Question: 3.4.19 Computer Challenge. Let N be a positive integer and let Z1; : : : ;ZN be independent random variables, each having the geometric distribution

3.4.19 Computer Challenge. Let N be a positive integer and let Z1; : : : ;ZN be independent random variables, each having the geometric distribution

k Pr(Z = k) = (2) (-2). for k = 1, 2,....

Since these are discrete random variables, the maximum among them may be unique, or there may be ties for the maximum. Let pN be the probability that the maximum is unique. How does pN behave when N is large? (Alternative formulation: You toss N dimes. Those that are heads you set aside; those that are tails you toss again. You repeat this until all of the coins are heads. Then, pN is the probability that the last toss was of a single coin.)

k Pr(Z = k) = (2) (-2). for k = 1, 2,....

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