# 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 $\sigma^2$.