distributions3 (version 0.1.2)

Gamma: Create a Gamma distribution

Description

Several important distributions are special cases of the Gamma distribution. When the shape parameter is 1, the Gamma is an exponential distribution with parameter \(1/\beta\). When the \(shape = n/2\) and \(rate = 1/2\), the Gamma is a equivalent to a chi squared distribution with n degrees of freedom. Moreover, if we have \(X_1\) is \(Gamma(\alpha_1, \beta)\) and \(X_2\) is \(Gamma(\alpha_2, \beta)\), a function of these two variables of the form \(\frac{X_1}{X_1 + X_2}\) \(Beta(\alpha_1, \alpha_2)\). This last property frequently appears in another distributions, and it has extensively been used in multivariate methods. More about the Gamma distribution will be added soon.

Usage

Gamma(shape, rate = 1)

Arguments

shape

The shape parameter. Can be any positive number.

rate

The rate parameter. Can be any positive number. Defaults to 1.

Value

A Gamma object.

Details

We recommend reading this documentation on https://alexpghayes.github.io/distributions3/, where the math will render with additional detail.

In the following, let \(X\) be a Gamma random variable with parameters shape = \(\alpha\) and rate = \(\beta\).

Support: \(x \in (0, \infty)\)

Mean: \(\frac{\alpha}{\beta}\)

Variance: \(\frac{\alpha}{\beta^2}\)

Probability density function (p.m.f):

$$ f(x) = \frac{\beta^{\alpha}}{\Gamma(\alpha)} x^{\alpha - 1} e^{-\beta x} $$

Cumulative distribution function (c.d.f):

$$ f(x) = \frac{\Gamma(\alpha, \beta x)}{\Gamma{\alpha}} $$

Moment generating function (m.g.f):

$$ E(e^{tX}) = \Big(\frac{\beta}{ \beta - t}\Big)^{\alpha}, \thinspace t < \beta $$

See Also

Other continuous distributions: Beta(), Cauchy(), ChiSquare(), Erlang(), Exponential(), FisherF(), Frechet(), GEV(), GP(), Gumbel(), LogNormal(), Logistic(), Normal(), RevWeibull(), StudentsT(), Tukey(), Uniform(), Weibull()

Examples

Run this code
# NOT RUN {
set.seed(27)

X <- Gamma(5, 2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))
# }

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