Given the vector $\mathbf{d}$, where $\mathbf{d}\in\mathbb{C}^{N\times 1}$, we have two variables

$X = \mid\mathrm{F}[d]\mid^2,\quad\quad X\ge 0$

$Y = a+b (\mathrm{d}^H\mathrm{d})\quad Y\ge 0$

where :

$\mid.\mid$: denotes the magnitude operator,

$\mathrm{F}[.]$ is the Fourier Transform,

$(.)^H$ is the conjugate transpose operator

both $a$ and $b$ are real constants.

I have got the probability distribution function for $X$, $f_X(x)$ and for $Y$, similarly denoted by $f_Y(y)$. What is the joint probability function F(x,y) and how do I get the Expectation of a function of both X and Y denoted by $g(X,Y)$.