Editing Student's t-distribution (section)

==={{anchor|Three-parameter version|location-scale}}Location-scale {{mvar|t}}&nbsp;distribution===
====Location-scale transformation====
Student's {{mvar|t}}&nbsp;distribution generalizes to the three parameter ''location-scale {{mvar|t}}&nbsp;distribution'' <math>\operatorname{\ell st}(\mu,\ \tau^2,\ \nu)\ </math> by introducing a [[location parameter]] <math>\ \mu\ </math> and a [[scale parameter]] <math>\ \tau ~.</math> With
:<math>\ T \sim t_\nu\ </math>
and [[location-scale family]] transformation
:<math>\ X = \mu + \tau\ T\ </math>
we get
:<math>\ X \sim \operatorname{\ell st}(\mu,\ \tau^2,\ \nu) ~.</math>

The resulting distribution is also called the ''non-standardized Student's {{mvar|t}}&nbsp;distribution''.

====Density and first two moments====
The location-scale {{mvar|t}} distribution has a density defined by:<ref name="Jackman">{{cite book |title=Bayesian Analysis for the Social Sciences |url=https://archive.org/details/bayesianmodeling00jack |url-access=limited |author=Jackman, S. |series=Wiley Series in Probability and Statistics |publisher=Wiley |year=2009 |isbn=9780470011546 |page=[https://archive.org/details/bayesianmodeling00jack/page/n542 507] |doi=10.1002/9780470686621}}</ref>

:<math>p(x\mid \nu,\mu,\tau) = \frac{\Gamma \left(\frac{\nu + 1}{2} \right)}{\Gamma\left( \frac{\nu}{2}\right) \tau \sqrt{\pi \nu}} \left(1 + \frac{1}{\nu} \left(\frac{x-\mu}{\tau} \right)^2 \right)^{-(\nu+1)/2}</math>

Equivalently, the density can be written in terms of <math>\tau^2</math>:

:<math>\ p(x \mid \nu, \mu, \tau^2) = \frac{\Gamma( \frac{\nu + 1}{2})}{\Gamma\left(\frac{\nu}{2}\right)\sqrt{\pi \nu \tau^2}} \left(1 + \frac{1}{ \nu } \frac{(x - \mu)^2}{\tau^2} \right)^{-(\nu+1)/2}</math>

Other properties of this version of the distribution are:<ref name=Jackman/>

:<math>\begin{align}
\operatorname{\mathbb E}\{\ X\ \} &= \mu & \text{ for } \nu > 1\ ,\\
\operatorname{var}\{\ X\ \} &= \tau^2\frac{\nu}{\nu-2} & \text{ for } \nu > 2\ ,\\
\operatorname{mode}\{\ X\ \} &= \mu ~.
\end{align} </math>

====Special cases====
* If <math>\ X\ </math> follows a location-scale {{mvar|t}}&nbsp;distribution <math>\ X \sim \operatorname{\ell st}\left(\mu,\ \tau^2,\ \nu\right)\ </math> then for <math>\ \nu \rightarrow \infty\ </math> <math>\ X\ </math> is normally distributed <math>X \sim \mathrm{N}\left(\mu, \tau^2\right)</math> with mean <math>\mu</math> and variance <math>\ \tau^2 ~.</math>
* The location-scale {{mvar|t}}&nbsp;distribution <math>\ \operatorname{\ell st}\left(\mu,\ \tau^2,\ \nu=1 \right)\ </math> with degree of freedom <math>\nu=1</math> is equivalent to the [[Cauchy distribution]] <math>\mathrm{Cau}\left(\mu, \tau\right) ~.</math>
* The location-scale {{mvar|t}}&nbsp;distribution <math>\operatorname{\ell st}\left(\mu=0,\ \tau^2=1,\ \nu\right)\ </math> with <math>\mu=0</math> and <math>\ \tau^2=1\ </math> reduces to the Student's {{mvar|t}}&nbsp;distribution <math>\ t_\nu ~.</math>