# Probability

Isaac Meilijson
Department of Statistics and Operations Research
isaco@math.tau.ac.il
Office Hours: Wednesdays 15:10-16:00. 311 Schreiber.
+972-3-640-8826.

### Class Hours, Spring Semester 2005:

Sundays 11:10-12:00 at 111 Orenstein, Wednesdays 13:10-15:00 at 006 Melamed Hall

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### Course Content:

The goals of this course are:

To provide students of Mathematics, Statistics, Operations Research and Computer Science with a formal exposure to Probability, continuous and general probability models and random variables, as well as to some of the main limit theorems.

To prepare the Probability background for the proper study and understanding of Statistics, stochastic aspects of Operations Research, Stochastic Processes.

## Topics:

1. Probability space. Random variable, cumulative distribution function, density (when it exists), quantiles.
2. Review of common discrete distributions and introduction of continuous distributions such as Uniform, Exponential and Normal.
3. The distribution of a transformation of a univariate random variable.
4. Expected value of a random variable. Expected value of a function of a random variable. Variance, moments, moment generating function. Illustration on all the above distributions.
5. Joint distribution, conditional and marginal distributions, conditioning, conditional expectation. Correlation coefficient. The continuous version of the Law of Total Probability.
6. The distribution of a transformation of a multivariate random variable. The distribution of the sum, product and ratio of univariate random variables.
7. Types of convergence of sequences of random variables. Limit theorems, in particular the Central Limit Theorem.
8. The bivariate normal distribution.
9. Distributions related to the normal distribution (F, t, Chi-squared). Statistical motivation. Inter-relations between these distributions and the Exponential, Gamma and Poisson distributions.

### Helpful Reference Books

• Gnedenko, B. V. The Theory of Probability. Chelsea Publ. Co. 1962. (Classic. Good coverage, very clear).
• Ross, Sh. A First Course in Probability. (More modern, geared towards Engineering and Technology applications. Recommended text).
• Feller, W. An Introduction to Probability Theory and its Applications. Vol 1. Wiley 1968. (Excellent introduction to the subject, under discrete models only. Good reading, not as a text for this course).
• Hoel, P., Port, S., Stone C. Introduction to Probability Theory.