# Definition:Student's t-Distribution

## Definition

Let $X$ be a continuous random variable on a probability space $\struct {\Omega, \Sigma, \Pr}$.

Let $\Img X = \R$.

$X$ is said to have a **Student's $t$-distribution with $k$ degrees of freedom** if and only if it has probability density function:

- $\map {f_X} x = \dfrac {\map \Gamma {\frac {k + 1} 2} } {\sqrt {\pi k} \map \Gamma {\frac k 2} } \paren {1 + \dfrac {x^2} k}^{-\frac {k + 1} 2}$

for some $k \in \R_{>0}$.

This is written:

- $X \sim \StudentT k$

### Degrees of Freedom

The parameter $k$ is the **number of degrees of freedom** of $\StudentT k$.

## Also see

- Results about
**Student's $t$-distribution**can be found**here**.

## Source of Name

This entry was named for William Sealy Gosset.

## Historical Note

William Sealy Gosset's employer, Guinness, had previously had trade secrets disclosed within academic papers. Because of this, they disallowed entirely their employees from publishing academic papers, irrespective of their content.

However, after much convincing that his results regarding the $t$-distribution were of high mathematical importance, and that they were of no direct commercial use to rival breweries, he was allowed to publish.

To avoid the attention of other employees, Guinness allowed Gosset to publish under his pen name **Student**.

## Technical Note

The $\LaTeX$ code for \(\StudentT {k}\) is `\StudentT {k}`

.

When the argument is a single character, it is usual to omit the braces:

`\StudentT k`

## Sources

- 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**Student's t-distribution** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**t-distribution** - 2021: Richard Earl and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(6th ed.) ... (previous) ... (next): Appendix $15$: Probability distributions

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- 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 39$: Probability Distributions: Student's $t$ Distribution: $39.5$