Question

For independent X and Y, we have probability density function for them where pdf of X is f(x) = ne^-nx and pdf of Y is f(y) = me^-my. (x and y greater than 0). Let M1=max(X,Y) and M2=min(X,Y). Find cov(M2,M1).

Answer 1

Independent X and Y, we have probability density function for them where pdf of X is f(x) = ne^-nx and pdf of Y is f(y) = me^-my. (x and y greater than 0).

Cov(M₂,M₁) = E(M₂,M₁) = E(M₁)E(M₂)

Now M₂ M₁ = max(X,Y) min(X,Y)

= XY always.

E(M₂,M₁) = E(XY) = E(X) E(Y)

Again M₁ + M₂ = X + Y always

M₁ = (X + Y - M₂)

Now

.

Also,

Similarly,

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