39. A photographic supply business accepts orders on each of two different phone lines. On each line the waiting time until the rst call is exponentially distributed with mean 1 minute, and the two times are independent of one another. Let X be the shorter of the two waiting times and let Y be the longer. It can be shown that the joint pdf of X and Y is f (x, y) 

2 e(xy), 0  x  y  q.

a. Find the marginal density of X.

b. Find the conditional density of Y given X  x.

c. Find the probability that Y is greater than 2, given that X  1.

d. Are X and Y independent? Explain.

e. Find the conditional mean of Y given X  x. Is a linear function of x?

f. Find the conditional variance of Y given X  x.