# Multinomial distribution

The multinomial or Polynomialverteilung is a discrete probability distribution. They can be regarded as multivariate generalization of the binomial distribution.

## Definition and model

Let and with. Then the probability density of the multinomial distribution is given by

Here is the multinomial coefficient.

In the special case, there is the binomial distribution, more precisely, the joint distribution of and for a distributed random variable.

## Application and motivation

The multinomial distribution can be motivated by a starting urn model with replacement. In an urn varieties are balls. The proportion of varieties balls in the urn is. The urn will be times respectively from a sphere, listed their property (variety ) and then back down the ball in the urn.

Now We want to know the number of balls of each variety in this sample. Since the multinomial follow, the sample has the probability:

Assuming an urn with balls varieties, each with a ball of each type, so you get the classic dice You throw these times, it has possible outcomes and is interested in how likely it is that straight times occurs just times and so on. Next describe the respective probabilities of the cube faces and thus whether it is fair or unfair dice.

## Properties

For each of the random variable is a binomial distribution with parameters and, hence, has the expectation value

And variance

The covariance of two random variables and is calculated as

And for the correlation coefficient ( Pearson ) follows:

The multinomial distribution has in Bayesian statistics as a priori distribution, the Dirichlet distribution.

Discrete univariate distributions for finite sets: Benford | Bernoulli | beta - binomial | binomial | categorical | hypergeometric | Rademacher | generalized binomial | Zipf | Zipf - Mandelbrot

Discrete univariate distributions for infinite sets: Boltzmann | Conway - Maxwell - Poisson | negative binomial | extended negative binomial | Compound Poisson | discrete uniform | discrete phase -type | Gauss - Kuzmin | geometric | logarithmic | parabolic fractal | Poisson | Poisson - Gamma | Skellam | Yule- Simon | Zeta

Continuous univariate distributions with compact interval: Beta | Cantor | Kumaraswamy | raised cosine | triangle | U - square | steady uniform | Wigner semicircle

Continuous univariate distributions with half-open interval: Beta prime | Bose -Einstein | Burr | Chi-Square | Coxian | Erlang | Exponential | F | Fermi -Dirac | Folded normal | Fréchet | Gamma | Gamma Gamma | extreme | generalized inverse Gaussian | semi logistically | semi- normal | Hotelling's T-square | hyper- exponential | hypoexponential | inverse chi-square | scale - inverse- chi-square | inverse Normal | inverse gamma | Levy | log-normal | log- logistically | Maxwell -Boltzmann | Maxwell speed | Nakagami | not centered chi-square | Pareto | Phase -Type | Rayleigh | relativistic Breit-Wigner | Rice | Rosin -Rammler | shifted Gompertz | truncated normal | Type -2 Gumbel | Weibull | Wilks ' lambda

Continuous univariate distributions with unbounded interval: Cauchy | extreme | exponentially Power | Fishers z | Fisher - Tippett ( Gumbel ) | generalized hyperbolic | Hyperbolic- secant | Landau | Laplace | alpha- stable | logistics | normal ( Gaussian ) | normal - inverse Gauß'sch | skew - normal | Student's t | Type -1 Gumbel | Variance gamma | Voigt

Discrete multivariate distributions: Ewen | multinomial | Dirichlet compound multinomial

Continuous multivariate distributions: Dirichlet | generalized Dirichlet | multivariate normal | multivariate Student | normal scaled inverse gamma | Normal - Gamma

Multivariate matrix distributions: Inverse Wishart | matrix normal | Wishart

- Probability distribution