# Gaussian Integrals

The following formulas are useful for derivations involving Gaussian integrals.

## Representations of Delta Function

In three dimensions

The following property of the delta function is also very useful

## Bivariate Gaussian

The following notation is useful for the case of the Bivariate Gaussian distribution

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

## Conditional Probabilities

Here we look at the conditional probability distribution of the multivariate Gaussian distributions. For the bivariate Gaussian the conditional probability is

while the general case is

where we have that

and

with