Eleisha's Segment 17: The Multivariate Normal Distribution

To Calculate:

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

$y_{1}=x_{1}/x_{2},\qquad y_{2}=x_{2}^{2}$

2. Consider the 3-dimensional multivariate normal over $(x_{1},x_{2},x_{3})$ with $\mu =(-1,-1,-1)$ and

$\Sigma ^{-1}=\left({\begin{array}{ccc}5&-1&2\\-1&8&1\\2&1&4\end{array}}\right).$ (Note the matrix inverse notation.)

What are 2-dimensional $\mu$ and $\Sigma ^{-1}$ for

(a) the distribution on the slice$x_{3}=0$?

(b) the marginalization over $x_{3}$ ?

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.)

Back To: Eleisha Jackson