Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
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
up vote 1 down vote accepted

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.

share|cite|improve this answer
Thank you! Now it's kind of obvious:). – arkadiy Feb 5 '13 at 10:16

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.