Question

(a) When are two random variables X and Y independent?

(b) Show that if E(XY ) = E(X)E(Y ) then V ar(X + Y ) = V ar(X) + V ar(Y ).

Answer 1

Solution

(a) Two  random variables X and Y are independent if knowing the value of one of them does not change the probabilities for the other one. In other words, if X and Y are independent , then the joint density of X and Y is equal to the product of their marginal densities.

That is, f( x,y) = g(x)h(y) for all x, y

(b) Given E(XY) = E(X)E(Y)

Now Var( X+ Y ) = Var(X) + Var(Y) + 2Cov(X,Y)

   = Var(X) + Var(Y) + 2 [ E(XY) - E(X)E(Y)] Since E(XY) = E(X)E(Y) , then Cov(X,Y) = 0

Therefore , Var( X+ Y ) = Var(X) + Var(Y)

Hence proved.