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: $39.5$