Economic situations cause fluctuations in the prices of raw commodities and in finished products. Let X denote the price paid for a barrel of crude oil by initial carrier, and let Y denote the price paid by the refinery purchasing the product from the carrier. Assume that the joint density for (X,Y) is given by
fxy (x,y) = c
20 < x < y < 40.
a) Find the value of c that makes this a joint density for the two-dimensional random variable.
b) Find the probability that the carrier will pay at least $25 per barrel and the refinery will pay at most $30 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 per barrel.
d) Find the marginal densities for X and Y.
e) Are X and Y independent? Explain.
f) Find the curve of regression of X on Y. Is the regression linear?
g) Find the curve of regression of Y on X. Is this regression linear?