Question

3.3.13 Let X and Y be independent, with X ∼ Bernoulli(1/2) and Y ∼ Bernoulli(1/3).
Let Z = X + Y and W = X − Y. Compute Cov(Z, W) and Corr(Z, W).

Answer 1

$Cov(Z,W)=E[ZW]-E[Z]E[W]=E[(X+Y)(X-Y)]-E[X+Y]E[X-Y]=$

$=E[X^2-Y^2]-(E[X]+E[Y])(E[X]-E[Y])=E[X^2]-E[Y^2]-(E[X])^2+(E[Y])^2=Var(X)-Var(Y)=\frac14-\frac13\cdot\frac23=\frac14-\frac29=\frac1{36};$

$Corr(Z,W)=\frac{Cov(Z,W)}{\sqrt{Var(X)Var(Y)}}=\frac{\frac{1}{36}}{\sqrt{\frac{1}{4}\cdot\frac{2}9}}=\frac{1}{\sqrt{2}\sqrt{36}}=\frac{\sqrt{2}}{12}$ .