# Segment 19 Sanmit Narvekar

## Segment 19

#### To Calculate

1. Prove the assertion on lecture slide 5, namely that, for a multivariate normal distribution, the quantity $\displaystyle ({\mathbf x-\mathbf\mu})^T{\mathbf\Sigma}^{-1}({\mathbf x-\mathbf\mu})$ , where $\displaystyle \mathbf x$ is a random draw from the multivariate normal, is $\displaystyle \chi^2$ distributed.

First, we define (as before):

$\displaystyle \Sigma = L L^T$

$\displaystyle Ly = x - \mu$

Then:

$\displaystyle ({\mathbf x-\mathbf\mu})^T{\mathbf\Sigma}^{-1}({\mathbf x-\mathbf\mu})$

By making the appropriate substitutions

$\displaystyle = (Ly)^T (L L^T)^{-1} (Ly)$

Then we expand the transpose and inverse:

$\displaystyle = y^T L^T (L^T)^{-1} L^{-1} L y$

By regrouping, we can see that the terms involving L and L^T cancel with their inverses, giving:

$\displaystyle = y^T y = \sum_i y_i^2$

These are essentially t-values when mu is 0 and sigma is 1. Thus, it is chi-square distributed.

2. Suppose you measure a bunch of quantities $\displaystyle x_i$ , each of which is measured with a measurement accuracy $\displaystyle \sigma_i$ and has a theoretically expected value $\displaystyle \mu_i$ . Describe in detail how you might use a chi-square test statistic as a p-value test to see if your theory is viable? Should your test be 1 or 2 tailed?