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The pdf $f_Z(z)$ of the sum $Z=X+Y$ of any two jointly continuous random variables $X$ and $Y$ with joint pdf $f_{X,Y}(x,y)$ is as follows: $$\text{For all } z, -\infty < z < \infty, ~~ f_Z(z) = \int_{-\infty}^\infty f_{X,Y}(x,z-x) \, \mathrm dx.\tag{1}$$ For the special case when $X$ and $Y$ are nonnegative random variables (including as a special ...