# Segment 17 Sanmit Narvekar

Jump to navigation Jump to search

## Segment 17

#### To Calculate

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

$\displaystyle y_1 = x_1/x_2, \qquad y_2 = x_2^2$

The Jacobian matrix for the transformation above is:

$\displaystyle J = \left[ \begin{array}{cc} \frac{\partial y_1}{\partial x_1} & \frac{\partial y_1}{\partial x_2} \\ \frac{\partial y_2}{\partial x_1} & \frac{\partial y_2}{\partial x_2} \end{array} \right]$

Computing the respective entries gives:

$\displaystyle J = \left[ \begin{array}{cc} \frac{1}{x_2} & - \frac{x_1}{x_2^2} \\ 0 & 2x_2 \end{array} \right]$

And finally we take the determinant to get the Jacobian determinant:

$\displaystyle |J| = \left(\frac{1}{x_2}\right) (2x_2) - (0)\left(-\frac{x_1}{x_2^2}\right) = 2$

2. Consider the 3-dimensional multivariate normal over $\displaystyle (x_1,x_2,x_3)$ with $\displaystyle \mu = (-1,-1,-1)$ and

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

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

(a) the distribution on the slice $\displaystyle x_3=0$ ?

(b) the marginalization over $\displaystyle x_3$ ?

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.)