Mathematical and statistical functions for the Gamma distribution, which is commonly used as the prior in Bayesian modelling, the convolution of exponential distributions, and to model waiting times.

Returns an R6 object inheriting from class SDistribution.

The Gamma distribution parameterised with shape, \(\alpha\), and rate, \(\beta\), is defined by the pdf, $$f(x) = (\beta^\alpha)/\Gamma(\alpha)x^{\alpha-1}exp(-x\beta)$$ for \(\alpha, \beta > 0\).

The distribution is supported on the Positive Reals.

Gamma(shape = 1, rate = 1)

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`

,
`Gompertz`

,
`Gumbel`

,
`InverseGamma`

,
`Laplace`

,
`Logistic`

,
`Loglogistic`

,
`Lognormal`

,
`MultivariateNormal`

,
`Normal`

,
`Pareto`

,
`Poisson`

,
`Rayleigh`

,
`ShiftedLoglogistic`

,
`StudentTNoncentral`

,
`StudentT`

,
`Triangular`

,
`Uniform`

,
`Wald`

,
`Weibull`

Other univariate distributions:
`Arcsine`

,
`Arrdist`

,
`Bernoulli`

,
`BetaNoncentral`

,
`Beta`

,
`Binomial`

,
`Categorical`

,
`Cauchy`

,
`ChiSquaredNoncentral`

,
`ChiSquared`

,
`Degenerate`

,
`DiscreteUniform`

,
`Empirical`

,
`Erlang`

,
`Exponential`

,
`FDistributionNoncentral`

,
`FDistribution`

,
`Frechet`

,
`Geometric`

,
`Gompertz`

,
`Gumbel`

,
`Hypergeometric`

,
`InverseGamma`

,
`Laplace`

,
`Logarithmic`

,
`Logistic`

,
`Loglogistic`

,
`Lognormal`

,
`Matdist`

,
`NegativeBinomial`

,
`Normal`

,
`Pareto`

,
`Poisson`

,
`Rayleigh`

,
`ShiftedLoglogistic`

,
`StudentTNoncentral`

,
`StudentT`

,
`Triangular`

,
`Uniform`

,
`Wald`

,
`Weibull`

,
`WeightedDiscrete`

`distr6::Distribution`

-> `distr6::SDistribution`

-> `Gamma`

`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.

`distr6::Distribution$cdf()`

`distr6::Distribution$confidence()`

`distr6::Distribution$correlation()`

`distr6::Distribution$getParameterValue()`

`distr6::Distribution$iqr()`

`distr6::Distribution$liesInSupport()`

`distr6::Distribution$liesInType()`

`distr6::Distribution$median()`

`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.

```
Gamma$new(
shape = NULL,
rate = NULL,
scale = NULL,
mean = NULL,
decorators = NULL
)
```

`shape`

`(numeric(1))`

Shape parameter, defined on the positive Reals.`rate`

`(numeric(1))`

Rate parameter of the distribution, defined on the positive Reals.`scale`

`numeric(1))`

Scale parameter of the distribution, defined on the positive Reals.`scale = 1/rate`

. If provided`rate`

is ignored.`mean`

`(numeric(1))`

Alternative parameterisation of the distribution, defined on the positive Reals. If given then`rate`

and`scale`

are ignored. Related by`mean = shape/rate`

.`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.

`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).

`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.