distributions3 (version 0.2.1)

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)

Value

A Gamma object.

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: $x\in (0,\mathrm{\infty })$

Mean: $\frac{\alpha }{\beta }$

Variance: $\frac{\alpha }{{\beta }^2}$

Probability density function (p.m.f):

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

Cumulative distribution function (c.d.f):

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

Moment generating function (m.g.f):

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

See Also

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

Examples

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))