Let $X=(X_1,X_2,X_3)'\in N(\mu,\varLambda)$, where
$$\mu=\begin{pmatrix} 0\\ 0\\ 0 \end{pmatrix},\ \ \ \ \ \ \varLambda=\begin{pmatrix} a&a&a\\a&b&b\\a&b&c \end{pmatrix}, \ \ c>b>a>0.$$
How do I find the conditional probability distribution of $X_2$ given $X_3=z$?
Let $f_{X_1, X_2, X_3} (x_1, x_2, x_3)$ represent the joint probability distribution of $X_1, X_2$ and $X_3$; essentially, $f_{X_1, X_2, X_3}\sim N(\mu, \varLambda )$ and $f_{X_1, X_2, X_3} (x_1, x_2, x_3)$ is the value of $N(\mu, \varLambda)$ at $(X_1, X_2, X_3 ) = (x_1, x_2, x_3)$.
Then
$$f_{X_2 | X_3}(X_2 = x | X_3 = z) = \frac{ \int \limits_{t = -\infty}^{+\infty} f_{X_1, X_2, X_3} (t, x, z) dt}{ \int \limits_{h = -\infty}^{+\infty} \int \limits_{g = -\infty}^{+\infty} f_{X_1, X_2, X_3} (h, g, z) \, dh \, dg } $$