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