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.
Returns an R6 object inheriting from class SDistribution.
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 \(a,b,c \ \in \ R\), \(a \le c \le b\).
The distribution is supported on \([a, b]\).
Tri(lower = 0, upper = 1, mode = 0.5, symmetric = FALSE)
N/A
N/A
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
distr6::Distribution -> distr6::SDistribution -> Triangular
nameFull name of distribution.
short_nameShort name of distribution for printing.
descriptionBrief description of the distribution.
aliasAlias of the distribution.
packagesPackages required to be installed in order to construct the distribution.
propertiesReturns distribution properties, including skewness type and symmetry.
Inherited methods
distr6::Distribution$cdf()distr6::Distribution$confidence()distr6::Distribution$correlation()distr6::Distribution$getParameterValue()distr6::Distribution$iqr()distr6::Distribution$liesInSupport()distr6::Distribution$liesInType()distr6::Distribution$parameters()distr6::Distribution$pdf()distr6::Distribution$prec()distr6::Distribution$print()distr6::Distribution$quantile()distr6::Distribution$rand()distr6::Distribution$setParameterValue()distr6::Distribution$stdev()distr6::Distribution$strprint()distr6::Distribution$summary()distr6::Distribution$workingSupport()
new()Creates a new instance of this R6 class.
Triangular$new(
lower = NULL,
upper = NULL,
mode = NULL,
symmetric = NULL,
decorators = NULL
)lower(numeric(1))
Lower limit of the Distribution, defined on the Reals.
upper(numeric(1))
Upper limit of the Distribution, defined on the Reals.
mode(numeric(1))
Mode of the distribution, if symmetric = TRUE then determined automatically.
symmetric(logical(1))
If TRUE then the symmetric Triangular distribution is constructed, where the mode is
automatically calculated. Otherwise mode can be set manually. Cannot be changed after
construction.
decorators(character())
Decorators to add to the distribution during construction.
Triangular$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)$descriptionmean()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.
mode()The mode of a probability distribution is the point at which the pdf is a local maximum, a distribution can be unimodal (one maximum) or multimodal (several maxima).
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).
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.
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.
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.
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.
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.
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.
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.
## ------------------------------------------------
## 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."