Hi I am trying to solve the following problem:
Let $F_X:\mathbb{R}\to[0,1]$ and $F_Y:\mathbb{R}\to[0,1]$ be unnivariate Cumulative Distribution Functions (CDFs) and suppose $-1\le\alpha\le 1$. Define $F_{XY}^{(\alpha)}:\mathbb{R}^2\to [0,1]$ by $$F_{XY}^{(\alpha)}(x,y)=F_X(x)F_Y(y)\{1+\alpha[1-F_X(x)][1-F_Y(y)]\}$$ The collection $\{F_{XY}^{(\alpha)}:-1\le\alpha\le 1\}$ is called the Farlie-MOrgenstern family of bivariate CDFs corresponding to $F_X$ and $F_Y$.
(a) Show that the marginal cdfs of X and Y are given by $F_X$ and $F_Y$.
(b) What value of $\alpha$ corresponds to X and Y being independent.
(c) Suppose X and Y follows exponential(1) distribution, then show that the joint pdfs is given by $$f_{XY}^{(\alpha)}(x,y)=\{1+\alpha[(1-2e^{-x})(1-2e^{-y})]\}e^{-(x+y)}I(x>0)I(y>0)$$
(d) For the special case of (c) derive the correlation between X and Y as a function of $\alpha$.
So far I have done (a),(b) and (c) and I am stuck with (d). I would highly appreciate if you could help. Thanks in advance.