## Rose-Hulman - Department of Mathematics - Course Syllabus MA381 - Introduction to Probability with Applications to Statistics - 2010-11

 Description & Prerequisites Course Goals Texts and Other Materials Course Topics Course Policies Links last posted to site: 05/30/12

Parts of the web page to be completed or determined by the instructor are in green.

### Catalogue Description and Prerequisites

MA 381 Introduction to Probability with Applications to Statistics 4R-0L-4C F,W,S Pre: MA 113
Introduction to probability theory; axioms of probability, sample spaces, and probability laws (including conditional probabilities). Univariate random variables (discrete and continuous) and their expectations including these distributions: binomial, Poisson, geometric, uniform, exponential, and normal. Introduction to moment generating functions. Introduction to jointly distributed random variables. Univariate and joint transformations of random variables. The distribution of linear combinations of random variables and an introduction to the Central Limit Theorem. Applications of probability to statistics.

Prerequisite: Calculus III - MA113

### Course Goals

• An understanding of the ideas of probability and probability modeling, including sample spaces, axioms, discrete and continuous random variables, univariate and joint distributions, moment generating functions, the central limit theorem, and basic statistical inference
• An understanding of the special language, notation, and point of view of probability
• An understanding of the concepts of probability necessary to undertake basic modeling and decision making in math, science, and engineering
• The ability to solve standard computational problems in probability, which includes using the computer as a tool for mathematical analysis and problem solving
• An understanding of the relationship between random variables and their distribution functions
• The ability to recognize special models, such as Bernoulli trials or Poisson processes
• An understanding of how probability is applied to inferential statistics
• The ability to communicate in mathematical terms

### Textbook and other required materials

Textbook: Fundamentals of Probability with Stochastic Processes, 3rd edition, by Saeed Ghahramani
Computer Usage:  Maple may be used during the course to help with various calculations.

### Course Topics

The course will cover the following chapters and topics from the textbook:

1. Sample Space, Events, Axioms of Probability
2. Combinatorial Methods Basic Counting Principles, combinations, permutations
3. Conditional Probability, laws of multiplication and total probability, Bayes' Formula, Independence
4. Random Variables, distribution functions, discrete random variables, expectation of discrete random variables, variances and moments of discrete random variables, standardized random variables.
5. Special Discrete Distributions: Bernoulli and Binomial, Poisson (including Poisson Process), Geometric
6. Continuous Random Variables: Probability density functions, density function of a function of a random variable, expectation and variances
7. Special Continuous Distributions: Uniform, Normal , Exponential
8. Bivariate Distributions: Joint distributions, independent random variables, conditional distributions Transformations of 2 random variables
9. Expected values of sums of random variables, covariance, correlation, conditioning on random variables
10. Moment generating functions, sums of independent random variables, Markov and Chebyshev inequalities, Central Limit Theorem
11. Estimators and Confidence Intervals  topics selected from
• Estimators of mean μ,  variance σ 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
• Confidence interval for difference of two means from different normal populations

### Course Requirements and Policies

The following policies and requirements will apply to all sections and classes:

#### Computer Usage

• Maple will be used to help with various calculations during the course.
• However, emphasis will be placed on the mathematical derivation and simplification of formulas and expressions without the use of technology.

#### Final Exam Policy

The final exam will cover all material from the course.  The use of Maple may be allowed, but not emphasized.  For full credit to be awarded for any given problem, all work must be shown.