2017 September 27,

Math 265: Day 8

Carleton College, Prof. Joshua R. Davis

Due at the start of class on Day 11

Complete these problems. Write them up carefully, in the order assigned, for handing in with the rest of your homework.

• 4.3, 4.8
• Give an example of a discrete random variable X whose expectation is undefined. That is, the sum of P(X = x), over the possible values x, equals 1, but the sum of x P(X = x) diverges. Explain/prove your answer.
• When I visit my collaborators in Madison, Wisconsin, I have two parking options. The legal option is to pay $25 for a parking garage spot very close to my destination. The illegal option to to park farther away, spending no money on the parking spot but spending $10 worth of my time in walking to my destination. In the illegal case, there is also a probability p that I will be required to pay a $200 fine. What is the expected cost of the illegal option, in terms of p? And at which values of p is the illegal option better or worse than the legal option (based solely on expectation)?
• Let X ~ Binom(n, p). This random variable counts the number of successes in n independent Bernoulli trials, each of probability p. So we expect X to take a value somewhere around p n. Assume for the sake of simplicity that p n is an integer. (This would hold if p = 1 / 6 and n = 600, for example.) The probability that X is exactly p n is P(X = p n). Using Stirling's approximation for the factorial, show that P(X = p n) is approximately (2 π p (1 - p) n)^-1/2. Under what conditions on n and p is this a good approximation?