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''with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s0907102.png" /> degrees of freedom, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s0907104.png" />-distribution''
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''with  $  f $
 +
degrees of freedom, $  t $-
 +
distribution''
  
 
The probability distribution of the random variable
 
The probability distribution of the random variable
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s0907105.png" /></td> </tr></table>
+
$$
 +
t _ {f}  =
 +
\frac{U}{\sqrt {\chi _ {f}  ^ {2} / f } }
 +
,
 +
$$
 +
 
 +
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
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s0907106.png" /> is a random variable subject to the standard normal law <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s0907107.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s0907108.png" /> is a random variable not depending on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s0907109.png" /> and subject to the [[Chi-squared distribution| "chi-squared" distribution]] with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s09071010.png" /> degrees of freedom. The distribution function of the random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s09071011.png" /> is expressed by the formula
+
$$
 +
{\mathsf P} \{ t _ {f} \leq  x \}  S _ {f} ( x) =
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s09071012.png" /></td> </tr></table>
+
$$
 +
= \
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s09071013.png" /></td> </tr></table>
+
\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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s09071014.png" />, then
+
In particular, if $  f= 1 $,  
 +
then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s09071015.png" /></td> </tr></table>
+
$$
 +
S _ {1} ( x)  =
 +
\frac{1}{2}
 +
+
 +
\frac{1} \pi
 +
  \mathop{\rm arctan}  x
 +
$$
  
 
is the distribution function of the [[Cauchy distribution|Cauchy distribution]]. The probability density of the Student distribution is symmetric about 0, therefore
 
is the distribution function of the [[Cauchy distribution|Cauchy distribution]]. The probability density of the Student distribution is symmetric about 0, therefore
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s09071016.png" /></td> </tr></table>
+
$$
 +
S _ {f} ( t) + S _ {f} (- t)  = 1
 +
\  \textrm{ for  any  }  t \in \mathbf R  ^ {1} .
 +
$$
 +
 
 +
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 ) ,
 +
$$
  
The moments <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s09071017.png" /> of a Student distribution exist only for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s09071018.png" />, the odd moments are equal to 0, and, in particular <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s09071019.png" />. The even moments of a Student distribution are expressed by the formula
+
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.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s09071020.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {f\rightarrow \infty }  S _ {f} ( x)  = \
 +
\Phi ( x)  = \
  
in particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s09071021.png" />. The distribution function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s09071022.png" /> of the random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s09071023.png" /> is expressed in terms of the [[Beta-distribution|beta-distribution]] function in the following way:
+
\frac{1}{\sqrt {2 \pi } }
 +
\int\limits _ {- \infty } ^ { x }  e ^ {- t  ^ {2} /2 }  dt.
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s09071024.png" /></td> </tr></table>
+
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
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s09071025.png" /> is the incomplete beta-function, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s09071026.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s09071027.png" />, then the Student distribution converges to the standard normal law, i.e.
+
$$
 +
\overline{X}\;  =
 +
\frac{1}{n}
 +
\sum _ { i= } 1 ^ { n }  X _ {i} \  \textrm{ and } \ \
 +
s  ^ {2}  =
 +
\frac{1}{n-}
 +
1 \sum _ { i= } 1 ^ { n }  ( X _ {i} - \overline{X}\; )  ^ {2}
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s09071028.png" /></td> </tr></table>
+
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
  
Example. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s09071029.png" /> be independent, identically, normally <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s09071030.png" />-distributed random variables, where the parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s09071031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s09071032.png" /> are unknown. Then the statistics
+
$$
 +
n-
 +
\frac{1}{\sigma  ^ {2} }
 +
s ^ {2}  = \chi _ {n-} 1  ^ {2}
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s09071033.png" /></td> </tr></table>
+
is distributed according to the  "chi-squared" law with  $  f= n- 1 $
 +
degrees of freedom, then by virtue of their independence, the fraction
  
are the best unbiased estimators of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s09071034.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s09071035.png" />; here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s09071036.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s09071037.png" /> are stochastically independent. Since the random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s09071038.png" /> is subject to the standard normal law, while
+
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s09071039.png" /></td> </tr></table>
+
\frac{\sqrt n ( \overline{X}\; - a) / \sigma }{\sqrt {\chi _ {n-} 1  ^ {2} / ( n- 1) } }
  
is distributed according to the "chi-squared"  law with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s09071040.png" /> degrees of freedom, then by virtue of their independence, the fraction
+
 
 +
\frac{\sqrt n ( \overline{X}\; - a) }{s}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s09071041.png" /></td> </tr></table>
+
$$
  
is subject to the Student distribution with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s09071042.png" /> degrees of freedom. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s09071043.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s09071044.png" /> be the solutions of the equations
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s09071045.png" /></td> </tr></table>
+
$$
 +
S _ {n-} 1 \left (
 +
\frac{\sqrt n ( \overline{X}\; - a) }{s}
 +
\right )  = \
 +
\left \{
  
Then the statistics <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s09071046.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s09071047.png" /> are the lower and upper bounds of the confidence set for the unknown mathematical expectation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s09071048.png" /> of the normal law <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s09071049.png" />, and the confidence coefficient of this confidence set is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s09071050.png" />, i.e.
+
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.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s09071051.png" /></td> </tr></table>
+
$$
 +
{\mathsf P} \left \{ \overline{X}\; -
 +
\frac{s}{\sqrt n }
 +
t _ {f} ( P) < a < \overline{X}\; +
 +
\frac{s}{\sqrt
 +
n }
 +
t _ {f} ( P) \right \}  = 2P- 1.
 +
$$
  
 
The Student distribution was first used by W.S. Gosset (pseudonym Student).
 
The Student distribution was first used by W.S. Gosset (pseudonym Student).

Revision as of 08:24, 6 June 2020


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

$$ S _ {n-} 1 \left ( \frac{\sqrt n ( \overline{X}\; - a) }{s} \right ) = \ \left \{ Then the statistics $ \overline{X}\; - ( s/ \sqrt n ) t _ {f} ( P) $ and $ \overline{X}\; + ( s/ \sqrt n ) t _ {f} ( P) $ are the lower and upper bounds of the confidence set for the unknown mathematical expectation $ a $ of the normal law $ N( a, \sigma ^ {2} ) $, and the confidence coefficient of this confidence set is equal to $ 2P- 1 $, i.e. $$ {\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
How to Cite This Entry:
Student distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Student_distribution&oldid=48882
This article was adapted from an original article by M.S. Nikulin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article