Under what condition are $U$ and $V$ uncorrelated? Let $X$ and $Y$ be independent random variables with finite variances, and let $U = X + Y$ and $V = XY$. Under what condition are $U$ and $V$ uncorrelated?

MY ATTEMPT

We say that two variables are uncorrelated if their covariance equals zero. Bearing this in mind, we have
\begin{align*}
\textbf{Cov}(U,V) &amp; = \textbf{E}(UV) - \textbf{E}(U)\textbf{E}(V)\\\\
&amp; = \textbf{E}(X^{2}Y + XY^{2}) - \textbf{E}(X+Y)\textbf{E}(XY)\\\\
&amp; = \textbf{E}(X^{2})\textbf{E}(Y) + \textbf{E}(X)\textbf{
E}(Y^{2}) - \textbf{E}(X)^{2}\textbf{E}(Y) - \textbf{E}(X)\textbf{E}(Y)^{2}\\\\
&amp; = \textbf{E}(Y)[\textbf{E}(X^{2}) - \textbf{E}(X)^{2}] + \textbf{E}(X)[\textbf{E}(Y^{2}) - \textbf{E}(Y)^{2}]\\\\
&amp; = \textbf{E}(Y)\textbf{Var}(X) + \textbf{E}(X)\textbf{Var}(Y) = 0
\end{align*}

Is there a specific name for this last expression? Any contribution is appreciated. Thanks :)
First of all, there is no specific name of what you've got. Second of all, assuming r.v. $X$ is not degenerated we have the following condition 
$$
\mathbf{E} Y = - \mathbf{E} X \frac{\mathbf{Var} Y}{\mathbf{Var} X}.
$$
First of all, there is no specific name of what you've got. Second of all, assuming r.v. $X$ is not degenerated we have the following condition 
$$
\mathbf{E} Y = - \mathbf{E} X \frac{\mathbf{Var} Y}{\mathbf{Var} X}.
$$