17. In a sequence of dependent Bernoulli trials, the conditional probability of success at the i th trial, given that all preceding trials have resulted in failure, is pi (i = 1, 2, . . . ). Give an expression in terms of the pi for the probability that the first success occurs at the nth trial.

Suppose that pi = 1/(i +1) and that the time intervals between successive trials are independent random variables, the interval between the (n − 1)th and the nth trials being exponentially distributed with density nα exp(−nαx), where α is a given constant. Show that the expected time to achieve the first success is finite if and only if α > 0. (Oxford 1975F)