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).

References

[1]	H. Cramér, "Mathematical methods of statistics" , Princeton Univ. Press (1946)
[2]	L.N. Bol'shev, N.V. Smirnov, "Tables of mathematical statistics" , Libr. math. tables , 46 , Nauka (1983) (In Russian) (Processed by L.S. Bark and E.S. Kedrova)
[3]	"Student" (W.S. Gosset), "The probable error of a mean" Biometrika , 6 (1908) pp. 1–25