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`

`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.

`properties`

Returns distribution properties, including skewness type and symmetry.

`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)$description
```

`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.

`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."
```