# Three random variables, two are independent

Suppose we have 3 random variables $X,Y,Z$ such that $Y\perp Z$ and let $W = X+Y$. How can we infer from this that $$\int f_{WXZ}(x+y,x,z)\mathrm{d}x = \int f_{WX}(x+y,x)f_{Z}(z)\mathrm{d}x$$ Any good reference where I could learn about independence relevant to this question is also welcome.

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I assume you mean $f_{WX}(y+x,x)$? –  nbubis Feb 4 '13 at 10:12
Correct. Thank you. –  arkadiy Feb 4 '13 at 10:13
There seems to be no guarantee that $(W,X,Z)$, or even $(W,X)$, has a density. Consider the case $X=Y$, for example. –  Did Feb 4 '13 at 18:35
It was assumed that all densities exist (with respect to the Lebesgue measure) and that all random variables are different. Do you refer to something else? If yes, could you explain more, please? –  arkadiy Feb 4 '13 at 19:04

After integrating out $x$, the left-hand side is the joint density for $Y$ and $Z$ at $(y,z)$, and the right-hand side is the density for $Y$ at $y$ multiplied by the density for $Z$ at $z$. These are the same because $Y$ and $Z$ were assumed to be independent.