= Solution
If $\rho=0$, the <bivariate normal density> in part (a) factorizes as
$$
f(x_1,x_2)=
\frac{e^{-(x_1-\mu_1)^2/(2\sigma_1^2)}}{\sqrt{2\pi}\sigma_1}
\frac{e^{-(x_2-\mu_2)^2/(2\sigma_2^2)}}{\sqrt{2\pi}\sigma_2}
=f_1(x_1)f_2(x_2).
$$
Integrating over any product of measurable sets proves <independence of random variables>. Conversely, independent variables with finite second moments have $\mathbb E[X_1X_2]=\mathbb E[X_1]\mathbb E[X_2]$, so their <covariance> is zero and $\rho=\operatorname{Cov}(X_1,X_2)/(\sigma_1\sigma_2)=0$. This proves <independence of uncorrelated jointly normal variables>; zero correlation alone would not imply independence without the joint normal assumption.
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