(Rene) Segment 17: The multivariate normal distribution

Contents

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 