View source for Segment 17. The Multivariate Normal Distribution

====Watch this segment====
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The direct YouTube link is [http://youtu.be/t7Z1a_BOkN4 http://youtu.be/t7Z1a_BOkN4]

Links to the slides: [http://wpressutexas.net/coursefiles/17.MultivariateNormal.pdf PDF file] or [http://wpressutexas.net/coursefiles/17.MultivariateNormal.ppt 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.)

====Class Activity====

[[Media:somematrices.txt|Some 3x3 Matrices]]

[http://wpressutexas.net/coursewiki/images/f/f4/Multivar_normal.pdf MVN Exercise]

[[Media:MultivarGaussExample.nb.txt|Bill's Mathematica notebook for problem 2 (above)]].  (Download file, rename as MultivarGaussExample.nb, then open in Mathematica.)