Since X and Y take on only two values, we may choose a,b,c,d so that The followings take only values 0 and 1.

U = \(\frac {X+a}{b}\) , V = \(\frac {Y+c}{d}\)

If E(XY) = E(X)E(Y),

then E(UV) = E(U)E(V)

If U and V are independent, so are X and Y.

Thus we can prove the independence for U and V taking on values 0 and 1 with E(UV) = E(U)E(V)

E(UV) = P(U=1, V=1) = E(U)E(V) = P(U=1)P(V=1)

P(U=1, V=0) = P(U=1) - P(U=1, V=1) = P(U=1)(1-P(V=1)) = P(U=1)P(V=0)

Similarly,

P(U=0, V=1) = P(U=0)P(V=1) P(U=0, V=0) = P(U=0)P(V=0)

Discussion week7