Question

3.3.14 Let X and Y be independent, with X ∼ Bernoulli(1/2) and Y ∼ N(0, 1). Let
Z = X +Y and W = X −Y. Compute Var(Z), Var(W), Cov(Z, W), and Corr(Z, W

Answer 1

Given that X is Bernoulli random variable with parameter 1/2

$X\sim Bernoulli(\frac{1}{2})$ then $E(X)=p=\frac{1}{2}$ and $V(X)=pq=\frac{1}{2}\cdot\frac{1}{2}=\frac{1}{4}$

We know that $V(X)=E(X^2)-(E(X))^2$

than $\frac{1}{4}=E(X^2)-(\frac{1}{2})^2 \implies E(X^2)=\frac{1}{4}+\frac{1}{4}=\frac{1}{2}$

Y is a standard normal random variable that is

$Y\sim N(0,1)\implies E(Y)=0$ and $V(Y)=1$

$V(Y)=E(Y^2)-(E(Y))^2 \implies 1=E(Y^2)-0^2\implies E(Y^2)=1$

X and Y are independent variables and given that $Z = X + Y$ then

$E(Z)=E(X)+E(Y)=\frac{1}{2}+0=\frac{1}{2}$

$V(Z)=V(X)+V(Y)+2Cov(X,Y)$

Since X and Y are independent Cov(X, Y)=0 then

$V(Z)=\frac{1}{4}+1=\frac{5}{4}$

$W=X-Y\ so\ E(W)=E(X)-E(Y)=\frac{1}{2}-0=\frac{1}{2}$

$V(W)=V(X)+V(Y)+2Cov(X,Y)=\frac{1}{4}+1=\frac{5}{4}$

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

$Cov(Z,W)=E(X^2-Y^2)-\frac{1}{2}\cdot\frac{1}{2}=E(X^2)-E(Y^2)-\frac{1}{4}=\frac{1}{2}-1-\frac{1}{4}=-\frac{3}{4}$

$Corr(Z,W)=\frac{Cov(Z,W)}{\sqrt{V(Z)}\sqrt{V(W)}}=\frac{-\frac{3}{4}}{\sqrt{\frac{5}{4}}\sqrt{\frac{5}{4}}}=-\frac{3}{5}$

Answer: $\frac{5}{4},\frac{5}{4},-\frac{3}{4}, -\frac{3}{5}$ .