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.

Grading policy

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

Some exams

TEXTBOOKS

(in addition to lecture notes available on this site)

Available on the Web (only), in English, include some interactive pages

• SIEGRIST K., "Virtual Laboratories in Probability and Statistics"
  http://www.math.uah.edu/stat/

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".

ADDITIONAL LITERATURE

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).