# Student t-distribution

Calculates cumulative distribution function value and probability density function value for Student t-distribution. Quantile calculator evaluates Student quantiles for given probability and specified number of degrees of freedom.

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#### Timur

Created: 2015-11-29 06:40:48, Last updated: 2021-03-03 07:43:57

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Student t-distribution arises when estimating the mean of a normally distributed population in situations where the sample size is small, and population deviation is unknown. William Sealy Gosset developed the distribution at the beginning of the XX century, who published his works under the pseudonym Student.

## Probability density function

Probability density function has the following form:
$f(t) = \frac{\Gamma(\frac{n+1}{2})} {\sqrt{n\pi}\,\Gamma(\frac{n}{2})} \left(1+\frac{t^2}{n} \right)^{\!-\frac{n+1}{2}},\!$
where n - is degrees of freedom and $\Gamma$ - Gamma function

## Cumulative distribution function

Cumulative distribution function can be expressed using Gamma and hypergeometric function:
$\tfrac{1}{2} + t\frac{\Gamma \left( \tfrac{1}{2}(n+1) \right)} {\sqrt{\pi n}\,\Gamma \left(\tfrac{n}{2}\right)} {}_2F_1 \left ( \tfrac{1}{2},\tfrac{1}{2}(n+1); \tfrac{3}{2}; -\tfrac{t^2}{n} \right)$

#### Stdent t-distribution

Digits after the decimal point: 5
Probability density function value

Cumulative distribution function value

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CDF Graph
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## Quantile function

$\alpha$-quantile Student is a number $t_{\alpha,n}$ which conforms to $F_n\left(t_{\alpha,n}\right) = 1- \alpha$, where Fn - Student-t cumulative distribution function.
Inverse cumulative distribution function (quantile function) doesn't have a simple form; commonly, we use pre-calculated values from the tables published by Gosset and other researchers.

The following calculator approximates quantile function value with the aid of the jStat statistics package:

#### Student t-distribution quantile function

Digits after the decimal point: 2
Quantile

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