The following formulas are useful for derivations involving Gaussian integrals.
In three dimensions
\[\delta(\vec{x}) = \frac{1}{(2\pi)^3} \int\limits_{-\infty}^{\infty} d^3k \ e^{i\vec{k}\cdot \vec{x}}.\]The following property of the delta function is also very useful
\[\delta(y-x) = \delta(x'-x+y-x')\\ = \sum_{n=0}^{\infty} \frac{(y-x')^n}{n!} \left( \frac{\partial}{\partial x'} \right) \delta(x'-x) \\ = \sum_{n=0}^{\infty} \frac{(y-x')^n}{n!} \left( - \frac{\partial}{\partial x} \right) \delta(x'-x)\]The following notation is useful for the case of the Bivariate Gaussian distribution
\[\vec{\mu} = \begin{bmatrix} \mu_1 \\ \mu_2 \end{bmatrix}, \\ \Sigma = \begin{bmatrix} \sigma^2_1 & \rho \sigma_1 \sigma_2 \\ \rho \sigma_1 \sigma_2 & \sigma_2^2 \end{bmatrix}\\ \vec{x} = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}, \\\]where \(\vec{\mu}\) is the vector of mean values, \(\vec{x}\) is the vector of variables and \(\Sigma\) is the covariance matrix. The joint probability distribution of \(x_1\) and \(x_2\) is
\[P(x_1,x_2) = \frac{1}{\sqrt{(2\pi)^2|\Sigma|}} \text{Exp}\left[ \left(\vec{\mu}-\vec{x}\right)^{T}\Sigma^{-1} \left(\vec{\mu}-\vec{x} \right) \right]\]Here we look at the conditional probability distribution of the multivariate Gaussian distributions. For the bivariate Gaussian the conditional probability is
\[P(x_1| x_2) = \frac{1}{\sqrt{2\pi(1-\rho^2)\sigma^2_1}} \text{Exp}\left( -\frac{1}{2(1-\rho^2)\sigma^2_1}\left(x_1 -\mu_1 - \frac{\sigma_1}{\sigma_2}\rho(x_2-\mu_2) \right)^2 \right),\]while the general case is
\[P(\vec{x} |\vec{y}) = \frac{1}{\sqrt{(2\pi)^2|\Sigma|}} \text{Exp}\left[ \left(\hat{\mu}-\vec{x}\right)^{T}\hat{\Sigma}^{-1} \left(\hat{\mu}-\vec{x} \right) \right]\]where we have that
\[\hat{\mu} = \vec{\mu}_1+\Sigma_{12}\Sigma^{-1}_{22}(\vec{y}-\vec{\mu}_2)\]and
\[\hat{\Sigma} = {\bf \Sigma}_{11} - {\bf \Sigma}_{12}{\bf \Sigma}^{-1}_{22}{\bf \Sigma}_{21}\]with
\[\hat{\mu} = \begin{bmatrix} \mu_1 \\ \mu_2 \end{bmatrix}, \\ {\bf \Sigma} = \begin{bmatrix} {\bf \Sigma_{11}} & {\bf \Sigma_{12}} \\ {\bf \Sigma_{21}} & {\bf \Sigma_{22}} \end{bmatrix}\\ \hat{x} = \begin{bmatrix} \vec{x} \\ \vec{y} \end{bmatrix},\]