Suppose that (X, Y ) can be described by the joint CDF:

FX,Y (x, y) = 1 - $\displaystyle e^{-x}$ - $\displaystyle e^{-y}$+$\displaystyle e^{-(x+y+(theta)xy)}$

Show That

FX,Y (x, y) = $\displaystyle e^{-x-y-(theta)xy}$[(1 + $\displaystyle \theta$x)(1 + $\displaystyle \theta$y) − $\displaystyle \theta$] , x, y > 0, $\displaystyle \theta$ [0, 1].

Suppose that $\displaystyle \theta$ = 0. Explain why X and Y are independent?

Show that fX(x) = e−x, x > 0. (use $\displaystyle \lambda$ = 1 + $\displaystyle \theta$x

Calculate fY|X(y|x) and use it to compute E[Y|X = x]

N.B (where i have used the word "theta", obviously i have meant to use the symbol, but i was having problems doing latex math inside of latex math so used the word, so on each occasion where it is used, it should read "$\displaystyle \theta$xy" hope it doesnt cause confusion)