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''negative multinomial distribution''
 
''negative multinomial distribution''
  
The joint [[Probability distribution|probability distribution]] (cf. also [[Joint distribution|Joint distribution]]) of random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066230/n0662301.png" /> that take non-negative integer values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066230/n0662302.png" /> defined by the formula
+
The joint [[Probability distribution|probability distribution]] (cf. also [[Joint distribution|Joint distribution]]) of random variables $  X _ {1} \dots X _ {k} $
 
+
that take non-negative integer values $  m = 0, 1 \dots $
<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/n/n066/n066230/n0662303.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
defined by the formula
  
<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/n/n066/n066230/n0662304.png" /></td> </tr></table>
+
$$ \tag{* }
 +
{\mathsf P} \{ X _ {1} = m _ {1} \dots X _ {k} = m _ {k} \} =
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066230/n0662305.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066230/n0662306.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066230/n0662307.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066230/n0662308.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066230/n0662309.png" />) are parameters. A negative multinomial distribution is a multi-dimensional [[Discrete distribution|discrete distribution]] — a distribution of a random vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066230/n06623010.png" /> with non-negative integer components.
+
$$
 +
= \
  
The [[Generating function|generating function]] of the negative polynomial distribution with parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066230/n06623011.png" /> has the form
+
\frac{\Gamma ( r+ m _ {1} + \dots + m _ {k} ) }{\Gamma ( r) m _ {1} ! \dots m _ {k} ! }
 +
p _ {0}  ^ {r} p _ {1} ^ {m _ {1} } \dots p _ {k} ^ {m _ {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/n/n066/n066230/n06623012.png" /></td> </tr></table>
+
where  $  r > 0 $
 +
and  $  p _ {0} \dots p _ {k} $(
 +
$  0 < p _ {i} < 1 $,
 +
$  i = 0 \dots k $;  
 +
$  p _ {0} + \dots + p _ {k} = 1 $)
 +
are parameters. A negative multinomial distribution is a multi-dimensional [[Discrete distribution|discrete distribution]] — a distribution of a random vector  $  ( X _ {1} \dots X _ {k} ) $
 +
with non-negative integer components.
  
A negative multinomial distribution arises in the following multinomial scheme. Successive independent trials are carried out, and in each trial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066230/n06623013.png" /> different outcomes with labels <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066230/n06623014.png" /> are possible, having probabilities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066230/n06623015.png" />, respectively. The trials continue up to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066230/n06623016.png" />-th appearance of the outcome with label 0 (here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066230/n06623017.png" /> is an integer). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066230/n06623018.png" /> is the number of appearances of the outcome with label <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066230/n06623019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066230/n06623020.png" />, during the trials, then formula (*) expresses the probability of the appearance of outcomes with labels <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066230/n06623021.png" />, equal, respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066230/n06623022.png" /> times, up to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066230/n06623023.png" />-th appearance of the outcome 0. A negative multinomial distribution in this sense is a generalization of a [[Negative binomial distribution|negative binomial distribution]], coinciding with it when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066230/n06623024.png" />.
+
The [[Generating function|generating function]] of the negative polynomial distribution with parameters  $  r, p _ {0} \dots p _ {k} $
 +
has the form
  
If a random vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066230/n06623025.png" /> has, conditionally on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066230/n06623026.png" />, a [[Multinomial distribution|multinomial distribution]] with parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066230/n06623027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066230/n06623028.png" /> and if the parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066230/n06623029.png" /> is itself a random variable having a negative binomial distribution with parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066230/n06623030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066230/n06623031.png" />, then the marginal distribution of the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066230/n06623032.png" />, given the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066230/n06623033.png" />, is the negative multinomial distribution with parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066230/n06623034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066230/n06623035.png" />.
+
$$
 +
P( z _ {1} \dots z _ {k} )  = p _ {0}  ^ {r} \left ( 1 - \sum
 +
_ { i= } 1 ^ { k }  z _ {i} p _ {i} \right )  ^ {-} r .
 +
$$
  
 +
A negative multinomial distribution arises in the following multinomial scheme. Successive independent trials are carried out, and in each trial  $  k+ 1 $
 +
different outcomes with labels  $  0 \dots k $
 +
are possible, having probabilities  $  p _ {0} \dots p _ {k} $,
 +
respectively. The trials continue up to the  $  r $-
 +
th appearance of the outcome with label 0 (here  $  r $
 +
is an integer). If  $  X _ {i} $
 +
is the number of appearances of the outcome with label  $  i $,
 +
$  i = 1 \dots k $,
 +
during the trials, then formula (*) expresses the probability of the appearance of outcomes with labels  $  1 \dots k $,
 +
equal, respectively,  $  m _ {1} \dots m _ {k} $
 +
times, up to the  $  r $-
 +
th appearance of the outcome 0. A negative multinomial distribution in this sense is a generalization of a [[Negative binomial distribution|negative binomial distribution]], coinciding with it when  $  k= 1 $.
  
 +
If a random vector  $  ( X _ {0} \dots X _ {k} ) $
 +
has, conditionally on  $  n $,
 +
a [[Multinomial distribution|multinomial distribution]] with parameters  $  n > 1 $,
 +
$  p _ {0} \dots p _ {k} $
 +
and if the parameter  $  n $
 +
is itself a random variable having a negative binomial distribution with parameters  $  r > 0 $,
 +
$  0 < \pi < 1 $,
 +
then the marginal distribution of the vector  $  ( X _ {1} \dots X _ {k} ) $,
 +
given the condition  $  X _ {0} = r $,
 +
is the negative multinomial distribution with parameters  $  r $,
 +
$  p _ {0} ( 1- \pi ) \dots p _ {k} ( 1- \pi ) $.
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Neyman,  "Proceedings of the international symposium on discrete distributions" , Montreal  (1963)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Neyman,  "Proceedings of the international symposium on discrete distributions" , Montreal  (1963)</TD></TR></table>

Revision as of 08:02, 6 June 2020


negative multinomial distribution

The joint probability distribution (cf. also Joint distribution) of random variables $ X _ {1} \dots X _ {k} $ that take non-negative integer values $ m = 0, 1 \dots $ defined by the formula

$$ \tag{* } {\mathsf P} \{ X _ {1} = m _ {1} \dots X _ {k} = m _ {k} \} = $$

$$ = \ \frac{\Gamma ( r+ m _ {1} + \dots + m _ {k} ) }{\Gamma ( r) m _ {1} ! \dots m _ {k} ! } p _ {0} ^ {r} p _ {1} ^ {m _ {1} } \dots p _ {k} ^ {m _ {k} } , $$

where $ r > 0 $ and $ p _ {0} \dots p _ {k} $( $ 0 < p _ {i} < 1 $, $ i = 0 \dots k $; $ p _ {0} + \dots + p _ {k} = 1 $) are parameters. A negative multinomial distribution is a multi-dimensional discrete distribution — a distribution of a random vector $ ( X _ {1} \dots X _ {k} ) $ with non-negative integer components.

The generating function of the negative polynomial distribution with parameters $ r, p _ {0} \dots p _ {k} $ has the form

$$ P( z _ {1} \dots z _ {k} ) = p _ {0} ^ {r} \left ( 1 - \sum _ { i= } 1 ^ { k } z _ {i} p _ {i} \right ) ^ {-} r . $$

A negative multinomial distribution arises in the following multinomial scheme. Successive independent trials are carried out, and in each trial $ k+ 1 $ different outcomes with labels $ 0 \dots k $ are possible, having probabilities $ p _ {0} \dots p _ {k} $, respectively. The trials continue up to the $ r $- th appearance of the outcome with label 0 (here $ r $ is an integer). If $ X _ {i} $ is the number of appearances of the outcome with label $ i $, $ i = 1 \dots k $, during the trials, then formula (*) expresses the probability of the appearance of outcomes with labels $ 1 \dots k $, equal, respectively, $ m _ {1} \dots m _ {k} $ times, up to the $ r $- th appearance of the outcome 0. A negative multinomial distribution in this sense is a generalization of a negative binomial distribution, coinciding with it when $ k= 1 $.

If a random vector $ ( X _ {0} \dots X _ {k} ) $ has, conditionally on $ n $, a multinomial distribution with parameters $ n > 1 $, $ p _ {0} \dots p _ {k} $ and if the parameter $ n $ is itself a random variable having a negative binomial distribution with parameters $ r > 0 $, $ 0 < \pi < 1 $, then the marginal distribution of the vector $ ( X _ {1} \dots X _ {k} ) $, given the condition $ X _ {0} = r $, is the negative multinomial distribution with parameters $ r $, $ p _ {0} ( 1- \pi ) \dots p _ {k} ( 1- \pi ) $.

Comments

References

[a1] J. Neyman, "Proceedings of the international symposium on discrete distributions" , Montreal (1963)
How to Cite This Entry:
Negative polynomial distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Negative_polynomial_distribution&oldid=14676
This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article