70. Let Ar 1,X2, ... be a sequence of independent identically distributed continuous random variables. We say that a record occurs at time ç if Xn > maxiA^,..., ×ç-ë). That is, Xn is a record if it is larger than each of Xl9 Xn_x. Show

(i) P[a record occurs at time ç} = l/n (ii) ^[number of records by time ç] = Óº=é 1/'
(iii) Var(number of records by time ri) = Ó?=é(' ~~ I ) / ' 2 (iv) Let TV = min[/?: ç > 1 and a record occurs at time n}. Show Å [Í] = oo.
Hint: For (ii) and (iii) represent the number of records as the sum of indicator (that is, Bernoulli) random variables.