Chapter 3

Required:

I. Complete the following 7 problems:

• Exercise 3-8, p. 94
• Exercise 3-15, p. 99
• Exercise 3-27, p. 109
  • For (a), use both the direct definition of the expected value of \(X\) and the survival function trick. See II. below for the survival function trick.
• Exercise 3-29, p. 110
• Sample Exam Problem 3-2, p. 128
• Sample Exam Problem 3-5, p. 129
• Sample Exam Problem 3-14, p. 131

II. Prove the survival function trick for discrete random variables:

Theorem: Let \(N\) be a nonnegative discrete random variable with finite mean. Then \[ E[N] = \sum_{n = 0}^{\infty} P(N > n).\]

Hints:

• This theorem is easiest to prove by starting from the right-hand side.
• Start by expanding \(P(N > n)\) in terms of the probability mass function of \(N\).
• It will be useful to write each expanded term on a single line, and align the different lines based on the probability mass function terms in a term's expansion.

III. Prove the following property for a discrete random variable \(X\) with finite mean.

Theorem: Let \(X\) be a random variable with mean \(E[X]\), and let \(a, b\) be real (non-random) constants. Then \[E[a X + b] = a E[X] + b.\]

Hint:

• Use the result for the expectation of a function \(g\) of a random variable \(X\), and take advantage of properties of sums and probability mass functions.

IV. Complete two problems from Problem Solving Session 1 that you did not solve during the problem solving session and that were not presented by the student presenters.