Editing Student's t-distribution (section)

{{short description|Probability distribution}}
{{About|the mathematics of Student's {{mvar|t}}-distribution|its uses in statistics|Student's t-test}}
{{Infobox probability distribution
| name       = Student's {{mvar|t}}
| type       = density
| pdf_image  = [[File:student t pdf.svg|325px]]
| cdf_image  = [[File:student t cdf.svg|325px]]
| parameters = <math>\nu > 0</math> [[Degrees of freedom (statistics)|degrees of freedom]] ([[Real number|real]], almost always a positive [[integer]])
| support    = <math>x \in (-\infty, \infty)</math>
| pdf        = <math>\frac{\Gamma \left(\frac{\nu + 1}{2}\right)}{\sqrt{\pi\nu}\, \Gamma\left(\frac{\nu}{2}\right)} \left(1 + \frac{x^2}{\nu}\right)^{-\frac{\nu+1}{2}}</math>
| cdf        = <math>\begin{align}
  & \frac{1}{2} + x \Gamma\left(\frac{\nu + 1}{2}\right) \times \\
  &\quad \frac{{}_{2}F_1\!\left(\frac{1}{2}, \frac{\nu + 1}{2}; \frac{3}{2}; -\frac{x^2}{\nu}\right)}
              {\sqrt{\pi\nu}\, \Gamma\left(\frac{\nu}{2}\right)},
 \end{align}</math>
 where <math>{}_{2}F_1</math> is the [[hypergeometric function]]
| mean       = <math>0</math> for <math>\nu > 1,</math> otherwise [[indeterminate form|undefined]]
| median     = <math>0</math>
| mode       = <math>0</math>
| variance   = <math>\frac{\nu}{\nu  -2}</math> for <math>\nu > 2,</math> {{math|∞}} for <math>1 < \nu \le 2,</math> otherwise [[indeterminate form|undefined]]
| skewness   = <math>0</math> for <math>\ \nu > 3\ ,</math> otherwise [[indeterminate form|undefined]]
| kurtosis   = <math>\frac{6}{\nu - 4}</math> for <math>\nu > 4,</math> ∞ for <math>2 < \nu \le 4,</math> otherwise [[indeterminate form|undefined]]
| entropy    = <math>\begin{align}
  & \frac{\nu + 1}{2} \left[\psi\left(\frac{\nu + 1}{2}\right) -
                            \psi\left(\frac{\nu}{2}\right)\right] \\
  &\quad + \ln\left[\sqrt{\nu}\, \mathrm{B}\left(\frac{\nu}{2}, \frac{1}{2}\right)\right]~\text{(nats)},
\end{align}</math><br/>
where
: <math>\psi</math> is the [[digamma function]],
: <math>\mathrm{B}</math> is the [[beta function]]
| mgf        = undefined
| char       = <math>\frac{\big(\sqrt{\nu}\, |t|\big)^{\nu/2}\, K_{\nu/2}\big(\sqrt{\nu}\, |t|\big)}{\Gamma(\nu/2)\, 2^{\nu/2-1}}</math> for <math>\nu > 0</math>,<br/>
where <math>K_\nu</math> is the [[Bessel function|modified Bessel function of the second kind]]<ref>{{cite web |last=Hurst |first=Simon |title=The characteristic function of the Student {{mvar|t}} distribution |series=Financial Mathematics Research Report |volume=No.&nbsp;FMRR006-95 |id=Statistics Research Report No.&nbsp;SRR044-95 |url=http://wwwmaths.anu.edu.au/research.reports/srr/95/044/ |url-status=dead |archive-url=https://web.archive.org/web/20100218072259/http://wwwmaths.anu.edu.au/research.reports/srr/95/044/ |archive-date=February 18, 2010 }}</ref>
| ES         = <math>\mu + s\left(\frac{\big(\nu + [T^{-1}(1 - p)]^2\big) \times \tau\big(T^{-1}(1 - p)\big)}{(\nu - 1)(1 - p)}\right),</math>
where <math>T^{-1}</math> is the inverse standardized Student&nbsp;{{mvar|t}} [[cumulative distribution function|CDF]], and <math>\tau</math> is the standardized Student&nbsp;t [[probability density function|PDF]].<ref name=norton>{{cite journal |last1=Norton |first1=Matthew |last2=Khokhlov |first2=Valentyn |last3=Uryasev |first3=Stan |year=2019 |title=Calculating CVaR and bPOE for common probability distributions with application to portfolio optimization and density estimation |journal=[[Annals of Operations Research]] |volume=299 |issue=1–2 |pages=1281–1315 |publisher=Springer|doi=10.1007/s10479-019-03373-1 |arxiv=1811.11301 |s2cid=254231768 |url=http://uryasev.ams.stonybrook.edu/wp-content/uploads/2019/10/Norton2019_CVaR_bPOE.pdf |access-date=2023-02-27}}</ref>
}}

In [[probability]] theory and [[statistics]], '''Student's {{mvar|t}}&nbsp;distribution''' (or simply the '''{{mvar|t}}&nbsp;distribution''') <math>t_\nu </math> is a continuous [[probability distribution]] that generalizes the [[Normal distribution#Standard normal distribution|standard normal distribution]]. Like the latter, it is symmetric around zero and bell-shaped.

However, <math>t_\nu</math> has [[Heavy-tailed distribution|heavier tails]], and the amount of probability mass in the tails is controlled by the parameter <math>\nu</math>. For <math>\nu = 1</math> the Student's {{mvar|t}} distribution <math>t_\nu</math> becomes the standard [[Cauchy distribution]], which has very [[fat-tailed distribution|"fat" tails]]; whereas for <math>\nu \to \infty</math> it becomes the standard normal distribution <math>\mathcal{N}(0, 1),</math> which has very "thin" tails.

The name "Student" is a pseudonym used by [[William Sealy Gosset]] in his scientific paper publications during his work at the [[Guinness Brewery]] in [[Dublin, Ireland]].

The Student's {{mvar|t}}&nbsp;distribution plays a role in a number of widely used statistical analyses, including [[Student's t-test|Student's {{mvar|t}}-test]] for assessing the [[statistical significance]] of the difference between two sample means, the construction of [[confidence interval]]s for the difference between two population means, and in linear [[regression analysis]].

In the form of the ''location-scale {{mvar|t}}&nbsp;distribution'' <math>\operatorname{\ell st}(\mu, \tau^2, \nu)</math> it generalizes the [[normal distribution]] and also arises in the [[Bayesian analysis]] of data from a normal family as a [[Compound probability distribution|compound distribution]] when marginalizing over the variance parameter.