UCLA Department of Mathematics

Math 170B: General Course Outline

Catalog description

170B. Probability Theory. (Formerly numbered 150B.) Lecture, three hours; discussion, one hour. Prerequisite: course M170A or Statistics M152A/M100A. Convergence in distribution, normal approximations, laws of large numbers, Poisson processes, random walks.

Textbook

Sheldon Ross, A First Course in Probability, 5th Ed., Macmillan

Reviews and Exams

The following schedule, with textbook sections and topics, is based on 25 lectures. The remaining 3 classroom meetings are for leeway, review, and midterm exams. These are scheduled by the individual instructor.

Schedule of Lectures

Lecture Section Topics
1 (1.4, 7.2) Quick review of basics, including inclusion-exclusion formula and expectation of sum of random variables
2 7.3 Variance of sum of random variables
3 1.5, (2.5) Multinomial coefficient, example 5h on p. 42 (*)
4 1.6, (2.5) Balls in urns, runs of success (example 51, p. 47) (*)
5 (2.5) Matching (example 5j on p. 44) (*)
6 4.9.3, (7.2-3) Hypergeometric distribution (includes example 2g of Sec. 7.2 and example 3d of 7.3) (*)
7 4.9.2 Negative binomial distribution (includes example 2f of Sec. 7.2) (*); Banach match problem
8-9 (6.1), (7.3) Multinomial distribution; correlation coefficient
10-11 7.2, 7.3 More examples of uses of indicators
12-14 3.5 Multiple conditioning
15-17 6.4, 6.5, 7.4 Conditional distribution, expectation and variance
18-19 7.6 Moment generating function
20 7.7, (7.6) Multivariate normal and joint moment generating function
21-22 6.6 Order statistics
23 8.2 Chebyshev's inequality and the weak law of large numbers
24-25 8.3 Central limit theorem
If time permits two or three lectures on fluctuations in coin tossing, from Chapter 3 of Feller's book An Introduction to Probability Theory and its Applications, Vol. 1, (3rd edition), provide a nice ending for the course. The choice of topics from Chapters 5-8 in the syllabus is flexible.

(*) It is proposed that computations of expectations and variances using indicators be done in connection to new problems and models being introduced.

( ) Sections indicated withing parenthesis are relevant to the lecture but only as far as a small part of it is concerned.

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UCLA Department of Mathematics

6363 Math Sciences, Los Angeles, CA 90095-1555
Phone: (310) 206-1286 Fax: (310) 206-6673
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