Question

3.3.12 Suppose you flip four fair coins. Let X be the number of heads showing, and
let Y be the number of tails showing. Compute Cov(X, Y) and Corr(X, Y).

Answer 1

Solution

Y~Bin(4,0.5) and X~Bin(4,0.5)

Y=4-X

E(X)=np= 4(0.5)=2

E(Y)=np=4(0.5)=2

Var(X)=np(1-p)=1

Var(Y)=1

$Var(X) = E(x^2)-{E(x)}^2$

$E(x^2)=1+2^2=5$

$$Cov(XY) =E(XY) - E(X) E(Y)$$ $$=E(X(4-X))-(2)(2)$$ $$=4E(X)-E(X^2)-4$$ $$=4(2)-5-4=-1$$

Cov(XY) =-1

Corr(X, Y)

$$Corr(XY) ={Cov(XY) \over \sigma_x \sigma_y}$$ $$={-1 \over (1*1)} =-1$$

Corr(X, Y) =-1