Let joint cumulative probability density function of Random variable X,Y
$$F_{1,2}(x,y) = x^2(1-e^{-2y})\;\; \text{when}\;\;0\le x\lt1, y\ge0$$ and
$$= (1-e^{-2y}) \;\; \text{when}\;\; x\ge 1, y\ge0$$and
$$=0 \;\; \text{when} \;\;y \lt 0$$
in this case, I'd like to reversely get the joint pdf of X,Y.
Is there any typical way or algorithm to get the joint pdf from joint cdf?
Best Answer
Yes the typical way is differentiation: $$ f(x,y) = \partial_x\partial_y F(x,y).$$ One must be careful in general cause a PDF doesn't always exist, but here taking this derivative will do the trick. (The discontinuity across the line $x=1$ isn't a big deal. The support of the PDF just drops suddenly to zero when you cross into the half plane $x>1$.)