# Multivariate Normal: Conditional Density Derivation

We derive the classical result: what is the density of a multivariate normal conditioned on some proper subset of its components?

That is, if

where

then we want to characterize

# Preliminary Results

We’ll want a couple of preliminary results before establishing the primary result.

#### Marginal Distribution

Using the moment generating function, it is easy to show that the marginal distribution of is

#### Block Matrix Inverse

The key piece of the derivation relies on being able to compute the inverse of a partitioned, 2x2 matrix. To establish a formula for the inverse, use gaussian elimination.

where .

Hence,

# Conditional Distribution

Let , .

Then the quadratic term in is

where .

In particular, we can write the conditional density as

Equivalently,