### Probability Theory

#### Semester 2, 2005/2006

Lecturer
Prof. Boris Tsirelson, School of Mathematical Sciences.
Time and place
Sunday 11-12 Shenkar 104;
Wednesday 13-15 Melamed auditorium (006).
Instructor
Shiri Alon.
Prerequisites
(Infinitesimal) calculus 2; introduction to probability theory.
The final exam.

#### TOPICS

1. General probability spaces.
2. Random variables. Cumulative distribution function, density (if exists), quantiles. Examples, discrete and continuous (uniform, exponential, normal).
3. One-dimensional transformations of distributions.
4. Expectation of a random variable (and its function). Variance, moments, moment generating function. Special cases.
5. Joint distributions and independence, marginal distributions, joint density.
6. Infinite sequences of random variables. Borel-Cantelli lemma. Modes of convergence. Convergence of expectations. Strong law of large numbers, central limit theorem.
7. Joint distributions: conditioning, correlation, and transformations. Conditional distributions, conditional expectation. Total probability formula (continuous case). Regression and correlation. Examples. Multidimensional transformations of distributions. Distribution and density of sum, product and quotient of one-dimensional random variables.

#### Exercises

Problems and solutions (courtesy of the instructor)

#### Some exams

##### Books, in English
• [BT] BERTSEKAS D. (Dimitri), TSITSIKLIS J., "Introduction to probability" (2002).
• FELLER W. (William), "An introduction to probability theory and its applications" Vols. 1,2 (1950/57/68).
• HOEL P. (Paul), PORT S., STONE C., "Introduction to probability theory" (1971).
• ROSS S.M. (Sheldon), "A first course in probability" (1994/98).
##### In Hebrew
• DVORETZKI A. (Arie), "Mavo leTorat a-Histabrut".
##### In English
• ASH R.B. (Robert), "Basic probability theory" (1970).
• BLUM J.R. (Julius), Rosenblatt J.I., "Probability and statistics" (1972).
• CAPINSKI M. (Marek), Zastawniak T., "Probability through problems" (2001).
• FELDMAN D. (Dorian), FOX M., "Probability: the mathematics of uncertainty" (1991).
• FRISTEDT B. (Bert), GRAY L., "A modern approach to probability theory" (1996,1997).
• MELSA J.L. (James), SAGE A.P., "An introduction to probability and stochastic processes" (1973).
• MEYER P.L. (Paul), "Introductory probability and statistical applications" (1965,70).
• NEUTS M.F. (Marcel), "Probability" (1973).
• PITMAN J. (Jim), "Probability" (1993).
• PORT S.C. (Sidney), "Theoretical probability for applications" (1994).
• SNELL J.L. (Laurie), "Introduction to probability" (1988).
• USPENSKY J. (James), "Introduction to mathematical probability" (1937).
##### In Hebrew
• MERZBACH E. (Ely), SHIMRON A. "Torat a-Histabrut" (1993).