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.
Returns an R6 object inheriting from class SDistribution.
distr6::Distribution -> distr6::SDistribution -> Multinomial
nameFull name of distribution.
short_nameShort name of distribution for printing.
descriptionBrief description of the distribution.
packagesPackages required to be installed in order to construct the distribution.
new()Creates a new instance of this R6 class.
Multinomial$new(size = 10, probs = c(0.5, 0.5), decorators = NULL)
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.
mean()The arithmetic mean of a (discrete) probability distribution X is the expectation $$E_X(X) = \sum p_X(x)*x$$ with an integration analogue for continuous distributions.
Multinomial$mean(...)
...Unused.
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.
Multinomial$variance(...)
...Unused.
skewness()The skewness of a distribution is defined by the third standardised moment, $$sk_X = E_X[\frac{x - \mu}{\sigma}^3]$$ where \(E_X\) is the expectation of distribution X, \(\mu\) is the mean of the distribution and \(\sigma\) is the standard deviation of the distribution.
Multinomial$skewness(...)
...Unused.
kurtosis()The kurtosis of a distribution is defined by the fourth standardised moment, $$k_X = E_X[\frac{x - \mu}{\sigma}^4]$$ where \(E_X\) is the expectation of distribution X, \(\mu\) is the mean of the distribution and \(\sigma\) is the standard deviation of the distribution. Excess Kurtosis is Kurtosis - 3.
Multinomial$kurtosis(excess = TRUE, ...)
excess(logical(1))
If TRUE (default) excess kurtosis returned.
...Unused.
entropy()The entropy of a (discrete) distribution is defined by $$- \sum (f_X)log(f_X)$$ where \(f_X\) is the pdf of distribution X, with an integration analogue for continuous distributions.
Multinomial$entropy(base = 2, ...)
base(integer(1))
Base of the entropy logarithm, default = 2 (Shannon entropy)
...Unused.
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.
Multinomial$mgf(t, ...)
t(integer(1))
t integer to evaluate function at.
...Unused.
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.
Multinomial$cf(t, ...)
t(integer(1))
t integer to evaluate function at.
...Unused.
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.
Multinomial$pgf(z, ...)
z(integer(1))
z integer to evaluate probability generating function at.
...Unused.
clone()The objects of this class are cloneable with this method.
Multinomial$clone(deep = FALSE)
deepWhether to make a deep clone.
The Multinomial distribution parameterised with number of trials, \(n\), and probabilities of success, \(p_1,...,p_k\), is defined by the pmf, $$f(x_1,x_2,\ldots,x_k) = n!/(x_1! * x_2! * \ldots * x_k!) * p_1^{x_1} * p_2^{x_2} * \ldots * p_k^{x_k}$$ for \(p_i, i = {1,\ldots,k}; \sum p_i = 1\) and \(n = {1,2,\ldots}\).
The distribution is supported on \(\sum x_i = N\).
cdf and quantile
are
omitted as no closed form analytic expression could be found, decorate with FunctionImputation for a numerical imputation.
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