Correlation between $XY$ and $Y$ Let $X$ and $Y$ be independent random variables with means $\mu_X$ and $\mu_Y$ and variances $\sigma_X^2$ and $\sigma_Y^2$. Find an expression of correlation of $XY$ and $Y$ in terms of these means and variances.
When I follow the definition for correlation, I get a zero in the denominator, which does not make sense. Please help. 

The denominator is $\sigma_{XY}\sigma_Y$.
Let us compute $\sigma_{XY}$, since that is undoubtedly the source of the problem. We get
$$\sigma_{XY}^2=E((XY)^2)-(E(XY))^2.$$
By independence, 
$$E((XY)^2)=E(X^2)E(Y^2)=(\sigma_X^2+\mu_X^2)(\sigma_Y^2+\mu_Y^2).$$
And $(E(XY))^2=\mu_X^2\mu_Y^2$. Subtract. We get
$$\sigma_{XY}^2=\sigma_X^2\sigma_Y^2+\mu_X^2\sigma_Y^2+\mu_Y^2 \sigma_X^2.$$
The calculation for the numerator is quite a bit easier.
