61. Let X1, X2,... be a sequence of independent identically distributed continuous random variables. We say that a record occurs at time n if Xn > 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] = n i=1 1/i;

(c) Var(number of records by time n) = n i=1 (i − 1)/i 2;

(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.