Definition:Student's t-Distribution

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Definition

Student's $t$-distribution is the probability distribution of a random variable which is proportional to the ratio of:

a standard normal distribution

and:

the square root of a $\chi$-squared variable with $k$ degrees of freedom.


Let $S = \set {x_1, x_2, \ldots, x_n}$ be a sample of size $n$ from a normal distribution with expectation $\mu$.

The statistic:

$t := \dfrac {\overline x - \mu} {s / \sqrt n}$

where:

$\overline x$ is the mean of $S$
$s^2 := \ds \sum_{i \mathop = 1}^n \dfrac {\paren {x_i - \overline x}^2} {n - 1}$

has a Student's $t$-distribution with $n - 1$ degrees of freedom.


This is written:

$X \sim \StudentT k$


Degrees of Freedom

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


Also known as

Student's $t$-distribution is also known just as the $t$-distribution.


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

Student's $t$-distribution was devised by William Sealy Gosset in $1908$.

Note on Gosset's Pen Name

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 and $t$-test 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

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