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

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

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.