# (Rene) Segment 17: The multivariate normal distribution

## Contents

### Problems

#### To Calculate

1. Calculate the Jacobian determinant of the transformation of variables defined by

Vice versa we have:




The determinant is 2.

2. Consider the 3-dimensional multivariate normal over with and

. (Note the matrix inverse notation.)

What are 2-dimensional and for

(a) the distribution on the slice ?

(b) the marginalization over ?

Hint: The answers are all simple rationals, but I had to use Mathematica to work them out.

a) We have that . Here,  is computed using Mathematica,




Since . Here, we have that




Since the expectation of is zero, we can solve a system of equations to find the expectation and . Furthermore, the new covariance matrix is simply,




where is the 2 by 2 upper left part of the inverse of the covariance matrix .

b) Let , =

Then the exponent in the multivariate normal distribution can be written as,





Marginalizing over is the same a marginalizing over . Hence,




Consequently, the mean stays unaltered, and the inverse of the covariance matrix is