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''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| "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|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|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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Cramér,   "Mathematical methods of statistics" , Princeton Univ. Press  (1946)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">   "Student"  (W.S. Gosset),   "The probable error of a mean"  ''Biometrika'' , '''6'''  (1908)  pp. 1–25</TD></TR></table>