## MATH 362K Syllabus

PROBABILITY I

**Prerequisite and degree relevance:** M408D with a grade of at least C-. A student may not receive credit for M316 after completing M362K with a grade of C or better.

**Course description:** This is an introductory course in the mathematical theory of probability, thus it is fundamental to further work in probability and statistics. Principles of set theory and a set of axioms for probability are used to derive some probability density and/or distribution functions. Special counting techniques are developed to handle some problems. Properties associated with a random variable are developed for the usual elementary distributions. Problem solving is required, and some theorem proving can be done, but the course emphasizes computation and intuition.

**Suggested Textbook:** A First Course in Probability, eighth edition, by Sheldon Ross.

The following course outline refers to section numbers in Ross' book, and assumes a MWF lecture format (it must be modified for TTh classes)

**Some Alternate Textbooks: **

- Charles M. Grinstead and J. Laurie Snell, Introduction to Probability, 2nd revised ed., AMS 1977. This book has an interesting style that is different from the more standard format of Ross. It introduces some important ideas in examples and exercises, so the instructor needs to know what not to omit. There is too much emphasis on computation for this course, but otherwise itis very well written, with many good examples and exercises.
- Saeed Ghahramani, Fundamentals of Probability. Prentice Hall, 1996. Similar to Ross' text.

**Background:** M362K is required of all undergraduate mathematics majors, and it is a prerequisite for courses in statistics. However, many of the students are majoring in other subjects(e.g., computer science or economics), and have little preparation in abstract mathematics. Calculus skills (integration and infinite series) tend to be weak, even at this level. Similarly, you cannot expect students to have any background in proofs, and should not expect competence in this. The course tends to be relatively easier for the first three to four weeks, so some students get the wrong impression as to its difficulty. Clarifying this early for the students can avoid unpleasant surprises later.

**Course Content:** Emphasize problem solving and intuition. Some advanced concepts should be presented without proof, so as to devote more attention to the examples. Basic combinatorics: Counting principle, permutations, combinations. Basic concepts: Sample spaces, events, basic axioms and theorems of probability, finite sample spaces with equally likely probabilities. Conditional probability: Reduced sample space, independence, Bayes' Theorem. Random variables: Discrete and continuous random variables, discrete probability functions and continuous probability density functions, distribution functions, expectation, variance, functions of random variables. Special distributions: Bernoulli, Binomial, Poisson, and Geometric discrete random variables. Uniform, Normal, and Exponential continuous random variables. Approximation of Binomial by Poisson or Normal. Jointly distributed random variables: Joint distribution functions, independence, conditional distributions, expectation, covariance Sums of independent random variables: expectation, variance. Inequalities and Limit theorems: Markov's and Chebyshev's inequalities, Weak and Strong Law of Large Numbers, Central Limit Theorem.

- 1.1-1.4: 3 lectures, Limit this material to one week.
- 2.1-2.5; 2.7: 4 lectures, Do not get bogged down in 2.5; limit it to about one lecture.
- 3.1-3.4: 4 lectures, Students like tree diagrams for Bayes' Theorem, and need more help and examples to learn how to extract information from word problems.
- 4.1-4.5: 4 lectures.
- 4.6-4.7; 4.8.1: 3 lectures, Omit 4.6.2 and 4.7.1. One could delay 4.7 to 5.4.1. Sections 4.8.2 and 4.8.3 are optional.
- 5.1-5.5; 5.7: 7 lectures. Omit 5.5.1; Section 5.6.1 is optional.
- 6.1-6.5: 4 lectures.
- 7.1-7.2; 7.4: 2 lectures, Omit 7.2.1, 7.2.2. Sections 7.5, 7.7, 7.8 are optional, as is correlation.
- 8.1-8.4: 3 lectures, Do not let any optional material crowd out the limit theorems. Emphasize intuitive understanding of the Central Limit Theorem by examples, and omit the proof, especially if optional 7.7 is not covered. One or two topics are optional.

There are a wealth of examples in the text, so the instructor has time to present only some of them. The outline above allows room for 34 lectures, 3 in class exam days, and 3 review days, for a total of 40 days. A typical semester has 42 MWF class days in the fall and 44 in the spring, so a few days for make-up or optional material are provided. It is likely that an instructor will find no time for any of the optional material.

T. Arbogast, J. Luecke, and M. Smith, August 2008