Multinomial distribution

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

References

[JK]	N.L. Johnson, S. Kotz, "Discrete distributions" , Wiley (1969) MR0268996 Zbl 0292.62009