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''polynomial distribution''
 
''polynomial distribution''
  
The joint distribution of random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065330/m0653301.png" /> that is defined for any set of non-negative integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065330/m0653302.png" /> satisfying the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065330/m0653303.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065330/m0653304.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065330/m0653305.png" />, by the formula
+
{{MSC|60E99}}
  
<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/m/m065/m065330/m0653306.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
[[Category:Distribution theory]]
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065330/m0653307.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065330/m0653308.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065330/m0653309.png" />) are the parameters of the distribution. A multinomial distribution is a multivariate discrete distribution, namely the distribution for the random vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065330/m06533010.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065330/m06533011.png" /> (this distribution is in essence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065330/m06533012.png" />-dimensional, since it is degenerate in the Euclidean space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065330/m06533013.png" /> dimensions). A multinomial distribution is a natural generalization of a [[Binomial distribution|binomial distribution]] and coincides with the latter for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065330/m06533014.png" />. The name of the distribution is given because the probability (*) is the general term in the expansion of the multinomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065330/m06533015.png" />. The multinomial distribution appears in the following probability scheme. Each of the random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065330/m06533016.png" /> is the number of occurrences of one of the mutually exclusive events <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065330/m06533017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065330/m06533018.png" />, in repeated independent trials. If in each trial the probability of event <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065330/m06533019.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065330/m06533020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065330/m06533021.png" />, then the probability (*) is equal to the probability that in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065330/m06533022.png" /> trials the events <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065330/m06533023.png" /> will appear <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065330/m06533024.png" /> times, respectively. Each of the random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065330/m06533025.png" /> has a binomial distribution with mathematical expectation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065330/m06533026.png" /> and variance <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065330/m06533027.png" />.
+
The joint distribution of random variables $  X _ {1} \dots X _ {k} $
 +
that is defined for any set of non-negative integers  $  n _ {1} \dots n _ {k} $
 +
satisfying the condition  $  n _ {1} + \dots + n _ {k} = n $,  
 +
$  n _ {j} = 0 \dots n $,  
 +
$  j = 1 \dots k $,  
 +
by the formula
  
The random vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065330/m06533028.png" /> has mathematical expectation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065330/m06533029.png" /> and covariance matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065330/m06533030.png" />, where
+
$$ \tag{* }
 +
{\mathsf P} \{ X _ {1} = n _ {1} \dots X _ {k} = n _ {k} \}  = \
 +
\frac{n!}{n _ {1} ! \dots n _ {k} ! } p _ {1} ^ {n _ {1} } \dots p _ {k} ^ {n _ {k} } ,
 +
$$
  
<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/m/m065/m065330/m06533031.png" /></td> </tr></table>
+
where  $  n, p _ {1} \dots p _ {k} $(
 +
$  p _ {j} \geq  0 $,
 +
$  \sum p _ {j} = 1 $)
 +
are the parameters of the distribution. A multinomial distribution is a multivariate discrete distribution, namely the distribution for the random vector  $  ( X _ {1} \dots X _ {k} ) $
 +
with  $  X _ {1} + \dots + X _ {k} = n $(
 +
this distribution is in essence  $  ( k- 1) $-
 +
dimensional, since it is degenerate in the Euclidean space of  $  k $
 +
dimensions). A multinomial distribution is a natural generalization of a [[Binomial distribution|binomial distribution]] and coincides with the latter for  $  k = 2 $.
 +
The name of the distribution is given because the probability (*) is the general term in the expansion of the multinomial  $  ( p _ {1} + \dots + p _ {k} )  ^ {n} $.
 +
The multinomial distribution appears in the following probability scheme. Each of the random variables  $  X _ {i} $
 +
is the number of occurrences of one of the mutually exclusive events  $  A _ {j} $,
 +
$  j = 1 \dots k $,
 +
in repeated independent trials. If in each trial the probability of event  $  A _ {j} $
 +
is  $  p _ {j} $,
 +
$  j = 1 \dots k $,
 +
then the probability (*) is equal to the probability that in  $  n $
 +
trials the events  $  A _ {1} \dots A _ {k} $
 +
will appear  $  n _ {1} \dots n _ {k} $
 +
times, respectively. Each of the random variables  $  X _ {j} $
 +
has a binomial distribution with mathematical expectation  $  np _ {j} $
 +
and variance  $  np _ {j} ( 1- p _ {j} ) $.
  
(the rank of the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065330/m06533032.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065330/m06533033.png" /> because <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065330/m06533034.png" />). The characteristic function of a multinomial distribution is
+
The random vector  $  ( X _ {1} \dots X _ {k} ) $
 +
has mathematical expectation  $  ( np _ {1} \dots np _ {k} ) $
 +
and covariance matrix $  B = \| b _ {ij} \| $,
 +
where
  
<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/m/m065/m065330/m06533035.png" /></td> </tr></table>
+
$$
 +
b _ {ij}  = \left \{
 +
\begin{array}{ll}
 +
np _ {i} ( 1- p _ {i} ),  & i = j,  \\
 +
- np _ {i} p _ {j} ,  & i \neq j,  \\
 +
\end{array}
 +
\ \
 +
i, j = 1 \dots k
 +
\right .$$
  
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065330/m06533036.png" />, the distribution of the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065330/m06533037.png" /> with normalized components
+
(the rank of the matrix  $  B $
 +
is  $  k- 1 $
 +
because  $  \sum_{i=1}  ^ {k} n _ {i} = n $).  
 +
The characteristic function of a multinomial distribution is
  
<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/m/m065/m065330/m06533038.png" /></td> </tr></table>
+
$$
 +
f( t _ {1} \dots t _ {k} )  = \left ( p _ {1} e ^ {it _ {1} } + \dots + p _ {k} e ^ {it _ {k} } \right )  ^ {n} .
 +
$$
 +
 
 +
For  $  n \rightarrow \infty $,
 +
the distribution of the vector  $  ( Y _ {1} \dots Y _ {k} ) $
 +
with normalized components
 +
 
 +
$$
 +
Y _ {i}  = \
 +
 
 +
\frac{X _ {i} - np _ {i} }{\sqrt {np _ {i} ( 1- p _ {i} ) } }
 +
 
 +
$$
  
 
tends to a certain multivariate [[Normal distribution|normal distribution]], while the distribution of the sum
 
tends to a certain multivariate [[Normal distribution|normal distribution]], while the distribution of the sum
  
<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/m/m065/m065330/m06533039.png" /></td> </tr></table>
+
$$
 +
\sum_{i=1} ^ { k }  ( 1 - p _ {i} ) Y _ {i}  ^ {2}
 +
$$
  
(which is used in mathematical statistics to construct the [["Chi-squared" distribution| "chi-squared" test]]) tends to the [[Chi-squared test| "chi-squared" distribution]] with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065330/m06533040.png" /> degrees of freedom.
+
(which is used in mathematical statistics to construct the [[Chi-squared distribution| "chi-squared" test]]) tends to the [[Chi-squared test| "chi-squared" distribution]] with $  k- 1 $
 +
degrees of freedom.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"H. Cramér,   "Mathematical methods of statistics" , Princeton Univ. Press (1946)</TD></TR></table>
+
{|
 
+
|valign="top"|{{Ref|C}}|| H. Cramér, "Mathematical methods of statistics" , Princeton Univ. Press (1946) {{MR|0016588}} {{ZBL|0063.01014}}
 
+
|}
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N.L. Johnson,   S. Kotz,   "Discrete distributions" , Wiley (1969)</TD></TR></table>
+
{|
 +
|valign="top"|{{Ref|JK}}|| N.L. Johnson, S. Kotz, "Discrete distributions" , Wiley (1969) {{MR|0268996}} {{ZBL|0292.62009}}
 +
|}

Latest revision as of 13:11, 6 January 2024


polynomial distribution

2020 Mathematics Subject Classification: Primary: 60E99 [MSN][ZBL]

The joint distribution of random variables $ X _ {1} \dots X _ {k} $ that is defined for any set of non-negative integers $ n _ {1} \dots n _ {k} $ satisfying the condition $ n _ {1} + \dots + n _ {k} = n $, $ n _ {j} = 0 \dots n $, $ j = 1 \dots k $, by the formula

$$ \tag{* } {\mathsf P} \{ X _ {1} = n _ {1} \dots X _ {k} = n _ {k} \} = \ \frac{n!}{n _ {1} ! \dots n _ {k} ! } p _ {1} ^ {n _ {1} } \dots p _ {k} ^ {n _ {k} } , $$

where $ n, p _ {1} \dots p _ {k} $( $ p _ {j} \geq 0 $, $ \sum p _ {j} = 1 $) are the parameters of the distribution. A multinomial distribution is a multivariate discrete distribution, namely the distribution for the random vector $ ( X _ {1} \dots X _ {k} ) $ with $ X _ {1} + \dots + X _ {k} = n $( this distribution is in essence $ ( k- 1) $- dimensional, since it is degenerate in the Euclidean space of $ k $ dimensions). A multinomial distribution is a natural generalization of a binomial distribution and coincides with the latter for $ k = 2 $. The name of the distribution is given because the probability (*) is the general term in the expansion of the multinomial $ ( p _ {1} + \dots + p _ {k} ) ^ {n} $. The multinomial distribution appears in the following probability scheme. Each of the random variables $ X _ {i} $ is the number of occurrences of one of the mutually exclusive events $ A _ {j} $, $ j = 1 \dots k $, in repeated independent trials. If in each trial the probability of event $ A _ {j} $ is $ p _ {j} $, $ j = 1 \dots k $, then the probability (*) is equal to the probability that in $ n $ trials the events $ A _ {1} \dots A _ {k} $ will appear $ n _ {1} \dots n _ {k} $ times, respectively. Each of the random variables $ X _ {j} $ has a binomial distribution with mathematical expectation $ np _ {j} $ and variance $ np _ {j} ( 1- p _ {j} ) $.

The random vector $ ( X _ {1} \dots X _ {k} ) $ has mathematical expectation $ ( np _ {1} \dots np _ {k} ) $ and covariance matrix $ B = \| b _ {ij} \| $, where

$$ b _ {ij} = \left \{ \begin{array}{ll} np _ {i} ( 1- p _ {i} ), & i = j, \\ - np _ {i} p _ {j} , & i \neq j, \\ \end{array} \ \ i, j = 1 \dots k \right .$$

(the rank of the matrix $ B $ is $ k- 1 $ because $ \sum_{i=1} ^ {k} n _ {i} = n $). The characteristic function of a multinomial distribution is

$$ f( t _ {1} \dots t _ {k} ) = \left ( p _ {1} e ^ {it _ {1} } + \dots + p _ {k} e ^ {it _ {k} } \right ) ^ {n} . $$

For $ n \rightarrow \infty $, the distribution of the vector $ ( Y _ {1} \dots Y _ {k} ) $ with normalized components

$$ Y _ {i} = \ \frac{X _ {i} - np _ {i} }{\sqrt {np _ {i} ( 1- p _ {i} ) } } $$

tends to a certain multivariate normal distribution, while the distribution of the sum

$$ \sum_{i=1} ^ { k } ( 1 - p _ {i} ) Y _ {i} ^ {2} $$

(which is used in mathematical statistics to construct the "chi-squared" test) tends to the "chi-squared" distribution with $ k- 1 $ degrees of freedom.

References

[C] H. Cramér, "Mathematical methods of statistics" , Princeton Univ. Press (1946) MR0016588 Zbl 0063.01014

Comments

References

[JK] N.L. Johnson, S. Kotz, "Discrete distributions" , Wiley (1969) MR0268996 Zbl 0292.62009
How to Cite This Entry:
Multinomial distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Multinomial_distribution&oldid=11687
This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article