Let X and Y be random variables with positive variance. The correlation of X and Y is defined as
ρ(X,Y)=V(X)V(Y)cov(X,Y).
(a) Using Exercise 17(c), show that
0≤V(σ(X)X+σ(Y)Y)=2(1+ρ(X,Y)).
(b) Now show that
0≤V(σ(X)X−σ(Y)Y)=2(1−ρ(X,Y)).
(c) Using (a) and (b), show that
−1≤ρ(X,Y)≤1.