(Rene) 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

Vice versa we have:


The determinant is 2.


2. Consider the 3-dimensional multivariate normal over with and

. (Note the matrix inverse notation.)

What are 2-dimensional and for

(a) the distribution on the slice ?

(b) the marginalization over ?

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

a) We have that . Here,  is computed using Mathematica,


Since . Here, we have that


Since the expectation of is zero, we can solve a system of equations to find the expectation and . Furthermore, the new covariance matrix is simply,


where is the 2 by 2 upper left part of the inverse of the covariance matrix .

b) Let , =

Then the exponent in the multivariate normal distribution can be written as,


       
       
       

Marginalizing over is the same a marginalizing over . Hence,


Consequently, the mean stays unaltered, and the inverse of the covariance matrix is


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