23. A gas station sells regular and premium fuel. The two storage tanks holding the two types of gasoline are refilled every week. The proportions of the available supplies of regular and premium gasoline that are sold during a given week in the summer is an outcome of a bivariate random variable having the joint density function f xð Þ¼ ; y 2 5ð Þ 3x þ 2y I½ 0;1 ðxÞI½ 0;1 ðyÞ; where x ¼ proportion of regular fuel sold and y ¼ proportion of premium fuel sold.

(a) Find the marginal density function of X. What is the probability that greater than 75 percent of the available supply of regular fuel is sold in a given week?

(b) Define the regression curve of Yon X, i.e., define E(Y |

x). What is the expected value of Y, given that x ¼ .75? Are Y and X independent random variables?

(c) Regular gasoline sells for $1.25/gal and premium gasoline sells for $1.40/gal. Each storage tank holds 1,000 gal of gasoline. What is the expected revenue generated by the sale of gasoline during a week in the summer, given that x ¼ .75?