Independence and Simple Correlation. (a) Show that if X and Y are independent, then E(XY ) = E(X)E(Y ) = xy where x = E(X)

Independence and Simple Correlation.

(a) Show that if X and Y are independent, then E(XY ) = E(X)E(Y ) = μxμy where μx = E(X)

and μy = E(Y ). Therefore, cov(X, Y) = E(X − μx)(Y − μy) = 0.

(b) Show that if Y = a + bX, where a and b are arbitrary constants, then ρxy = 1 if b > 0 and

−1 if b < 0.