Random Variable with its own RV's joint distribution How to derive the joint distribution: $f(x, x)$, I thought it would be just $f(x)$, but if I do the double integral of $f(x)$, it would not be $1$. I can not just multiply $f(x)$ twice because $X$ and $X$ is highly dependent. 

You can think about it this way:
consider $y=x$ now 
$f(y = x|X=x) = 1,$ and zero otherwise
$f(y,x) = f(x), y=x$ zero otherwise .
