# Multinomial: Multinomial Distribution Class In distr6: The Complete R6 Probability Distributions Interface

## Description

Mathematical and statistical functions for the Multinomial distribution, which is commonly used to extend the binomial distribution to multiple variables, for example to model the rolls of multiple dice multiple times.

## Details

The Multinomial distribution parameterised with number of trials, n, and probabilities of success, p_1,...,p_k, is defined by the pmf,

f(x_1,x_2,…,x_k) = n!/(x_1! * x_2! * … * x_k!) * p_1^{x_1} * p_2^{x_2} * … * p_k^{x_k}

for p_i, i = {1,…,k}; ∑ p_i = 1 and n = {1,2,…}.

## Value

Returns an R6 object inheriting from class SDistribution.

## Distribution support

The distribution is supported on ∑ x_i = N.

## Default Parameterisation

Multinom(size = 10, probs = c(0.5, 0.5))

## Omitted Methods

`cdf` and `quantile` are omitted as no closed form analytic expression could be found, decorate with `FunctionImputation` for a numerical imputation.

N/A

## Super classes

`distr6::Distribution` -> `distr6::SDistribution` -> `Multinomial`

## Public fields

`name`

Full name of distribution.

`short_name`

Short name of distribution for printing.

`description`

Brief description of the distribution.

`packages`

Packages required to be installed in order to construct the distribution.

## Active bindings

`properties`

Returns distribution properties, including skewness type and symmetry.

## Methods

#### Public methods

Inherited methods

#### Method `new()`

Creates a new instance of this R6 class.

##### Usage
`Multinomial\$new(size = NULL, probs = NULL, decorators = NULL)`
##### Arguments
`size`

`(integer(1))`
Number of trials, defined on the positive Naturals.

`probs`

`(numeric())`
Vector of probabilities. Automatically normalised by `probs = probs/sum(probs)`.

`decorators`

`(character())`
Decorators to add to the distribution during construction.

#### Method `mean()`

The arithmetic mean of a (discrete) probability distribution X is the expectation

E_X(X) = ∑ p_X(x)*x

with an integration analogue for continuous distributions.

##### Usage
`Multinomial\$mean(...)`
`...`

Unused.

#### Method `variance()`

The variance of a distribution is defined by the formula

var_X = E[X^2] - E[X]^2

where E_X is the expectation of distribution X. If the distribution is multivariate the covariance matrix is returned.

##### Usage
`Multinomial\$variance(...)`
`...`

Unused.

#### Method `skewness()`

The skewness of a distribution is defined by the third standardised moment,

sk_X = E_X[((x - μ)/σ)^3]

where E_X is the expectation of distribution X, μ is the mean of the distribution and σ is the standard deviation of the distribution.

##### Usage
`Multinomial\$skewness(...)`
`...`

Unused.

#### Method `kurtosis()`

The kurtosis of a distribution is defined by the fourth standardised moment,

k_X = E_X[((x - μ)/σ)^4]

where E_X is the expectation of distribution X, μ is the mean of the distribution and σ is the standard deviation of the distribution. Excess Kurtosis is Kurtosis - 3.

##### Usage
`Multinomial\$kurtosis(excess = TRUE, ...)`
##### Arguments
`excess`

`(logical(1))`
If `TRUE` (default) excess kurtosis returned.

`...`

Unused.

#### Method `entropy()`

The entropy of a (discrete) distribution is defined by

- ∑ (f_X)log(f_X)

where f_X is the pdf of distribution X, with an integration analogue for continuous distributions.

##### Usage
`Multinomial\$entropy(base = 2, ...)`
##### Arguments
`base`

`(integer(1))`
Base of the entropy logarithm, default = 2 (Shannon entropy)

`...`

Unused.

#### Method `mgf()`

The moment generating function is defined by

mgf_X(t) = E_X[exp(xt)]

where X is the distribution and E_X is the expectation of the distribution X.

##### Usage
`Multinomial\$mgf(t, ...)`
##### Arguments
`t`

`(integer(1))`
t integer to evaluate function at.

`...`

Unused.

#### Method `cf()`

The characteristic function is defined by

cf_X(t) = E_X[exp(xti)]

where X is the distribution and E_X is the expectation of the distribution X.

##### Usage
`Multinomial\$cf(t, ...)`
##### Arguments
`t`

`(integer(1))`
t integer to evaluate function at.

`...`

Unused.

#### Method `pgf()`

The probability generating function is defined by

pgf_X(z) = E_X[exp(z^x)]

where X is the distribution and E_X is the expectation of the distribution X.

##### Usage
`Multinomial\$pgf(z, ...)`
##### Arguments
`z`

`(integer(1))`
z integer to evaluate probability generating function at.

`...`

Unused.

#### Method `setParameterValue()`

Sets the value(s) of the given parameter(s).

##### Usage
```Multinomial\$setParameterValue(
...,
lst = list(...),
error = "warn",
resolveConflicts = FALSE
)```
##### Arguments
`...`

`ANY`
Named arguments of parameters to set values for. See examples.

`lst`

`(list(1))`
Alternative argument for passing parameters. List names should be parameter names and list values are the new values to set.

`error`

`(character(1))`
If `"warn"` then returns a warning on error, otherwise breaks if `"stop"`.

`resolveConflicts`

`(logical(1))`
If `FALSE` (default) throws error if conflicting parameterisations are provided, otherwise automatically resolves them by removing all conflicting parameters.

#### Method `clone()`

The objects of this class are cloneable with this method.

##### Usage
`Multinomial\$clone(deep = FALSE)`
##### Arguments
`deep`

Whether to make a deep clone.

## References

McLaughlin, M. P. (2001). A compendium of common probability distributions (pp. 2014-01). Michael P. McLaughlin.

Other discrete distributions: `Bernoulli`, `Binomial`, `Categorical`, `Degenerate`, `DiscreteUniform`, `EmpiricalMV`, `Empirical`, `Geometric`, `Hypergeometric`, `Logarithmic`, `NegativeBinomial`, `WeightedDiscrete`
Other multivariate distributions: `Dirichlet`, `EmpiricalMV`, `MultivariateNormal`