Hey downthesun01.

The definition of covariance is Cov[X,Y] = E[XY] - E[X]E[Y].

If X and Y are independent then P(A and B) = P(A)P(B) and from that you can prove that E[XY] = E[X]E[Y] which would prove that Cov[X,Y] = 0 for this particular case of X and Y being independent.

The rest of the argument expands out Cov[U,V] in terms of the Xi terms and note that Cov[Xi,Xj] = 0 for i != j (which is what they used above for independent random variables) and when i = j then you calculate E[Xi^2] which can be calculated in the normal way if you know the probability density function.

Thats the basic idea behind the independence and the algebra that is used.