Let X and Y be random variables with positive variance. The correlation of X and Y is defined as \rho(X, Y)=\frac{\operatorname{cov}(X, Y)}{\sqrt{V(X) V(Y)}} =) Using Exercise 17(c), show that 0 \leq V\left(\frac{X}{\sigma(X)}+\frac{Y}{\sigma(Y)}\right)=2(1+\rho(X, Y)) (b) Now show that 0 \leq V\left(\frac{X}{\sigma(X)}-\frac{Y}{\sigma(Y)}\right)=2(1-\rho(X, Y)) Using (a) and (b), show that -1 < p(X,Y) <1.

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