Editing Multinomial distribution (section)

=== Expected value and variance ===
The [[Expected value|expected]] number of times the outcome ''i'' was observed over ''n'' trials 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 [[covariance]]s:

:<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.

When these expressions are combined into a matrix with ''i, j'' element <math>\operatorname{cov} (X_i,X_j),</math> the result is a ''k'' &times; ''k'' [[Positive-definite matrix#Negative-definite, semidefinite and indefinite matrices|positive-semidefinite]] [[covariance matrix]] of rank ''k''&nbsp;&minus;&nbsp;1. In the special case where ''k''&nbsp;=&nbsp;''n'' and where the ''p''<sub>''i''</sub> are all equal, the covariance matrix is the [[centering matrix]].

The entries of the corresponding [[Correlation matrix#Correlation matrices|correlation matrix]] are

:<math>\rho(X_i,X_i) = 1.</math>

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

Note that the number of trials ''n'' drops out of this expression.

Each of the ''k'' components separately has a binomial distribution with parameters ''n'' and ''p''<sub>''i''</sub>, for the appropriate value of the subscript ''i''.

The [[Support (mathematics)|support]] of the multinomial distribution is the set

: <math>\{(n_1,\dots,n_k)\in \mathbb{N}^k \mid  n_1+\cdots+n_k=n\}.\,</math>

Its number of elements is

: <math>{n+k-1 \choose k-1}.</math>