27. Let g and h be two probability density functions with distribution functions G and H, respectively. Show that for 1 1,

27. Let g and h be two probability density functions with distribution functions G and H, respectively. Show that for −1 ≤ α ≤ 1, the function f(x, y) = g(x)h(y)

????

1 + α[2G(x) − 1][2H(y) − 1]

is a joint probability density function of two random variables. Moreover, prove that g and h are the marginal probability density functions of f.

Note: This exercise gives an infinite family of joint probability density functions all with the same marginal probability density function of X and of Y .