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I have the following question. Let $X$ and $Y$ independent random variables. We define $ Z \equiv X + Y$ and $W \equiv X/Y $ Are $Z$ and $W$ independent and how can I prove it? Thanks

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  • $\begingroup$ The problem is: Let $X$, $Y$ be independent random variables such that $X$ and $Y$ $\sim Exp(1)$. Find the probability joint density function of $W$ and $Z$ where $Z \equiv Y+X$ and $W \equiv X/Y$. There are two ways to solve. One, find directly $f(z, w)$ with a jacobian, and the other is find $f(z)$ and $f(w)$ separated and the multiply both to obtain the joint density. The multiplication is valid if and only if $W$ and $Z$ are independent $\endgroup$ – Agustín Cugno May 17 '18 at 4:01
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Let $X=1$. Then $Z=1+Y$ and $W=\frac 1 Y$. Any random variable $Y$ is independent of $X$, but $W$ and $Z$ need not be independent. For example, if $Y$ takes values 1 and -1 with probability 1/2 each then $\frac 1 Y =Y$ and it is obvious that $W$ and $Z$ are not be independent.

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