Segment 17. The Multivariate Normal Distribution

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Links to the slides: PDF file or PowerPoint file


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

Some 3x3 Matrices

MVN Exercise

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