Multinomial: Multinomial Distribution Class

Description Details Value Distribution support Default Parameterisation Omitted Methods Also known as Super classes Public fields Active bindings Methods References See Also

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.

Also known as

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(...)
Arguments
...

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(...)
Arguments
...

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(...)
Arguments
...

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.

See Also

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

Other multivariate distributions: Dirichlet, EmpiricalMV, MultivariateNormal


distr6 documentation built on Sept. 6, 2021, 9:10 a.m.