Let [math]X[/math] and [math]Y[/math] be random variables with positive variance. The correlation of [math]X[/math] and [math]Y[/math] is defined as

[[math]] \rho(X,Y) = \frac{{\rm cov}(X,Y)}{\sqrt{V(X)V(Y)}}\ . [[/math]]

• Using Exercise(c), show that
  [[math]] 0 \leq V\left( \frac X{\sigma(X)} + \frac Y{\sigma(Y)} \right) = 2(1 + \rho(X,Y))\ . [[/math]]
• Now show that
  [[math]] 0 \leq V\left( \frac X{\sigma(X)} - \frac Y{\sigma(Y)} \right) = 2(1 - \rho(X,Y))\ . [[/math]]
• Using (a) and (b), show that
  [[math]] -1 \leq \rho(X,Y) \leq 1\ . [[/math]]