# Student distribution

with $f$ degrees of freedom, $t$- distribution

The probability distribution of the random variable

$$t _ {f} = \frac{U}{\sqrt {\chi _ {f} ^ {2} / f } } ,$$

where $U$ is a random variable subject to the standard normal law $N( 0, 1)$ and $\chi _ {f} ^ {2}$ is a random variable not depending on $U$ and subject to the "chi-squared" distribution with $f$ degrees of freedom. The distribution function of the random variable $t _ {f}$ is expressed by the formula

$${\mathsf P} \{ t _ {f} \leq x \} = S _ {f} ( x) =$$

$$= \ \frac{1}{\sqrt {\pi _ {f} } } \frac{\Gamma ( ( f+ 1 ) / 2 ) }{\Gamma ( f / 2 ) } \int\limits _ {- \infty } ^ { x } \left ( 1 + \frac{u ^ {2} }{f} \right ) ^ {- ( f+ 1 ) / 2 } du,\ | x | < \infty .$$

In particular, if $f= 1$, then

$$S _ {1} ( x) = \frac{1}{2} + \frac{1} \pi \mathop{\rm arctan} x$$

is the distribution function of the Cauchy distribution. The probability density of the Student distribution is symmetric about 0, therefore

$$S _ {f} ( t) + S _ {f} (- t) = 1 \ \textrm{ for any } t \in \mathbf R ^ {1} .$$

The moments $\mu _ {r} = {\mathsf E} t _ {f} ^ {r}$ of a Student distribution exist only for $r < f$, the odd moments are equal to 0, and, in particular ${\mathsf E} t _ {f} = 0$. The even moments of a Student distribution are expressed by the formula

$$\mu _ {2r} = f ^ { r } \frac{\Gamma ( ( r + 1 ) / 2 ) \Gamma ( f / 2 - r ) }{\sqrt \pi \Gamma ( f / 2 ) } ,\ \ 2 \leq 2r < f ;$$

in particular, $\mu _ {2} = {\mathsf D} \{ t _ {f} \} = f/( f- 2)$. The distribution function $S _ {f} ( x)$ of the random variable $t _ {f}$ is expressed in terms of the beta-distribution function in the following way:

$$S _ {f} ( x) = 1 - \frac{1}{2} I _ {f/( f+ x ^ {2} ) } \left ( \frac{f}{2} , \frac{1}{2} \right ) ,$$

where $I _ {z} ( a, b)$ is the incomplete beta-function, $0 \leq z \leq 1$. If $f \rightarrow \infty$, then the Student distribution converges to the standard normal law, i.e.

$$\lim\limits _ {f\rightarrow \infty } S _ {f} ( x) = \ \Phi ( x) = \ \frac{1}{\sqrt {2 \pi } } \int\limits _ {- \infty } ^ { x } e ^ {- t ^ {2} /2 } dt.$$

Example. Let $X _ {1} \dots X _ {n}$ be independent, identically, normally $N( a, \sigma ^ {2} )$- distributed random variables, where the parameters $a$ and $\sigma ^ {2}$ are unknown. Then the statistics

$$\overline{X}\; = \frac{1}{n} \sum _ { i= } 1 ^ { n } X _ {i} \ \textrm{ and } \ \ s ^ {2} = \frac{1}{n-} 1 \sum _ { i= } 1 ^ { n } ( X _ {i} - \overline{X}\; ) ^ {2}$$

are the best unbiased estimators of $a$ and $\sigma ^ {2}$; here $\overline{X}\;$ and $s ^ {2}$ are stochastically independent. Since the random variable $\sqrt n ( \overline{X}\; - a)/ \sigma$ is subject to the standard normal law, while

$$n- \frac{1}{\sigma ^ {2} } s ^ {2} = \chi _ {n-} 1 ^ {2}$$

is distributed according to the "chi-squared" law with $f= n- 1$ degrees of freedom, then by virtue of their independence, the fraction

$$\frac{\sqrt n ( \overline{X}\; - a) / \sigma }{\sqrt {\chi _ {n-} 1 ^ {2} / ( n- 1) } } = \frac{\sqrt n ( \overline{X}\; - a) }{s}$$

is subject to the Student distribution with $f= n- 1$ degrees of freedom. Let $t _ {f} ( P)$ and $t _ {f} ( 1- P) = - t _ {f} ( P)$ be the solutions of the equations

$$S _ {n-} 1 \left ( \frac{\sqrt n ( \overline{X}\; - a) }{s} \right ) = \ \left \{ \begin{array}{ll} P, & 0.5 < P < 1, \\ 1- P, & f = n- 1. \\ \end{array}$$

Then the statistics $\overline{X}\; - ( s/ \sqrt n ) t _ {f} ( P)$ and $\overline{X}\; + ( s/ \sqrt n ) t _ {f} ( P)$ are the lower and upper bounds of the confidence set for the unknown mathematical expectation $a$ of the normal law $N( a, \sigma ^ {2} )$, and the confidence coefficient of this confidence set is equal to $2P- 1$, i.e.

$${\mathsf P} \left \{ \overline{X}\; - \frac{s}{\sqrt n } t _ {f} ( P) < a < \overline{X}\; + \frac{s}{\sqrt n } t _ {f} ( P) \right \} = 2P- 1.$$

The Student distribution was first used by W.S. Gosset (pseudonym Student).

How to Cite This Entry:
Student distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Student_distribution&oldid=49611
This article was adapted from an original article by M.S. Nikulin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article