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.