# Segment 17. The Multivariate Normal Distribution

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### Problems

#### To Calculate

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

<math>y_1 = x_1/x_2, \qquad y_2 = x_2^2</math>

2. Consider the 3-dimensional multivariate normal over <math>(x_1,x_2,x_3)</math> with <math>\mu = (-1,-1,-1)</math> and

<math>\Sigma^{-1} = \left( \begin{array}{ccc}

5 & -1 & 2 \\ -1 & 8 & 1 \\ 2 & 1 & 4

\end{array} \right)</math>. (Note the matrix inverse notation.)

What are 2-dimensional <math>\mu</math> and <math>\Sigma^{-1}</math> for

(a) the distribution on the slice <math>x_3=0</math>?

(b) the marginalization over <math>x_3</math>?

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.)