probability or statistics - Obtaining joint distributions and conditional distributions using Mathematica

I have two multivariate Gaussian distributions $p(x)$ and $p(z)$ with mean vectors $m_x$ and $m_z$, and covariance matrices $\Sigma_x$ and $\Sigma_z$. my model is a simple linear model $x = W z+n$ where $n$ is a noise vector with mean $0$ and diagonal covariance matrix of the form $\sigma^2 I$, where $I$ is the identity matrix.

I observe the variable $x$. now how can I calculate the joint distribution $p(x,z)$ and the conditional distributions $p(x|z)$ and $p(z|x)$?