Help with probability please?

Question

Suppose X and Y have joint density f(x,y) = e^(-x-y) for x,y > 0. What is the covariance of X and Y?

Answer 1

cov = E[XY] - E[X] E[Y]

E[XY] = double integral (y from 0 to ∞)(x from 0 to ∞) xye^(-x-y) dxdy

In this case, the double integral can be split (because e^(-x-y) = e^(-x)*e^(-y)) as

integral (y from 0 to ∞) ye^(-y) dy [integral (x from 0 to ∞) xe^(-x) dx]

which is exactly

E[Y] E[X]

so cov(X,Y) = 0

Answer 2

Looks like the distribution requires that X and Y are independent so cov(X,Y)=0

If you use the equation

Cov(X,Y) = E[XY] - E[X]E[Y]

Then each term would solve to1 and the covariance vanishes.