# Gamma: Create a Gamma distribution In alexpghayes/distributions: Probability Distributions as S3 Objects

## 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/β. 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(α_1, β) and X_2 is Gamma(α_2, β), a function of these two variables of the form \frac{X_1}{X_1 + X_2} Beta(α_1, α_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

 1 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.

## 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 = α and rate = β.

Support: x \in (0, ∞)

Mean: \frac{α}{β}

Variance: \frac{α}{β^2}

Probability density function (p.m.f):

f(x) = \frac{β^{α}}{Γ(α)} x^{α - 1} e^{-β x}

Cumulative distribution function (c.d.f):

f(x) = \frac{Γ(α, β x)}{Γ{α}}

Moment generating function (m.g.f):

E(e^(tX)) = \Big(\frac{β}{ β - t}\Big)^{α}, \thinspace t < β

## Value

A Gamma object.

Other continuous distributions: Beta(), Cauchy(), ChiSquare(), Exponential(), Frechet(), GEV(), GP(), Gumbel(), LogNormal(), Logistic(), Normal(), RevWeibull(), StudentsT(), Tukey(), Uniform(), Weibull()
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 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))