# (Rene) Segment 17: The multivariate normal distribution

### 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

#### To Think About

1. Prove the assertions in slide 5. That is, implement the ideas in the blue text.

2. How would you plot an error ellipsoid in 3 dimensions? That is, what would be the 3-dimensional version of the code in slide 8? (You can assume the plotting capabilities of your favorite programming language.)