# Mathematical Statistics Lesson of the Day – An Example of An Ancillary Statistic

Consider 2 random variables, $X_1$ and $X_2$, from the normal distribution $\text{Normal}(\mu, \sigma^2)$, where $\mu$ is unknown.  Then the statistic

$D = X_1 - X_2$

has the distribution

$\text{Normal}(0, 2\sigma^2)$.

The distribution of $D$ does not depend on $\mu$, so $D$ is an ancillary statistic for $\mu$.

Note that, if $\sigma^2$ is unknown, then $D$ is not ancillary for $\sigma^2$.

### 3 Responses to Mathematical Statistics Lesson of the Day – An Example of An Ancillary Statistic

1. msanregret says:

Will you be giving any examples on real life applications using ancillary statistics? Also, how are you able to write the notation you are using? (I’m curious for my own writing)

• Hi Mitchell,

1) I don’t know of real-life applications of ancillary statistics, so I don’t plan on writing about them. The combination of sufficient statistics and ancillary statistics are helpful to understand complete statistics, and that is my goal for talking about them.

2) This particular theme of WordPress (The Enterprise Theme) allows me to write LaTeX in my blog posts. If you want to use WordPress.com to write a blog, choose your theme carefully – not all themes allow LaTeX.