Mathematical and statistical functions for the Triangular distribution, which is commonly used to model population data where only the minimum, mode and maximum are known (or can be reliably estimated), also to model the sum of standard uniform distributions.

Value

Returns an R6 object inheriting from class SDistribution.

Details

The Triangular distribution parameterised with lower limit, $$a$$, upper limit, $$b$$, and mode, $$c$$, is defined by the pdf,

$$f(x) = 0, x < a$$
$$f(x) = 2(x-a)/((b-a)(c-a)), a \le x < c$$
$$f(x) = 2/(b-a), x = c$$
$$f(x) = 2(b-x)/((b-a)(b-c)), c < x \le b$$
$$f(x) = 0, x > b$$ for $$a,b,c \ \in \ R$$, $$a \le c \le b$$.

Distribution support

The distribution is supported on $$[a, b]$$.

Default Parameterisation

Tri(lower = 0, upper = 1, mode = 0.5, symmetric = FALSE)

N/A

N/A

References

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

Other continuous distributions: Arcsine, BetaNoncentral, Beta, Cauchy, ChiSquaredNoncentral, ChiSquared, Dirichlet, Erlang, Exponential, FDistributionNoncentral, FDistribution, Frechet, Gamma, Gompertz, Gumbel, InverseGamma, Laplace, Logistic, Loglogistic, Lognormal, MultivariateNormal, Normal, Pareto, Poisson, Rayleigh, ShiftedLoglogistic, StudentTNoncentral, StudentT, Uniform, Wald, Weibull

Other univariate distributions: Arcsine, Arrdist, Bernoulli, BetaNoncentral, Beta, Binomial, Categorical, Cauchy, ChiSquaredNoncentral, ChiSquared, Degenerate, DiscreteUniform, Empirical, Erlang, Exponential, FDistributionNoncentral, FDistribution, Frechet, Gamma, Geometric, Gompertz, Gumbel, Hypergeometric, InverseGamma, Laplace, Logarithmic, Logistic, Loglogistic, Lognormal, Matdist, NegativeBinomial, Normal, Pareto, Poisson, Rayleigh, ShiftedLoglogistic, StudentTNoncentral, StudentT, Uniform, Wald, Weibull, WeightedDiscrete

Super classes

distr6::Distribution -> distr6::SDistribution -> Triangular

Public fields

name

Full name of distribution.

short_name

Short name of distribution for printing.

description

Brief description of the distribution.

alias

Alias 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

Inherited methods

Method new()

Creates a new instance of this R6 class.

Triangular$new( lower = NULL, upper = NULL, mode = NULL, symmetric = NULL, decorators = NULL ) Arguments lower (numeric(1)) Lower limit of the Distribution, defined on the Reals. upper (numeric(1)) Upper limit of the Distribution, defined on the Reals. mode (numeric(1)) Mode of the distribution, if symmetric = TRUE then determined automatically. symmetric (logical(1)) If TRUE then the symmetric Triangular distribution is constructed, where the mode is automatically calculated. Otherwise mode can be set manually. Cannot be changed after construction. decorators (character()) Decorators to add to the distribution during construction. Examples Triangular$new(lower = 2, upper = 5, symmetric = TRUE)
Triangular$new(lower = 2, upper = 5, mode = 4, symmetric = FALSE) # You can view the type of Triangular distribution with$description
Triangular$new(symmetric = TRUE)$description
Triangular$new(symmetric = FALSE)$description

Method 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.

Arguments

which

(character(1) | numeric(1)
Ignored if distribution is unimodal. Otherwise "all" returns all modes, otherwise specifies which mode to return.

Method median()

Returns the median of the distribution. If an analytical expression is available returns distribution median, otherwise if symmetric returns self$mean, otherwise returns self$quantile(0.5).

Arguments

...

Unused.

Method 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.

Arguments

excess

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

...

Unused.

Method 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.

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.

Arguments

z

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

...

Unused.

Method clone()

The objects of this class are cloneable with this method.

Triangular$clone(deep = FALSE) Arguments deep Whether to make a deep clone. Examples  ## ------------------------------------------------ ## Method Triangular$new
## ------------------------------------------------

Triangular$new(lower = 2, upper = 5, symmetric = TRUE) #> Tri(lower = 2, mode = NULL, symmetric = TRUE, upper = 5) Triangular$new(lower = 2, upper = 5, mode = 4, symmetric = FALSE)
#> Tri(lower = 2, mode = 4, symmetric = FALSE, upper = 5)

# You can view the type of Triangular distribution with $description Triangular$new(symmetric = TRUE)$description #> [1] "Symmetric Triangular Probability Distribution." Triangular$new(symmetric = FALSE)\$description
#> [1] "Triangular Probability Distribution."