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Difference between revisions of "Negative polynomial distribution"

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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  $  X _ {1} \dots X _ {k} $
+
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 $
+
that take non-negative integer values  $  m = 0, 1, \dots $
 
defined by the formula
 
defined by the formula
  
 
$$ \tag{* }
 
$$ \tag{* }
{\mathsf P} \{ X _ {1} = m _ {1} \dots X _ {k} = m _ {k} \} =
+
{\mathsf P} \{ X _ {1} = m _ {1}, \dots, X _ {k} = m _ {k} \} =
 
$$
 
$$
  
Line 24: Line 24:
 
= \  
 
= \  
  
\frac{\Gamma ( r+ m _ {1} + \dots + m _ {k} ) }{\Gamma ( r) m _ {1} ! \dots m _ {k} ! }
+
\frac{\Gamma ( r+ m _ {1} + \dots + m _ {k} ) }{\Gamma ( r) m _ {1} ! \cdots m _ {k} ! }
  p _ {0}  ^ {r} p _ {1} ^ {m _ {1} } \dots p _ {k} ^ {m _ {k} } ,
+
  p _ {0}  ^ {r} p _ {1} ^ {m _ {1} } \cdots p _ {k} ^ {m _ {k} } ,
 
$$
 
$$
  
 
where  $  r > 0 $
 
where  $  r > 0 $
and  $  p _ {0} \dots p _ {k} $(
+
and  $  p _ {0}, \dots, p _ {k} $ ($  0 < p _ {i} < 1 $,  
$  0 < p _ {i} < 1 $,  
+
$  i = 0, \dots, k $;  
$  i = 0 \dots k $;  
 
 
$  p _ {0} + \dots + p _ {k} = 1 $)  
 
$  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} ) $
+
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.
 
with non-negative integer components.
  
The [[Generating function|generating function]] of the negative polynomial distribution with parameters  $  r, p _ {0} \dots p _ {k} $
+
The [[Generating function|generating function]] of the negative polynomial distribution with parameters  $  r, p _ {0}, \dots, p _ {k} $
 
has the form
 
has the form
  
 
$$  
 
$$  
P( z _ {1} \dots z _ {k} )  =  p _ {0}  ^ {r} \left ( 1 - \sum
+
P( z _ {1}, \dots, z _ {k} )  =  p _ {0}  ^ {r} \left ( 1 - \sum
_ { i= } 1 ^ { k }  z _ {i} p _ {i} \right )  ^ {-} r .
+
_ { 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 $
 
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 $
+
different outcomes with labels  $  0, \dots, k $
are possible, having probabilities  $  p _ {0} \dots p _ {k} $,  
+
are possible, having probabilities  $  p _ {0}, \dots, p _ {k} $,  
respectively. The trials continue up to the  $  r $-
+
respectively. The trials continue up to the  $  r $-th appearance of the outcome with label 0 (here  $  r $
th appearance of the outcome with label 0 (here  $  r $
 
 
is an integer). If  $  X _ {i} $
 
is an integer). If  $  X _ {i} $
 
is the number of appearances of the outcome with label  $  i $,  
 
is the number of appearances of the outcome with label  $  i $,  
$  i = 1 \dots k $,  
+
$  i = 1, \dots, k $,  
during the trials, then formula (*) expresses the probability of the appearance of outcomes with labels  $  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} $
+
equal, respectively,  $  m _ {1}, \dots, m _ {k} $
times, up to the  $  r $-
+
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 $.
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} ) $
+
If a random vector  $  ( X _ {0}, \dots, X _ {k} ) $
 
has, conditionally on  $  n $,  
 
has, conditionally on  $  n $,  
 
a [[Multinomial distribution|multinomial distribution]] with parameters  $  n > 1 $,  
 
a [[Multinomial distribution|multinomial distribution]] with parameters  $  n > 1 $,  
$  p _ {0} \dots p _ {k} $
+
$  p _ {0}, \dots, p _ {k} $
 
and if the parameter  $  n $
 
and if the parameter  $  n $
 
is itself a random variable having a negative binomial distribution with parameters  $  r > 0 $,  
 
is itself a random variable having a negative binomial distribution with parameters  $  r > 0 $,  
 
$  0 < \pi < 1 $,  
 
$  0 < \pi < 1 $,  
then the marginal distribution of the vector  $  ( X _ {1} \dots X _ {k} ) $,  
+
then the marginal distribution of the vector  $  ( X _ {1}, \dots, X _ {k} ) $,  
 
given the condition  $  X _ {0} = r $,  
 
given the condition  $  X _ {0} = r $,  
 
is the negative multinomial distribution with parameters  $  r $,  
 
is the negative multinomial distribution with parameters  $  r $,  
$  p _ {0} ( 1- \pi ) \dots p _ {k} ( 1- \pi ) $.
+
$  p _ {0} ( 1- \pi ), \dots, p _ {k} ( 1- \pi ) $.
  
 
====Comments====
 
====Comments====

Latest revision as of 16:50, 1 February 2022


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

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

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

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=47953
This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article