Consider a pair of jointly normal random variables $X_{1}, X_{2}$, with mean values $\mu_{1}$, $\mu_{2}$, variances $\sigma_{1}^{2}, \sigma_{2}^{2}$ and correlation coefficient $\rho$ with $|\rho|<1$.

(a) Write down the joint probability density function for $\left(X_{1}, X_{2}\right)$.

(b) Prove that $X_{1}, X_{2}$ are independent if and only if $\rho=0$.