Editing Student's t-distribution (section)

== Definitions ==

===Probability density function===
'''Student's {{mvar|t}}&nbsp;distribution''' has the [[probability density function]] (PDF) given by
: <math>f(t) = \frac{\Gamma\left(\frac{\nu+1}{2}\right)}{\sqrt{\pi\nu} \Gamma\left(\frac{\nu}{2}\right)} \left(1 + \frac{t^2}{\nu}\right)^{-(\nu + 1)/2},</math>
where <math>\nu</math> is the number of ''[[degrees of freedom (statistics)|degrees of freedom]]'', and <math>\Gamma</math> is the [[gamma function]]. This may also be written as
: <math>f(t) = \frac{1}{\sqrt{\nu}\,\mathrm{B}\left(\frac{1}{2}, \frac{\nu}{2}\right)} \left(1 + \frac{t^2}{\nu}\right)^{-(\nu+1)/2},</math>
where <math>\mathrm{B}</math> is the [[beta function]]. In particular for integer valued degrees of freedom <math>\nu</math> we have:

For <math>\nu > 1</math> and even,
: <math>\frac{\Gamma\left(\frac{\nu + 1}{2}\right)}{\sqrt{\pi\nu}\, \Gamma\left(\frac{\nu}{2}\right)} = \frac{1}{2\sqrt{\nu}} \cdot \frac{(\nu - 1) \cdot (\nu - 3) \cdots 5 \cdot 3}{(\nu - 2) \cdot (\nu - 4) \cdots 4 \cdot 2}.</math>

For <math>\nu > 1</math> and odd,
: <math>\frac{\Gamma\left(\frac{\nu + 1}{2}\right)}{\sqrt{\pi\nu}\, \Gamma\left(\frac{\nu}{2}\right)} = \frac{1}{\pi \sqrt{\nu}} \cdot \frac{(\nu - 1) \cdot (\nu - 3) \cdots 4 \cdot 2}{(\nu - 2) \cdot (\nu - 4) \cdots 5 \cdot 3}.</math>

The probability density function is [[Symmetric distribution|symmetric]], and its overall shape resembles the bell shape of a normally distributed variable with mean 0 and variance 1, except that it is a bit lower and wider. As the number of degrees of freedom grows, the {{mvar|t}}&nbsp;distribution approaches the normal distribution with mean 0 and variance 1. For this reason <math>{\nu}</math> is also known as the normality parameter.<ref>{{cite book |last=Kruschke |first=J. K. |author-link=John K. Kruschke |year=2015 |title=Doing Bayesian Data Analysis |edition=2nd |publisher=Academic Press |isbn=9780124058880 |oclc=959632184}}</ref>

The following images show the density of the {{mvar|t}}&nbsp;distribution for increasing values of <math>\nu .</math> The normal distribution is shown as a blue line for comparison. Note that the {{mvar|t}}&nbsp;distribution (red line) becomes closer to the normal distribution as <math>\nu</math> increases.

{{Multiple image
| align = center
| caption_align = center
| header = Density of the {{mvar|t}}&nbsp;distribution (red) for 1, 2, 3, 5, 10, and 30 degrees of freedom compared to the standard normal distribution (blue).<br>Previous plots shown in green.
| perrow = 3
| image1 = T distribution 1df enhanced.svg
| image2 = T distribution 2df enhanced.svg
| image3 = T distribution 3df enhanced.svg
| image4 = T distribution 5df enhanced.svg
| image5 = T distribution 10df enhanced.svg
| image6 = T distribution 30df enhanced.svg
| caption1 = 1 degree of freedom
| caption2 = 2 degrees of freedom
| caption3 = 3 degrees of freedom
| caption4 = 5 degrees of freedom
| caption5 = 10 degrees of freedom
| caption6 = 30 degrees of freedom
}}

===Cumulative distribution function===
The [[cumulative distribution function]] (CDF) can be written in terms of {{mvar|I}}, the regularized
[[incomplete beta function]]. For {{nobr|{{math| ''t'' > 0}} ,}}

:<math>F(t) = \int_{-\infty}^t\ f(u)\ \operatorname{d}u ~=~ 1 - \frac{1}{2} I_{x(t)}\!\left( \frac{\ \nu\ }{ 2 },\ \frac{\ 1\ }{ 2 } \right)\ ,</math>

where

:<math>x(t) = \frac{ \nu }{\ t^2+\nu\ } ~.</math>

Other values would be obtained by symmetry. An alternative formula, valid for <math>\ t^2 < \nu\ ,</math> is

:<math>\int_{-\infty}^t f(u)\ \operatorname{d}u ~=~ \frac{1}{2} + t\ \frac{\ \Gamma\!\left( \frac{\ \nu+1\ }{ 2 } \right)\ }{\ \sqrt{\pi\ \nu\ }\ \Gamma\!\left( \frac{ \nu }{\ 2\ }\right)\ } \ {}_{2}F_1\!\left(\ \frac{1}{2}, \frac{\ \nu+1\ }{2}\ ; \frac{ 3 }{\ 2\ }\ ;\ -\frac{~ t^2\ }{ \nu }\ \right)\ ,</math>

where <math>\ {}_{2}F_1(\ ,\ ;\ ;\ )\ </math> is a particular instance of the [[hypergeometric function]].

For information on its inverse cumulative distribution function, see {{slink|quantile function|Student's t-distribution}}.

===Special cases===
Certain values of <math>\ \nu\ </math> give a simple form for Student's t-distribution.
{| class="wikitable"
|-
! <math>\ \nu\ </math>
! PDF
! CDF
! notes
|-
! 1
| <math>\ \frac{\ 1\ }{\ \pi\ (1 + t^2)\ }\ </math>
| <math>\ \frac{\ 1\ }{ 2 } + \frac{\ 1\ }{ \pi }\ \arctan(\ t\ )\ </math>
| See [[Cauchy distribution]]
|-
! 2
| <math>\ \frac{ 1 }{\ 2\ \sqrt{2\ }\ \left(1+\frac{t^2}{2}\right)^{3/2}}\ </math>
| <math>\ \frac{ 1 }{\ 2\ }+\frac{ t }{\ 2\sqrt{2\ }\ \sqrt{ 1 + \frac{~ t^2\ }{ 2 }\ }\ }\ </math>
|
|-
! 3
| <math>\ \frac{ 2 }{\ \pi\ \sqrt{3\ }\ \left(\ 1 + \frac{~ t^2\ }{ 3 }\ \right)^2\ }\ </math>
| <math>\ \frac{\ 1\ }{ 2 } + \frac{\ 1\ }{ \pi }\ \left[ \frac{ \left(\ \frac{ t }{\ \sqrt{3\ }\ }\ \right) }{ \left(\ 1 + \frac{~ t^2\ }{ 3 }\ \right) } + \arctan\left(\ \frac{ t }{\ \sqrt{3\ }\ }\ \right)\ \right]\ </math>
|
|-
! 4
| <math>\ \frac{\ 3\ }{\ 8\ \left(\ 1 + \frac{~ t^2\ }{ 4 }\ \right)^{5/2}}\ </math>
| <math>\ \frac{\ 1\ }{ 2 } + \frac{\ 3\ }{ 8 } \left[\ \frac{ t }{\ \sqrt{ 1 + \frac{~ t^2\ }{ 4 } ~}\ } \right] \left[\ 1 - \frac{~ t^2\ }{\ 12\ \left(\ 1 + \frac{~ t^2\ }{ 4 }\ \right)\ }\ \right]\ </math>
|
|-
! 5
| <math>\ \frac{ 8 }{\ 3 \pi \sqrt{5\ }\left(1+\frac{\ t^2\ }{ 5 }\right)^3\ }\ </math>
| <math>\ \frac{\ 1\ }{ 2 } + \frac{\ 1\ }{\pi}{ \left[ \frac{ t }{\ \sqrt{5\ }\left(1 + \frac{\ t^2\ }{ 5 }\right)\ } \left(1 + \frac{ 2 }{\ 3 \left(1 + \frac{\ t^2\ }{ 5 }\right)\ }\right) + \arctan\left( \frac{ t }{\ \sqrt{\ 5\ }\ } \right)\right]}\ </math>
|
|-
! <math>\ \infty\ </math>
| <math>\ \frac{ 1 }{\ \sqrt{2 \pi\ }\ }\ e^{-t^2/2}</math>
| <math>\ \frac{\ 1\ }{ 2 }\ {\left[ 1 + \operatorname{erf}\left( \frac{ t }{\ \sqrt{2\ }\ } \right) \right]}\ </math>
| See ''[[Normal distribution]]'', ''[[Error function]]''
|}