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 | ||
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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 $) | ||
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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 $-th appearance of the outcome with label 0 (here $ r $ | 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 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 $-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 $. | 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} ) $ | + | 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) |
Negative polynomial distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Negative_polynomial_distribution&oldid=52031