Hints for Homework Assignment 7

General Remarks

Relevant reading: Review your class notes (in particular, the indicator method, and the maximum/minimum trick), read Section 3.2 through p. 170 and the summary of properties on p. 181. Several of the problems require techniques from 3.1 (finding distributions of r.v.'s, computing probabilities involving r.v.'s, given their distributions), so you may need to review that material as well. (See the hints for HW 6 for some general advice on doing problems of this type.)

Specific Comments

• Problem 1: This is a straightforward application of the properties of expectation.
• Problem 2: Part (a) is a simple type II problem and does not involve expectations. You are given (at least indirectly) the joint distribution of X and Y, and you have to compute a probability involving X and Y. Part (b) is a straightforward application of a formula (which one?). Part (c) can be done in several different ways: using a two variable version of the formula referred to above, using the independence of X and Y; or by writing Z=X+Y, computing the dist. of Z (this was already done, more generally, in HW 6) and then computing E(2^Z).
• Problem 3: Part (a) is straightforward (and was done in class). Part (b) is an application of the maximum trick (also done in class). (c) Once the distribution is known (from (b)), the expectation is easy to compute using the definition of E(X).
• Problem 4: Parts (a) and (b) are ordinary probability computations in a success/failure model of the type that came up in Chapters 1 and 2. For part (c) you first need to compute the distribution of X, which in turn reduces to ordinary probability computations. Be sure to check your work by adding the probabilities in this distribution.
• Problem 5: This is a type I problem. In case (i), it boils down to computing the three values P(X=2), P(X=3), and P(X=4). Each of these is a combinatorial probability computation that is best done by counting ordered tuples. As always, you should check the distribution you obtained by checking whether the probabilities involved add up to 1. Case (ii) (with replacement) is equivalent to working with independent S/F trials. This time, the values of X can be arbitrarily large. The computation of probabilities of the form P(X=k) reduces (after replacing X by its definition) to a familiar problem in connection with S/F sequences.
• Problem 6: This problem requires an understanding of the indicator method that goes beyond merely knowing the formula for E(X) in terms of P(A1), P(A2), etc. Review the derivation of that formula, via "indicator random variables", from your class notes (or from the text). If you understand what's behind the formula, you should be able to handle the problem. Note that, although part of the problem is taken from the exercises to Section 3.3, the problem itself requires no material beyond 3.2.
• Problem 7: First read to problem carefully to make sure you understand the definition of a record. For parts (a) and (b), a success/failure model would be inappropriate since the precise numbers that come up matter, not just whether or not the number is a 6 (or any other specified number). However, the probabilities are easily computed by splitting into the cases when the record is 2, 3, ..., 6 and evaluating the probabilities for each of these cases separately. Part (c) can be done with the indicator method and the results of part (a). If you don't know how to start, review the problems on the indicator method worked in class and on the last HW assignment.

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