A Seattle newspaper intends to administer two different surveys relating to two different anti-tax initiatives on the ballot in November. The proportion of surveys mailed that will actually be completed and returned to the newspaper can be represented as the outcome of a bivariate random variable \((X, Y)\) having the density function

\(f(x, y)=\frac{2}{3}(x+2 y) I_{[0,1]}(x) I_{[0,1]}(y)\), where \(x\) is the proportion of surveys relating to initiative I that are returned, and y refers to the proportion of surveys relating to initiative II that are returned.

(a) Are \(X\) and \(Y\) independent random variables?

(b) What is the conditional distribution of \(x\), given \(y=.50\) ? What is the probability that less than 50 percent of the initiative I surveys are returned, given that 50 percent of the initiative II surveys are returned?

(c) Define the regression curve of \(X\) on \(Y\). Graph the regression curve. What is the expected proportion of initiative I surveys returned, given that 50 percent of the initiative II surveys are returned?