Prove positive covariance of Two random variables

[oofiedoofie]

If X and Y are random variables that are positive and have non-zero variance such $X \perp Y$.

Prove the random variables $W = X+Y$ and $Z=XY$ are positively correlated (i.e. $Cov(W,Z) > 0$).

 

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[BigBeachBanana]

If X and Y are random variables that are positive and have non-zero variance such $X \perp Y$.

Prove the random variables $W = X+Y$ and $Z=XY$ are positively correlated (i.e. $Cov(W,Z) > 0$).

Show that $\text{Cov}(X+Y,XY) = \text{Var}(Y)\mathbb{E}[X]+\text{Var}(X)\mathbb{E}[Y]$