Multinomial distribution
Parameters 
number of trials (integer) event probabilities () 

Support 

pmf  
Mean  
Variance 

MGF  
CF  where 
PGF 
In probability theory, the multinomial distribution is a generalization of the binomial distribution. For example, it models the probability of counts for rolling a k sided dice n times. For n independent trials each of which leads to a success for exactly one of k categories, with each category having a given fixed success probability, the multinomial distribution gives the probability of any particular combination of numbers of successes for the various categories.
When n is 1 and k is 2 the multinomial distribution is the Bernoulli distribution. When k is 2 and number of trials are more than 1 it is the Binomial distribution. When n is 1 it is the categorical distribution.
The Bernoulli distribution is the probability distribution of whether a Bernoulli trial is a success. In other words, it models the number of heads from flipping a coin one time. The binomial distribution generalizes this to the number of heads from doing n independent flips of the same coin. For the multinomial distribution the analog to the Bernoulli Distribution is the categorical distribution. Instead of flipping one coin, the categorical distribution models the roll of one k sided die. So the multinomial distribution can model n independent rolls of a k sided die.
Let k be a fixed finite number. Mathematically, we have k possible mutually exclusive outcomes, with corresponding probabilities p_{1}, ..., p_{k}, and n independent trials. Since the k outcomes are mutually exclusive and one must occur we have p_{i} ≥ 0 for i = 1, ..., k and . Then if the random variables X_{i} indicate the number of times outcome number i is observed over the n trials, the vector X = (X_{1}, ..., X_{k}) follows a multinomial distribution with parameters n and p, where p = (p_{1}, ..., p_{k}). While the trials are independent, their outcomes X are dependent because they must be summed to n.
Note that, in some fields, such as natural language processing, the categorical and multinomial distributions are conflated, and it is common to speak of a "multinomial distribution" when a categorical distribution is actually meant. This stems from the fact that it is sometimes convenient to express the outcome of a categorical distribution as a "1ofK" vector (a vector with one element containing a 1 and all other elements containing a 0) rather than as an integer in the range ; in this form, a categorical distribution is equivalent to a multinomial distribution over a single trial.
Specification
Probability mass function
Suppose one does an experiment of extracting n balls of k different colours from a bag, replacing the extracted ball after each draw. Balls from the same colour are equivalent. Denote the variable which is the number of extracted balls of colour i (i = 1, ..., k) as X_{i}, and denote as p_{i} the probability that a given extraction will be in colour i. The probability mass function of this multinomial distribution is:
for nonnegative integers x_{1}, ..., x_{k}.
The probability mass function can be expressed using the gamma function as:
This form shows its resemblance to the Dirichlet distribution which is its conjugate prior.
Visualization
As slices of generalized Pascal's triangle
Just like one can interpret the binomial distribution as (normalized) 1D slices of Pascal's triangle, so too can one interpret the multinomial distribution as 2D (triangular) slices of Pascal's pyramid, or 3D/4D/+ (pyramidshaped) slices of higherdimensional analogs of Pascal's triangle. This reveals an interpretation of the range of the distribution: discretized equilaterial "pyramids" in arbitrary dimension—i.e. a simplex with a grid.
As polynomial coefficients
Similarly, just like one can interpret the binomial distribution as the polynomial coefficients of when expanded, one can interpret the multinomial distribution as the coefficients of when expanded. (Note that just like the binomial distribution, the coefficients must sum to 1.) This is the origin of the name "multinomial distribution".
Properties
The expected number of times the outcome i was observed over n trials is
The covariance matrix is as follows. Each diagonal entry is the variance of a binomially distributed random variable, and is therefore
The offdiagonal entries are the covariances:
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 positivesemidefinite matrix of rank k − 1. In the special case where k = n and where the p_{i} are all equal, the covariance matrix is the centering matrix.
The entries of the corresponding correlation matrix are
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
Its number of elements is
Matrix notation
In matrix notation,
and
with p^{T} = the row vector transpose of the column vector p.
Example
In a recent threeway election for a large country, candidate A received 20% of the votes, candidate B received 30% of the votes, and candidate C received 50% of the votes. If six voters are selected randomly, what is the probability that there will be exactly one supporter for candidate A, two supporters for candidate B and three supporters for candidate C in the sample?
Note: Since we’re assuming that the voting population is large, it is reasonable and permissible to think of the probabilities as unchanging once a voter is selected for the sample. Technically speaking this is sampling without replacement, so the correct distribution is the multivariate hypergeometric distribution, but the distributions converge as the population grows large.
Sampling from a multinomial distribution
First, reorder the parameters such that they are sorted in descending order (this is only to speed up computation and not strictly necessary). Now, for each trial, draw an auxiliary variable X from a uniform (0, 1) distribution. The resulting outcome is the component
{X_{j} = 1, X_{k} = 0 for k ≠ j } is one observation from the multinomial distribution with and n = 1. A sum of independent repetitions of this experiment is an observation from a multinomial distribution with n equal to the number of such repetitions.
To simulate a multinomial distribution
Various methods may be used to simulate a multinomial distribution. A very simple one is to use a random number generator to generate numbers between 0 and 1. First, we divide the interval from 0 to 1 in k subintervals equal in size to the probabilities of the k categories. Then, we generate a random number for each of n trials and use a logical test to classify the virtual measure or observation in one of the categories.
Example
If we have :
Categories  1  2  3  4  5  6 
Probabilities  0.15  0.20  0.30  0.16  0.12  0.07 
Superior limits of subintervals  0.15  0.35  0.65  0.81  0.93  1.00 
Then, with a software like Excel, we may use the following recipe:
Cells :  Ai  Bi  Ci  ...  Gi 
Formulae :  Rand()  =If($Ai<0.15;1;0)  =If(And($Ai>=0.15;$Ai<0.35);1;0)  ...  =If($Ai>=0.93;1;0) 
After that, we will use functions such as SumIf to accumulate the observed results by category and to calculate the estimated covariance matrix for each simulated sample.
Another way is to use a discrete random number generator. In that case, the categories must be labeled or relabeled with numeric values.
In the two cases, the result is a multinomial distribution with k categories. This is equivalent, with a continuous random distribution, to simulate k independent standardized normal distributions, or a multinormal distribution N(0,I) having k components identically distributed and statistically independent.
Since the counts of all categories have to sum to the number of trials, the counts of the categories are always negatively correlated.^{[1]}
Related distributions
 When k = 2, the multinomial distribution is the binomial distribution.
 Categorical distribution, the distribution of each trial; for k = 2, this is the Bernoulli distribution.
 The Dirichlet distribution is the conjugate prior of the multinomial in Bayesian statistics.
 Dirichletmultinomial distribution.
 Betabinomial model.
 Negative multinomial distribution
 Hardy–Weinberg principle (it is a trinomial distribution with probabilities )
References
 ↑ "1.7  The Multinomial Distribution  STAT 504". onlinecourses.science.psu.edu. Retrieved 20160911.
 Evans, Morton; Hastings, Nicholas; Peacock, Brian (2000). Statistical Distributions. New York: Wiley. pp. 134–136. ISBN 0471371246. 3rd ed.