Math 6010 • Linear Models

Let \(X\) and \(Y\) be two standard normal random variables (defined on
a common probability space and having positive variances).
\begin{enumerate}
\item[(a)] Prove that they are independent if and only if they are jointly
normal (i.e. \((X,Y)\) is a two-dimensional normal random variable) and
uncorrelated.
\item[(b)] Give an example of two standard normal random variables that are
uncorrelated but not jointly normal.
\end{enumerate}