Probability distribution of the subtraction of two dependent non-gaussian distributions I have two dependent,correlated non-gaussian random variable X and Y. I want to calculate probability distribution of Z=X-Y. For independent distribution, I know the equation is as follows,

What should be the equation for dependent random variables? Is there any crude way to know the approximate probabilities?
Thanks
Farhana
 A: Building upon W. Huber's [sufficient] suggestions, the fact that the variables $X$ and $Y$ are dependent does neither modify nor complexify the derivation of the density of $Z=X-Y$. It is still a consequence of the principle of a change of variables:


*

*The pair $(X,Y)$ has density $f_{X,Y}(x,y)$

*The transform $$(x,y)\longrightarrow(x,x-y)$$ has its inverse equal to $$(x,z)\longrightarrow(x,z-x)$$ and the associated Jacobian is$$J(x,z)=\left|\dfrac{d(x,y)}{d(x,z)}\right|=\left|\dfrac{d(x,z-x)}{d(x,z)}\right|$$ equal to $1$. 

*The change of variables formula provides the density of $(X,Z)$ as $$f_{X,Z}(x,z)=f_{X,Y}(x,z-x)J(x,z)$$

*Marginalisation then returns the density of $Z$ as $$f_Z(z)=\int f_{X,Z}(x,z)\text{d}x=\int f_{X,Y}(x,z-x)\text{d}x$$

A: in such case, you will end up with a 2 dimensional fz(x,y) given that x and y are not independent (2 deg of freedom). you might dicretize and calcualte block by block the xy plane, as no elegant formula might exist. if the dependence realtionshio is polynomic you might apply various convolutional / Fourrier transform tricks.
