Solution
= Solution
Completing the square in $x_1$ in the joint density shows that the <conditional distribution of a bivariate normal variable> is
$$
\boxed{
X_1\mid X_2=x_2
\sim N\left(
\mu_1+\rho\frac{\sigma_1}{\sigma_2}(x_2-\mu_2),
\sigma_1^2(1-\rho^2)\right).}
$$
Thus its conditional density is
$$
\boxed{\frac1{\sqrt{2\pi\sigma_1^2(1-\rho^2)}}
\exp\left[
-\frac{\left(x_1-\mu_1-\rho\frac{\sigma_1}{\sigma_2}
(x_2-\mu_2)\right)^2}
{2\sigma_1^2(1-\rho^2)}
\right].}
$$