Editing Student's t-distribution (section)

====Sampling distribution of t-statistic====
The {{mvar|t}}&nbsp;distribution arises as the sampling distribution
of the {{mvar|t}}&nbsp;statistic. Below the one-sample {{mvar|t}}&nbsp;statistic is discussed, for the corresponding two-sample {{mvar|t}}&nbsp;statistic see [[Student's t-test]].

=====Unbiased variance estimate=====
Let <math>\ x_1, \ldots, x_n \sim {\mathcal N}(\mu, \sigma^2)\ </math> be independent and identically distributed samples from a normal distribution with mean <math>\mu</math> and variance <math>\ \sigma^2 ~.</math> The sample mean and unbiased [[sample variance]] are given by:

: <math>
\begin{align}
 \bar{x} &= \frac{\ x_1+\cdots+x_n\ }{ n }\ , \\[5pt]
 s^2     &= \frac{ 1 }{\ n-1\ }\ \sum_{i=1}^n (x_i - \bar{x})^2 ~.
\end{align}
</math>

The resulting (one sample) {{mvar|t}}&nbsp;statistic is given by

: <math> t = \frac{\bar{x} - \mu}{\ s / \sqrt{n \ }\ } \sim t_{n - 1} ~.</math>

and is distributed according to a Student's {{mvar|t}}&nbsp;distribution with <math>\ n - 1\ </math> degrees of freedom.

Thus for inference purposes the {{mvar|t}}&nbsp;statistic is a useful "[[pivotal quantity]]" in the case when the mean and variance <math>(\mu, \sigma^2)</math> are unknown population parameters, in the sense that the {{mvar|t}}&nbsp;statistic has then a probability distribution that depends on neither <math>\mu</math> nor <math>\ \sigma^2 ~.</math>

=====ML variance estimate=====
Instead of the unbiased estimate <math>\ s^2\ </math> we may also use the maximum likelihood estimate
:<math>\ s^2_\mathsf{ML} = \frac{\ 1\ }{ n }\ \sum_{i=1}^n (x_i - \bar{x})^2\ </math>
yielding the statistic
: <math>\ t_\mathsf{ML} = \frac{\bar{x} - \mu}{\sqrt{s^2_\mathsf{ML}/n\ }} = \sqrt{\frac{n}{n-1}\ }\ t ~.</math>
This is distributed according to the location-scale {{mvar|t}}&nbsp;distribution:
: <math> t_\mathsf{ML} \sim \operatorname{\ell st}(0,\ \tau^2=n/(n-1),\ n-1) ~.</math>