1. Let U and X be random variables, where U is Uniformly distributed (0,1). Given U=u, X has an exponential distribution with parameter =1/u.
(a) Find the joint density of U and X.
(b) Find E[UX]
2. Let (Xn) be a sequence of random variables. Suppose that for every ab i.o.} = 0. Show that lim_n->infinity{Xn} exists a.e., but may be infinite.
[Hint: Consider rational a and b first.]