Conditional distribution of a bivariate normal variable
= Conditional distribution of a bivariate normal variable
Conditionally on $X_2=x_2$,
$$
X_1\sim N\left(
\mu_1+\rho\frac{\sigma_1}{\sigma_2}(x_2-\mu_2),
\sigma_1^2(1-\rho^2)\right).
$$
= Conditional distribution of a bivariate normal variable
Conditionally on $X_2=x_2$,
$$
X_1\sim N\left(
\mu_1+\rho\frac{\sigma_1}{\sigma_2}(x_2-\mu_2),
\sigma_1^2(1-\rho^2)\right).
$$