Question: Let X be the first time that a failure occurs in an infinite sequence of Bernoulli trials with probability p for success. Let pk =

Let X be the first time that a failure occurs in an infinite sequence of Bernoulli trials with probability p for success. Let pk = P(X = k) for k = 1, 2, . . . . Show that pk = pk−1q where q = 1 − p. Show that Pk pk = 1. Show that E(X) = 1/q. What is the expected number of tosses of a coin required to obtain the first tail?

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