Segment 17. The Multivariate Normal Distribution

From Computational Statistics (CSE383M and CS395T)
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Links to the slides: PDF file or PowerPoint file

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.)