Let X1, X2, be independent and identically distributed continuous random variables We say that a record occurs at time n, n > 0 and has value X, if X, > max(X, X-1), where x = -0.

(a) Let N, denote the total number of records that have occurred up to (and including) time n. Compute E[N] and Var(N).

(b) Let Tmin{n. n > 1 and a record occurs at n}. Compute P{T> n} and show that P{T y}. Show that T, is independent of X7. That is, the time of the first value greater than y is independent of that value. (It may seem more intuitive if you turn this last statement around.)