# Segment 17. The Multivariate Normal Distribution

#### Watch this segment

(Don't worry, what you see statically below is not the beginning of the segment. Press the play button to start at the beginning.)

The direct YouTube link is http://youtu.be/t7Z1a_BOkN4

Links to the slides: PDF file or PowerPoint file

### Problems

#### To Calculate

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

**Failed to parse (unknown error): y_1 = x_1/x_2, \qquad y_2 = x_2^2**

2. Consider the 3-dimensional multivariate normal over **Failed to parse (unknown error): (x_1,x_2,x_3)**
with **Failed to parse (unknown error): \mu = (-1,-1,-1)**
and

**Failed to parse (unknown error): \Sigma^{-1} = \left( \begin{array}{ccc} 5 & -1 & 2 \\ -1 & 8 & 1 \\ 2 & 1 & 4 \end{array} \right)**
. (Note the matrix inverse notation.)

What are 2-dimensional **Failed to parse (unknown error): \mu**
and **Failed to parse (unknown error): \Sigma^{-1}**
for

(a) the distribution on the slice **Failed to parse (unknown error): x_3=0**
?

(b) the marginalization over **Failed to parse (unknown error): x_3**
?

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

#### 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.)

#### Class Activity

Bill's Mathematica notebook for problem 2 (above). (Download file, rename as MultivarGaussExample.nb, then open in Mathematica.)