Covariance of $A$ and $B$

Question

If I know that $X$ and $Y$ are r.v that are positive and have non zero variance such that $X \perp Y$, how can I show that the random variables $W= X + Y$ and $Z = XY$ are positively correlated (i.e. $\operatorname{Cov}(W, Z) > 0$).

My solution: \begin{align} \operatorname{Cov}(W, Z)&= E[WZ] - E[W]*E[Z]\\ &= E[XY] - E[X] * E[Y]\\ &= E[X]^2 - 2E[X]^2 \end{align} vhich simplifies to $\operatorname{Cov}(W,Z) =-E[X]^2$.

Is this correct or am I wrong? Please let me know and what the right approach is.

Answer 1

Too long for a comment. You should more carefully expand $W=X+Y\,:$

\begin{align}{\rm Cov}(W,Z)&=\mathbb E[WZ]-\mathbb E[W]\,\mathbb E[Z]\\&=\mathbb E[X^2]\,\mathbb E[Y]+\mathbb E[X]\,\mathbb E[Y^2]-\mathbb E[X]^2\,\mathbb E[Y]-\mathbb E[X]\,\mathbb E[Y]^2\\&={\rm Var}[X]\,\mathbb E[Y]+{\rm Var}[Y]\,\mathbb E[X]\end{align}