Gamma: Create a Gamma distribution

In distributions3: Probability Distributions as S3 Objects

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.

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

Value

A Gamma object.

See Also

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

Examples

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))

distributions3 documentation built on Sept. 30, 2024, 9:37 a.m.