Imagine a TV game show where each contestant i spins an infinitely calibrated wheel of fortune, which assigns him/her with some real number with a value between 1 and 100. All values are equally likely and the value obtained by each contestant is independent of the value obtained by any other contestant.
(a) Find P(X1 < X2).
(b) Find P(X1 < X2,X1 < X3).
(c) Let N be the integer-valued random variable whose value is the index of the first contestant who is assigned a smaller number than contestant 1. As an illustration, if contestant 1 obtains a smaller value than contestants 2 and 3 but contestant 4 has a smaller value than contestant 1 (X4 < X1), then N = 4. Find P(N > n) as a function of n.
(d) Find E[N], assuming an infinite number of contestants.