## Department of Mathematics - RHIT Topics Outline for MA381 - Introduction to Probability with Applications to Statistics - 2010-11

last posted to site: 08/19/10

### Text and Text Materials

Text: Fundamentals of Probability with Stochastic Processes , by S. Ghahramani

### Text Topics - Optional topics in italics

Ch. 1   Axioms of Probability

Required: Ch. 1.1 - 1.4: Sample Space, Events, Axioms of Probability, Basic Theorems

Optional: Ch. 1.5 - 1.7

Ch. 2   Combinatorial Methods

Required:  Basic Counting Principles, combinations, permutations (no more than 2 days)

Ch. 3   Conditional Probability and Independence

Required:  Ch. 3.1 - 3.5: Conditional Probability, laws of multiplication and total probability, Bayes' Formula, Independence

Optional:  Ch. 3.6   Applications of probability to genetics

Ch. 4   Distribution Functions and Discrete Random Variables

Required:  Ch. 4.1 - 4.6  Random Variables, distribution functions, discrete random variables, expectation of discrete random variables, variances and moments of discrete random variables, standardized random variables.

Ch. 5   Special Discrete Distributions

Required:  Ch. 5.1 - first part of 5.3   Bernoulli and Binomial, Poisson (including Poisson Process), Geometric

Optional:  Negative Binomial, hypergeometric

Ch. 6    Continuous Random Variables

Required: Ch. 6.1 - 6.3   Probability density functions, density function of a function of a random variable, expectation and variances

Ch. 7   Special Continuous Distributions

Required: Ch. 7.1 - 7.3  Uniform, Normal , Exponential

Ch. 8   Bivariate Distributions

Required: Ch. 8.1 - 8.3   Joint distributions, independent random variables, conditional distributions

Optional: Ch. 8.4  Transformations of 2 random variables

Ch. 10  More expectations and variances

Required:  Ch. 10.1 - 10.4   expected values of sums of random variables, covariance, correlation, conditioning on random variables

Ch. 11  Sums of independent random variables

Required:  Ch. 11.1 - 11.3, 11.5   Moment generating functions, sums of independent random variables, Markov and Chebyshev inequalities, Central Limit Theorem

Supplement:  Estimators and Confidence Intervals  (do as much as possible in no more than 3 days)

• Estimators of mean µ ,  variance s 2 and Bernoulli parameter p
• Unbiased Estimators?
• Confidence intervals for mean µ given normal population with known variance
• Use of confidence intervals to do hypothesis testing
• Sample size calculations
• (optional) Confidence interval for difference of two means from different normal populations