11. Economic conditions cause fluctuations in the prices of raw commodities as well as in finished products. Let X denote the price paid for a barrel of crude oil by the initial carrier, and let Y denote the rice paid by the refinery purchasing the product from the carrier. Assume that the joint density for (X, Y) is given by

f_xy(x, y) = c ; 20 < x < y <40

(a)Find the value of c that makes this a joint density for a two-dimensional randomvariable.

(b)Find the probability that the carrier will pay at least $25.00 per barrel and the refinery will pay at most $30.00 per barrel for the oil.

(c)Find the probability that the price paid by the refinery exceeds that of the carrier by at least &10.00 per barrel.

(d)Find the marginal densities for X and Y.

(e)Find the probability that the price paid by the carrier is at least$25.00.

(f ) Find the probability that the price paid by the refinery is at most $30.00.

(g) Are X and Y independent? Explain.

43. Consider Exercise 11:

a. Find the curve of regression of X on Y. Is the regression linear?

b. Find the mean price paid by the carrier given that the refinery price is 30 dollars per barrel.

c. Find the curve of regression of Y on X. Is the regression linear?

d. FInd the mean price paid by the refinery for a barrel of crude oil given that the carrier paid 35 dollars per barrel.