Nick and Penny are independently performing independent Bernoulli trials. For

Chapter 4, Problem 25

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Nick and Penny are independently performing independent Bernoulli trials. For concreteness, assume that Nick is flipping a nickel with probability p1 of Heads and Penny is flipping a penny with probability p2 of Heads. Let X1, X2,... be Nicks results and Y1, Y2,... be Pennys results, with Xi Bern(p1) and Yj Bern(p2). (a) Find the distribution and expected value of the first time at which they are simultaneously successful, i.e., the smallest n such that Xn = Yn = 1. Hint: Define a new sequence of Bernoulli trials and use the story of the Geometric. (b) Find the expected time until at least one has a success (including the success). Hint: Define a new sequence of Bernoulli trials and use the story of the Geometric. (c) For p1 = p2, find the probability that their first successes are simultaneous, and use this to find the probability that Nicks first success precedes Pennys.