The following formulas are useful for derivations involving Gaussian integrals.
In three dimensions
The following property of the delta function is also very useful
The following notation is useful for the case of the Bivariate Gaussian distribution
where is the vector of mean values, is the vector of variables and is the covariance matrix. The joint probability distribution of and is
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