For a sequence of bernoulli trials let x1 be the number of, Ask an Expert

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For a sequence of Bernoulli trials, let X1 be the number of trials until the first success. For j ≥ 2, let Xj be the number of trials after the (j - 1)st success until the jth success. It can be shown that X1, X2, . . . is an independent trials process.

(a) What is the common distribution, expected value, and variance for Xj ?

(b) Let Tn = X1 + X2 + · · · + Xn. Then Tn is the time until the nth success. Find E(Tn) and V (Tn).

(c) Use the results of (b) to find the expected value and variance for the number of tosses of a coin until the nth occurrence of a head.

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