37. Teresa and Allison each have arrival times uniformly distributed between 12:00 and 1:00. Their times do not in uence each other. If Y is the rst of the two times and X is the second, on a scale of 0 to 1, then the joint pdf of X and Y is f (x, y)  2 for 0  y  x  1.

a. Find the marginal density of X.

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

c. Find the conditional probability that Y is between 0 and .3, given that X is .5.

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.