Let X1, X2,... be a sequence of independent identically distributed continuous random variables. We say that a record occurs at time n if X> max(X1, X-1). That is, X, is a record if it is larger than each of XX-1 Show (i) Pla record occurs at time n] = 1/n (ii) E[number of records by time n] =i-11/i (iii) Var(number of records by time n) = -1(i-1)/i (iv) Let N = min{n: n> 1 and a record occurs at time n]. Show E[N] = 0.

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