Multinomial distribution

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Multinomial

Probability mass function
Cumulative distribution function
Parameters	<math>n > 0</math> number of trials (integer) <math>p_1, \ldots p_k</math> event probabilities (<math>\Sigma p_i = 1</math>)
Support	<math>X_i \in \{0,\dots,n\}</math> <math>\Sigma X_i = n\!</math>
Probability mass function (pmf)	<math>\frac{n!}{x_1!\cdots x_k!} p_1^{x_1} \cdots p_k^{x_k}</math>
Cumulative distribution function (cdf)
Mean	<math>E\{X_i\} = np_i</math>
Median
Mode
Variance	<math>{\mathrm{Var
Skewness	{{{skewness}}}
Excess kurtosis	{{{kurtosis}}}
Entropy	{{{entropy}}}
Moment-generating function (mgf)	{{{mgf}}}
Characteristic function	{{{char}}}

(X_i) = n p_i (1-p_i)</math>
<math>{\mathrm{Cov}}(X_i,X_j) = - n p_i p_j</math>|

 skewness   =|
 kurtosis   =|
 entropy    =|
 mgf        =<math>\left( \sum_{i=1}^k p_i e^{t_i} \right)^n</math>|
 char       =|

}} In probability theory, the multinomial distribution is a generalization of the binomial distribution.

The binomial distribution is the probability distribution of the number of "successes" in n independent Bernoulli trials, with the same probability of "success" on each trial. In a multinomial distribution, each trial results in exactly one of some fixed finite number k of possible outcomes, with probabilities p₁, ..., p_k (so that p_i ≥ 0 for i = 1, ..., k and <math>\sum_{i=1}^k p_i = 1</math>), and there are n independent trials. Then let the random variables <math>X_i</math> indicate the number of times outcome number i was observed over the n trials. <math>X=(X_1,\ldots,X_k)</math> follows a multinomial distribution with parameters n and p, where p = (p₁, ..., p_k).

Contents 1 Specification 1.1 Probability mass function 2 Properties 3 Related distributions 4 See also 5 External links 6 References

[edit] Specification

[edit] Probability mass function

The probability mass function of the multinomial distribution is:

<math> \begin{align}

f(x_1,\ldots,x_k;n,p_1,\ldots,p_k) & {} = \Pr(X_1 = x_1\mbox{ and }\dots\mbox{ and }X_k = x_k) \\ \\ & {} = \begin{cases} { \displaystyle {n! \over x_1!\cdots x_k!}p_1^{x_1}\cdots p_k^{x_k}}, \quad & \mbox{when } \sum_{i=1}^k x_i=n \\ \\ 0 & \mbox{otherwise,} \end{cases} \end{align} </math>

for non-negative integers x₁, ..., x_k.

[edit] Properties

The expected value is

<math>\operatorname{E}(X_i) = n p_i.</math>

The covariance matrix is as follows. Each diagonal entry is the variance of a binomially distributed random variable, and is therefore

<math>\operatorname{var}(X_i)=np_i(1-p_i).</math>

The off-diagonal entries are the covariances:

<math>\operatorname{cov}(X_i,X_j)=-np_i p_j</math>

for i, j distinct.

All covariances are negative because for fixed N, an increase in one component of a multinomial vector requires a decrease in another component.

This is a k × k nonnegative-definite matrix of rank k − 1.

The off-diagonal entries of the corresponding correlation matrix are

<math>\rho(X_i,X_j) = -\sqrt{\frac{p_i p_j}{ (1-p_i)(1-p_j)}}.</math>

Note that the sample size drops out of this expression.

Each of the k components separately has a binomial distribution with parameters n and p_i, for the appropriate value of the subscript i.

The support of the multinomial distribution is the set :<math>\{(n_1,\dots,n_k)\in \mathbb{N}^{k}| n_1+\cdots+n_k=n\}.</math> Its number of elements is

<math>{n+k-1 \choose k} = \left\langle \begin{matrix}n \\ k \end{matrix}\right\rangle,</math>

the number of n-combinations of a multiset with k types, or multiset coefficient.

[edit] Related distributions

• When k = 2, the multinomial distribution is the binomial distribution.
• The Dirichlet distribution is the conjugate prior of the multinomial in Bayesian statistics.
• Multivariate Polya distribution

[edit] See also

• Multinomial theorem

[edit] External links

• Discrete Probability Distribution - Multinomial Distribution

[edit] References

Evans, Merran; Nicholas Hastings, Brian Peacock (2000). Statistical Distributions. New York: Wiley, 134-136. 3rd ed.. ISBN 0-471-37124-6. 

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