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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?

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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 $x>1$.)

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