Geometric Distribution In Example 2, we tossed a coin repeatedly until the first heads

Chapter 12, Problem 14

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Geometric Distribution In Example 2, we tossed a coin repeatedly until the first heads showed up. Assume that the probability of heads is p, where p (0, 1). Let Y be a random variable that counts the number of trials until the first heads shows up. (a) Show that P(Y = 1) = p, P(Y = 2) = (1 p)p, and P(Y = 3) = (1 p)2 p. (b) Explain why P(Y = j ) = (1 p) j1 p for j = 1, 2, . . . . This equation is called the geometric distribution. (c) Prove that # j1 P(Y = j ) = 1 as follows: (i) For 0 q < 1, define Sn = 1 + q + q2 + +qn Show that Sn qSn = 1 qn+1 and conclude from this equation that Sn = 1 qn+1 1 q (ii) Show that P(Y k) = #k j=1 P(Y = j ) = p #k j=1 (1 p) j1 Use your results in (i) to show that this formula simplifies to 1 (1 p)k and conclude from this equation that lim k P(Y k) = 1 which is equivalent to # j1 P(Y = j ) = 1