1. Let {Xn : n >=1} be a sequence of positive independent random variables with E(Xn) = c 2 (0; 1) for each n. Let Yn = X1X2...Xn, the product of the Xi's. Use Markov's inequality to prove that Yn ~ 0 in probability.
2. Let X1; : : : ;Xn be iid Unif(0; 1) random variables.
(a) De ne Mn = max(X1; : : : ;Xn). Find the CDF of Mn.
(b) Prove that n(1-Mn) ! Z in distribution, where Z has CDF FZ(z) = 1-e^-z.