The joint probability density function of X and Y is given by
f(x,y) = {
(1 + xy)/4, |x| < 1 and |y| < 1
0, otherwise
}
(a) Determine the marginal probability density functions of X and Y.
(b) Show that X and Y are not independent.
(c) Find the distribution function of X^2 and that of Y^2. (Hint: Use the marginal density functions and the method to transform random variables in chapter 08.)
(d) Find the joint distribution function of X^2 and Y^2. (Hint: F_(X^2,Y^2)(a,b) = P(X^2 <= a, Y^2 <= b) = P(-sqrt(a) <= X <= sqrt(a), -sqrt(b) <= Y <= sqrt(b)) if both a >= 0 and b >= 0.)
(e) Are X^2 and Y^2 independent?