Let $X,Y$ be two random variables, I'll denote $f_X(x),f_Y(y)$ as their density functions, and $f_{X,Y}(x,y)$ as the joint density function.
I know that if I only have $f_{X,Y}(x,y)$, I can fairly easy find $f_X(x)$ by $f_X(x)=\int f_{X,Y}(x,y)dy$ (and likewise $f_Y(y)$), but is it possible to find the joint distribution when I only have $f_X(x)$ and $f_Y(y)$? and if so, how?
I know that if $X,Y$ are independent, then $f_{X,Y}(x,y)=f_X(x)f_Y(y)$, but what about the general case, where I don't know if they are independent? how do I find $f_{X,Y}(x,y)$?
[Math] Finding the joint distribution of two random variables
probability
Best Answer
In general, there is no way of determining the joint density $f_{X,Y}(x,y)$ from knowledge of the marginal densities $f_X(x)$ and $f_Y(y)$ and nothing else. In special cases such as when $X$ and $Y$ are independent, the joint density is, of course, the product of the marginal densities.
As an example, consider random variables $X$ and $Y$ that are uniformly distributed on $[0,1]$. If you are also given that they are independent, then $(X,Y)$ is uniformly distributed on the unit square. But consider that $X$ and $Y$ might have joint density $(2x+2y-4xy)\mathbf 1_{\{(x,y)\colon 0 \leq x, y \leq 1\}}$ on the unit square. Thus, in general, the marginal densities do not determine the joint density.
Read about copulas and Sklar's theorem for further information.