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
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.