74. Let X1, X2, . . . be a sequence of independent identically distributed continuous random variables.Wesay that a record occurs at time n ifXn > max(X1, . . . ,Xn−1).

That is, Xn is a record if it is larger than each of X1, . . . ,Xn−1. Show

(a) P{a record occurs at time n} = 1/n;

(b) E[number of records by time n] = ni

=1 1/i;

(c) Var(number of records by time n) = ni

=1(i − 1)/i2;

(d) Let N = min{n: n > 1 and a record occurs at time n}. Show E[N] = ∞.

Hint: For (ii) and (iii) represent the number of records as the sum of indicator (that is, Bernoulli) random variables.