= Solution
With $\sigma_1,\sigma_2>0$ and $|\rho|<1$, the <bivariate normal distribution> has <probability density function>
$$
\boxed{f(x_1,x_2)=\frac{1}{2\pi\sigma_1\sigma_2\sqrt{1-\rho^2}}
\exp\left[-\frac{1}{2(1-\rho^2)}
\left(\frac{(x_1-\mu_1)^2}{\sigma_1^2}
-\frac{2\rho(x_1-\mu_1)(x_2-\mu_2)}{\sigma_1\sigma_2}
+\frac{(x_2-\mu_2)^2}{\sigma_2^2}\right)\right].}
$$
The strict correlation bound makes its <covariance matrix> positive definite, so this is an ordinary two-dimensional density rather than a distribution supported on a line.
Back to article page
