Probability Theory – Exam 1 Review

General Coverage

The exam covers the material in the first four chapters of the book. (Sections 1.6, 2.6, 4.9, and Subsection 4.8.4 are not included in the exam material)

Studying and Self-tests

Be sure you understand all the homework problems and examples we did in class (if you don’t, please see me in office hours to talk about your questions). Furthermore, for the exam you must know the definitions (conceptual and formulaic), terminologies, and notations in your bones (that is, there won’t be time to leisurely recall definitions during the exam). The self-test problems (at the end of each chapter) are one good way to test your knowledge (and solutions are provided in the back of the book). Some self-tests include questions on material we haven’t covered or include questions that are more involved than what I will put on the exam. Here are self-test problems to avoid:

Chapter 1: 13, 14 (material not covered)

Chapter 3: 16, 20, 27, 28, 29 (long and a bit grungey—interesting to do, but won’t be like exam questions)

Chapter 4: 4, 17, 20, 24, 25 (long and a bit grungey—interesting to do, but won’t be like exam questions)

Format of Exam

The exam will be similar to the homework—a mix of problems and theoretical exercises. You need not memorize proofs of propositions/theorems given in the book, but you do need to know how to use the definitions and techniques we’ve discussed to show a particular result (that is, similar to the theoretical exercises). Even for the problems, you must explain your reasoning (it can be brief, but it must be complete and understandable).

Specific Coverage

· Understanding and use of the basic principle of counting, permutations, and combinations

· Understanding and use of the binomial theorem and multinomial coefficients

· Basic set theory (intersection, union, complement, DeMorgan’s laws)

· Axioms of probability and propositions (4.1 – 4.4 in Chapter 2) following from the axioms (this includes the inclusion-exclusion principle)

· Calculating probabilities on sample spaces with equally likely outcomes (using counting methods)

· Understanding and use of conditional probability and the general multiplication rule

· Law of Total Probability and Bayes’ Formula (note: there is no need to memorize the Bayes’ formula, per se, but you need to understand the idea—recall a tree diagram is not a complete solution for these types of problems; you must use the appropriate mathematical notation)

· Independent events (both showing independence via the definition and using independence—to find intersection probabilities—if it’s provided in a problem)

· is itself a probability and the definition of conditional independence

· General discrete random variables (determining probability mass function and definition of the cumulative distribution function)

· Expected value of a discrete random variable and expectation of a function of a random variable

· Variance of a discrete random variable and

· Expected value and variance of the variable , where a and b are constants

· Binomial random variable (setting, probability mass function, mean and derivation of the mean—recall the technique we used in class to determine the expectation; if needed, the variance will be provided for you)

· Poisson random variable (setting, probability mass function, mean and derivation of the mean—if needed, the variance will be provided for you)

· Poisson approximation to binomial probabilities (when n is large and p is small)

· Hypergeometric random variable (setting, probability mass function, mean and derivation of the mean—if needed, the variance will be provided for you)

· Geometric random variable (setting, probability mass function, memory-less property, mean—if needed, the variance will be provided for you)