(iii) Let X and Y be independent normal random variables, each with mean 0, and with non-zero variances a2 and b2, respectively. Show that V = Y/X has probability density function fV (v) =

c

π(c2 + v2)

for −∞ < v < ∞, where c = b/a. Hence find P(|Y | < |X|)

(•

b) Now let X and Y be independent random variables, each uniformly distributed on the interval (0, 1). By considering the random variables U = Y and V = XY 2, or otherwise, find the probability density function of V .
(Oxford 2010)