Let X1, X2 be two independent random variables each with

Let X1, X2 be two independent random variables each with p.d.f. f1(x) = e?x for x > 0 and f1(x) = 0 for x ? 0. Let Z = X1 ? X2 and W = X1/X2. a. Find the joint p.d.f. of X1 and Z. b. Prove that the conditional p.d.f. of X1 given Z = 0 is g1(x1|0) =  2e?2x1 for x1 > 0, 0 otherwise. c. Find the joint p.d.f. of X1 and W. d. Prove that the conditional p.d.f. of X1 given W = 1 is h1(x1|1) =  4x1e?2x1 for x1 > 0, 0 otherwise. e. Notice that {Z = 0}={W = 1}, but the conditional distribution of X1 given Z = 0 is not the same as the conditional distribution of X1 given W = 1. This discrepancy is known as the Borel paradox. In light of the discussion that begins on page 146 about how conditional p.d.f.