A product density

Topic: Joint distributions · more questions on this topic
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A pair (X,Y) has joint density f(x,y)=2e^{-x-2y} for x>0 and y>0.

1. Calculate \mathbb{E}(XY).

2. Calculate \operatorname{Cov}(X+Y, X-Y).

Worked answer (click to show)

The joint density factors as e^{-x}\cdot 2e^{-2y} on the positive quadrant, so X\sim\operatorname{Exp}(1) and Y\sim\operatorname{Exp}(2) are independent.

1. \mathbb{E}(XY)=(\mathbb{E} X)(\mathbb{E} Y)=1\times 1/2=1/2.

2. The cross covariances cancel:

\operatorname{Cov}(X+Y,X-Y)=\operatorname{Var}(X)-\operatorname{Var}(Y)=1-\frac14=\frac34.