Prove positive covariance of Two random variables

[oofiedoofie]

If X and Y are random variables that are positive and have non-zero variance such [imath] X \perp Y [/imath].

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

 

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[imath]\;[/imath]

 

[BigBeachBanana]

If X and Y are random variables that are positive and have non-zero variance such [imath] X \perp Y [/imath].

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

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