Probability Theory – Final Exam Review

General Coverage and Office Hours

The final exam covers the material since the midterm: Chapter 5 (Sections 5.5.1, 5.6.6–5.6.4 omitted), Chapter 6 (only Sections 6.1–6.3), and Chapter 7 (only Sections 7.2—indirectly, 7.4, and 7.7). The exam is Friday at 6:30pm. This week I have my usual office hours Monday (3–4:30) and Tuesday (1:30–2:30). Plus, I will have office hours Thursday 2–3 and Friday 2–3. If these times don’t work for you, please let me know, and we can set up another appointment.

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). You must actively practice for the exam. That is, do not simply read through your solutions to example, homework and self-test problems. Actually re-work through the problems. 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 that are applicable:

Chapter 5: All problems (except for 5.14 and 5.19)

Chapter 6: 6.1, 6.2, 6.5, 6.6, 6.7 (not part d), 6.13

Chapter 7: 7.2, 7.3, 7.11, 7.12, 7.13, 7.29

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). Additional notes: Although you must do integration on the exam, it will be basic (i.e., no integration-by-parts); Tables 5.1, 7.1, and 7.2 will be included with the exam.

Specific Coverage

Previous material that won’t be directly tested, but may be necessary to solve part of a problem:

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

· Basic set theory (intersection, union, complement)

· Axioms of probability and propositions (4.1 – 4.3 in Chapter 2) following from the axioms

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

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

· Expected value of a 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

· Common discrete distributions

New material:

· Common continuous distributions; Normal approximation to the binomial (rule-of-thumb check to ensure accuracy, continuity correction, and probability determination)

· Simplification of a sum or integral by rearranging terms and then recognizing a specific pdf (pmf) integrated(summed) over all possible values

· Distribution of a function of a random variable: distribution-function technique and moment-generating-function technique

· Joint distributions (and calculating certain probabilities associated with the joint pdf—e.g., being able to determine appropriate limits of integration); marginal distributions

· Independent random variables—determining independence (proposition 2.1 in Chapter 6) and using independence to determine the joint pmf/pdf

· Distribution of sums of independent random variables (all the results on the summary sheet I gave you); you can use these results when solving problems (that is, you need not re-derive them); the only two results you might be asked to show are for independent gammas and Poissons (the two results we went through in class)

· Expectation of a function of two random variables (proposition 2.1 in Chapter 7); Expectation and variance of a sum of random variables (general results and specific situations—e.g., definition of appropriate indicator variables); Covariance and correlation (definitions and properties)

· Moment-generating functions (definition, properties, and uniqueness)