5.16 Correlation coefficient
Recall that Cov(X, Y) can be arbitrarily high or low depending on the
magnitude of X and Y. In practice, it is common to use a normalized
version on covariance called the correlation coefficient, $\rho(X, Y)$, where
$\rho(X,Y) = \frac{Cov(X, Y)}{\sigma_X \sigma_Y}$,
where $\sigma_X$ and $\sigma_Y$ represent the standard deviations of X and Y respectively.
Prove that the magnitude of Cov(X, Y) is bounded, specifically
$-1 \leq \rho(X,Y) \leq 1$.
[Hint 1: It helps to use the Cauchy-Schwarz inequality from Exercise 5.15.]
[Hint 2: Start by working with V = (X-E [X]) and W = (Y-E [Y]).]
