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

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):

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

Examples

Run this code


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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