sup port, reset lower truncation pt " }}{PARA 0 "" 0 "" {TEXT 23 41 "# \+ 'a' to minimum value of support " }}{PARA 0 "" 0 "" {TEXT 23 80 "# 9. If 'b' is greater than the max value of support, reset upper t runcation" }}{PARA 0 "" 0 "" {TEXT 23 44 "# pt 'b' to maximum v alue of support " }}{PARA 0 "" 0 "" {TEXT 23 78 "# 10. Find out whi ch segments of the PDF contain the truncation pts a and b" }}{PARA 0 " " 0 "" {TEXT 23 32 "# 11. Compute CDF(b) - CDF(a)" }}{PARA 0 "" 0 " " {TEXT 23 69 "# 12. Check for CDF(b) - CDF(a)= 0: if so, treat as \+ a special case" }}{PARA 0 "" 0 "" {TEXT 23 59 "# 13. Compute and re turn the truncated RV's support list" }}{PARA 0 "" 0 "" {TEXT 23 75 "# 14. Compute and return the truncated PDF in the list of 3 lists fo rmat" }}{PARA 0 "" 0 "" {TEXT 23 2 "# " }}{PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "Top" }}{PARA 0 "" 0 "" {TEXT 422 76 " \+ " }} {PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "UniformVariate()" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 340 16 "UniformVariate()" }}{PARA 0 " " 0 "" {TEXT 23 72 "-------------------------------------------------- ----------------------" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 64 "# Purpose: UniformVariate generates a standard unif orm variate " }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 18 "# Arguments: None" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }} {PARA 0 "" 0 "" {TEXT 23 82 "# Note: With no arguments, the call rand () returns a 12 digit non-negative random" }}{PARA 0 "" 0 "" {TEXT 23 18 "# integer. " }}{PARA 0 "" 0 "" {TEXT 23 2 "# " }}{PARA 0 " " 0 "" {HYPERLNK 17 "To Top" 1 "" "Top" }}{PARA 0 "" 0 "" {TEXT 23 0 " " }}{PARA 0 "" 0 "Valid(X)" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 342 8 "Valid(X)" }{TEXT 341 1 " " }}{PARA 0 "" 0 "" {TEXT 23 72 "--------- ---------------------------------------------------------------" }} {PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 78 "# Purpos e: This procedure is called by other APPL procedures to determine the " }}{PARA 0 "" 0 "" {TEXT 23 75 "# \"validity\" of a given r andom variable and its APPL format. More" }}{PARA 0 "" 0 "" {TEXT 23 45 "# specifically, it checks that the:" }}{PARA 0 "" 0 "" {TEXT 23 56 "# (a) appropriate number of arguments is given " }}{PARA 0 "" 0 "" {TEXT 23 53 "# (b) RV X is in the list o f 3 lists format" }}{PARA 0 "" 0 "" {TEXT 23 83 "# (c) pdf o f X sums to 1 and pdf values are nonnegative for discrete RV's" }} {PARA 0 "" 0 "" {TEXT 23 66 "# (d) discrete support values f or the pdf of X are valid" }}{PARA 0 "" 0 "" {TEXT 23 64 "# \+ (e) area under the pdf of X is 1 for continuous RV's" }}{PARA 0 "" 0 " " {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 76 "# Argument: The rando m variable of interest X in the list-of-lists format: " }}{PARA 0 "" 0 "" {TEXT 23 70 "# [[f(x)], [support], [\"Continuous\" or \+ \"Discrete\", \"XXX\"]]," }}{PARA 0 "" 0 "" {TEXT 23 56 "# \+ where XXX is CDF, CHF, HF, IDF, PDF, or SF " }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 12 "# Algorithm" }}{PARA 0 "" 0 " " {TEXT 23 61 "# 1. Check that the RV X is in the list of 3 lists \+ format" }}{PARA 0 "" 0 "" {TEXT 23 59 "# 2. Convert X to PDF form \+ if not already in that form " }}{PARA 0 "" 0 "" {TEXT 23 64 "# 3. \+ Check to see whether the RV X is continuous or discrete" }}{PARA 0 "" 0 "" {TEXT 23 81 "# 4. If the RV X is continuous, calculate the in tegral(f(x)dx) from 0 to +inf" }}{PARA 0 "" 0 "" {TEXT 23 76 "# \+ and the integral(abs(f(x))dx) from 0 to +inf. If int(f(x)) = 1 and " }}{PARA 0 "" 0 "" {TEXT 23 81 "# int(abs(f(x))) = 1, then f(x) \+ >= 0. Otherwise, f(x) is negative for some" }}{PARA 0 "" 0 "" {TEXT 23 54 "# value x in its support and is NOT a valid pdf" }} {PARA 0 "" 0 "" {TEXT 23 78 "# 5. If the RV X is discrete, determi ne if it is in the \"dotdot\" case. In" }}{PARA 0 "" 0 "" {TEXT 23 76 "# probproc.map, dotdot is a defined structure type for disc rete RV's:" }}{PARA 0 "" 0 "" {TEXT 23 81 "# `type/dotdot` := \+ \{constant .. constant, symbol .. constant, constant .. " }}{PARA 0 " " 0 "" {TEXT 23 41 "# symbol, symbol .. symbol\}: " }} {PARA 0 "" 0 "" {TEXT 23 82 "# 6. If X is in the \"dotdot\" case, \+ then first convert the support of X to the " }}{PARA 0 "" 0 "" {TEXT 23 33 "# standard \"dotdot\" form: " }}{PARA 0 "" 0 "" {TEXT 23 70 "# [anything .. anything, incremented by k, transformed b y g(x)]" }}{PARA 0 "" 0 "" {TEXT 23 39 "# 7. Transform the pdf f(x ) by g(x)" }}{PARA 0 "" 0 "" {TEXT 23 68 "# 8. Check that transfor med pdf is >= 0 for all x in its support" }}{PARA 0 "" 0 "" {TEXT 23 66 "# 9. Check that the transformed pdf sums to 1 over its support " }}{PARA 0 "" 0 "" {TEXT 23 74 "# 10. If X is in the \"no dotdot\" case, convert the support of X to the " }}{PARA 0 "" 0 "" {TEXT 23 76 "# standard \"no dotdot\" form: [[fraction1, fraction2, frac tion3, ... ," }}{PARA 0 "" 0 "" {TEXT 23 65 "# fractionN], [a1, a2, a3, ... , aN], [\"Discrete\", \"PDF\"]]" }}{PARA 0 "" 0 "" {TEXT 23 54 "# 11. Check that f(x) >= 0 for all x in its support" }} {PARA 0 "" 0 "" {TEXT 23 43 "# 12. Sum f(x) over all x in its suppo rt" }}{PARA 0 "" 0 "" {TEXT 23 121 "# 13. If X is continuous and th e area under its pdf is is 1, then report to the # user that X \+ is a valid RV. #" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "Top" }}{PARA 0 "" 0 "Variance(X)" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 343 11 "Variance(X)" }}{PARA 0 "" 0 "" {TEXT 23 72 "---------------------- --------------------------------------------------" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 57 "# Other APPL Procedures \+ Called: PDF, Mean, ExpectedValue" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }} {PARA 0 "" 0 "" {TEXT 23 52 "# Purpose: Returns the variance of a dis tribution. " }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 32 "# Arguments: X: Random Variable" }}{PARA 0 "" 0 "" {TEXT 23 2 "# " }}{PARA 0 "" 0 "" {TEXT 23 13 "# Algorithm:" }}{PARA 0 "" 0 "" {TEXT 23 54 "# 1. Check for the appropriate number of arguments" } }{PARA 0 "" 0 "" {TEXT 23 57 "# 2. Check that the RV X is in a lis t-of-lists format" }}{PARA 0 "" 0 "" {TEXT 23 58 "# 3. Convert X t o PDF form if not already in that form" }}{PARA 0 "" 0 "" {TEXT 23 52 "# 4. Compute and return the Variance of the RV X" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 68 "# NOTE: Has NOT b een updated since a discrete data structure has" }}{PARA 0 "" 0 "" {TEXT 23 72 "# been developed ... will need to redo the disc rete portion of" }}{PARA 0 "" 0 "" {TEXT 23 20 "# the code" }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {HYPERLNK 17 "To To p" 1 "" "Top" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "VerifyPDF(X)" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 344 13 "VerifyPDF(X) " }}{PARA 0 "" 0 "" {TEXT 23 72 "---------- --------------------------------------------------------------" }} {PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 45 "# Other \+ APPL Procedures Called: PDF, Convert" }}{PARA 0 "" 0 "" {TEXT 23 1 "# " }}{PARA 0 "" 0 "" {TEXT 23 61 "# Purpose: For a continuous random \+ variable X, verify that:" }}{PARA 0 "" 0 "" {TEXT 23 58 "# \+ (1) integral(f(x)dx) from 0 to +inf = 1, and" }}{PARA 0 "" 0 "" {TEXT 23 45 "# (2) f(x) >= 0 for all x >= 0. " }}{PARA 0 "" 0 "" {TEXT 23 67 "# * New Conjecture: If int(f(x)) = 1 and int(abs (f(x))) = 1, then " }}{PARA 0 "" 0 "" {TEXT 23 14 "# f(x) >= 0" }} {PARA 0 "" 0 "" {TEXT 23 14 "# " }}{PARA 0 "" 0 "" {TEXT 23 59 "# For a discrete random variable X, verify that:" }} {PARA 0 "" 0 "" {TEXT 23 54 "# (1) f(x) sums to 1 over its support, and" }}{PARA 0 "" 0 "" {TEXT 23 45 "# (2) f(x) > = 0 for all x >= 0. " }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 70 "# Argument: The random variable of interest X in th e list-of-sublists" }}{PARA 0 "" 0 "" {TEXT 23 21 "# format : " }}{PARA 0 "" 0 "" {TEXT 23 70 "# [[f(x)], [support], [ \"Continuous\" or \"Discrete\", \"XXX\"]]," }}{PARA 0 "" 0 "" {TEXT 23 56 "# where XXX is PDF, CDF, IDF, SF, HF, or CHF " }} {PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 12 "# Algor ithm" }}{PARA 0 "" 0 "" {TEXT 23 54 "# 1. Check for the appropriat e number of arguments" }}{PARA 0 "" 0 "" {TEXT 23 64 "# 2. Check t hat the RV X is in the list of 3 sublists format" }}{PARA 0 "" 0 "" {TEXT 23 59 "# 3. Convert X to PDF form if not already in that for m " }}{PARA 0 "" 0 "" {TEXT 23 64 "# 4. Check to see whether the R V X is continuous or discrete" }}{PARA 0 "" 0 "" {TEXT 23 71 "# 5. \+ If the RV X is continuous, calculate the integral(f(x)dx) from" }} {PARA 0 "" 0 "" {TEXT 23 68 "# 0 to +inf and the integral(abs(f (x))dx) from 0 to +inf. If " }}{PARA 0 "" 0 "" {TEXT 23 62 "# i nt(f(x)) = 1 and int(abs(f(x))) = 1, then f(x) >= 0." }}{PARA 0 "" 0 " " {TEXT 23 68 "# Otherwise, f(x) is negative for some value x i n its support" }}{PARA 0 "" 0 "" {TEXT 23 31 "# and is NOT a va lid pdf" }}{PARA 0 "" 0 "" {TEXT 23 68 "# 6. If the RV X is discre te, determine if it is in the \"dot\" or" }}{PARA 0 "" 0 "" {TEXT 23 67 "# \"no dot\" case. In startup.map, dot is a defined struct ure" }}{PARA 0 "" 0 "" {TEXT 23 44 "# type for discrete random \+ variables:" }}{PARA 0 "" 0 "" {TEXT 23 65 "# `type/dot` := \{co nstant .. constant, symbol .. constant," }}{PARA 0 "" 0 "" {TEXT 23 67 "# constant .. symbol, symbol .. symbol\}: \+ " }}{PARA 0 "" 0 "" {TEXT 23 68 "# 7. Convert the support of X t o its standard \"dot\" or \"no dot\" " }}{PARA 0 "" 0 "" {TEXT 23 14 " # form:" }}{PARA 0 "" 0 "" {TEXT 23 34 "# (a) standard \+ \"dot\" form: " }}{PARA 0 "" 0 "" {TEXT 23 70 "# [anything .. a nything, incremented by k, transformed by g(x)]" }}{PARA 0 "" 0 "" {TEXT 23 36 "# (b) standarc \"no dot\" form:" }}{PARA 0 "" 0 " " {TEXT 23 62 "# [[fraction1, fraction2, fraction3, ... , fract ionN], " }}{PARA 0 "" 0 "" {TEXT 23 54 "# [a1, a2, a3, ... , a N], [\"Discrete\", \"PDF\"]]" }}{PARA 0 "" 0 "" {TEXT 23 68 "# 8. \+ If X has the \"dot\" form, transform the pdf f(x) by g(x) if " }} {PARA 0 "" 0 "" {TEXT 23 44 "# f(x)'s support is NOT of the for m: " }}{PARA 0 "" 0 "" {TEXT 23 28 "# [a .. b, 1, x -> x]" }} {PARA 0 "" 0 "" {TEXT 23 68 "# 9. Check that transformed pdf is >= 0 for all x in its support" }}{PARA 0 "" 0 "" {TEXT 23 66 "# 10. C heck that the transformed pdf sums to 1 over its support" }}{PARA 0 " " 0 "" {TEXT 23 69 "# 11. If X is in the \"no dot\" case, check tha t f(x) >= 0 for all x" }}{PARA 0 "" 0 "" {TEXT 23 23 "# in its \+ support" }}{PARA 0 "" 0 "" {TEXT 23 43 "# 12. Sum f(x) over all x i n its support" }}{PARA 0 "" 0 "" {TEXT 23 68 "# 13. If X is continu ous and the area under its PDF is is 1, then" }}{PARA 0 "" 0 "" {TEXT 23 69 "# report to the user that X is a valid RV. Similarly, i f X is" }}{PARA 0 "" 0 "" {TEXT 23 66 "# discrete and its PDF i s nonnegative for all values in its" }}{PARA 0 "" 0 "" {TEXT 23 68 "# \+ support and sums to 1 over its support, then also report to" }} {PARA 0 "" 0 "" {TEXT 23 39 "# the user that X is a valid RV." }}{PARA 0 "" 0 "" {TEXT 23 5 "# " }}{PARA 0 "" 0 "" {HYPERLNK 17 "T o Top" 1 "" "Top" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "Weib ullVariate(lambda, kappa, m)" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 345 32 "WeibullVariate(lambda, kappa, m)" }}{PARA 0 "" 0 "" {TEXT 23 72 "------------------------------------------------------------------ ------" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 47 "# Other APPL Procedures Called: UniformVariate" }}{PARA 0 "" 0 "" {TEXT 23 14 "# " }}{PARA 0 "" 0 "" {TEXT 23 79 "# Purpose : Given the parameters lambda and kappa for the Weibull distribution, " }}{PARA 0 "" 0 "" {TEXT 23 79 "# WeibullVariate returns n \+ Weibull variates and the mean and variance" }}{PARA 0 "" 0 "" {TEXT 23 38 "# values of these variates " }}{PARA 0 "" 0 "" {TEXT 23 1 "#" }}{PARA 0 "" 0 "" {TEXT 23 50 "# Arguments: lambda: no nnegative shape parameter," }}{PARA 0 "" 0 "" {TEXT 23 53 "# \+ kappa: nonnegative scale parameter, and" }}{PARA 0 "" 0 "" {TEXT 23 78 "# m: integer indicating the number of Weibull varia tes to generate" }}{PARA 0 "" 0 "" {TEXT 23 2 "# " }}{PARA 0 "" 0 "" {HYPERLNK 17 "To Top" 1 "" "Top" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "233 \+ 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }