The bivariate normal distribution's conditional distributions are normal too.

Conditional normal distributions

If \((X,Y) \sim \NormalDistn(\mu_X, \sigma_X^2, \mu_Y, \sigma_Y^2, \rho)\) then the conditional distribution of \(Y\), given \(X=x\) is univariate normal,

\[ \NormalDistn\left(\mu_Y + \frac{\sigma_Y}{\sigma_X}\rho(x-\mu_X),\; (1-\rho^2)\sigma_Y^2\right) \]

(Proved in full version)

A similar result holds for the conditional distribution of \(X\), given that \(Y=y\).