62. Let X, the payoff from playing a certain game, have pmf

a. Verify that f (x; u) is a legitimate pmf, and determine the expected payoff. (Hint: Look back at the properties of a geometric random variable discussed in Chapter 3.)

b. Let X1, . . . , Xn be the payoffs from n independent games of this type. Determine the mle of u. (Hint:

Let Y denote the number of observations among the n that equal 1 [that is, Y  I(Yi  1), where I(A)  1 if the event A occurs and 0 otherwise], and write the likelihood as a single expression in terms of and y.)

c. What is the approximate variance of the mle when n is large?