[math] \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}[/math]

If [math]X[/math] and [math]Y[/math] are any two random variables, then the

covariance of [math]X[/math] and [math]Y[/math] is defined by {\rm Cov}[math](X,Y) = E((X - E(X))(Y - E(Y)))[/math]. Note that {\rm Cov}[math](X,X) = V(X)[/math]. Show that, if [math]X[/math] and [math]Y[/math] are independent, then {\rm Cov}[math](X,Y) = 0[/math]; and show, by an example, that we can have {\rm Cov}[math](X,Y) = 0[/math] and [math]X[/math] and [math]Y[/math] not independent.