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.

Gamma(shape, rate = 1)

## Arguments

shape The shape parameter. Can be any positive number. 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):

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

Other continuous distributions: Beta, Cauchy, ChiSquare, Exponential, FisherF, LogNormal, Logistic, Normal, StudentsT, Tukey, Uniform, Weibull

## Examples


set.seed(27)

X <- Gamma(5, 2)
X#> Gamma distribution (shape = 5, rate = 2)
random(X, 10)#>   4.727510 3.628168 1.512156 4.771854 2.257310 3.645070 5.083710 2.509344
#>   1.093361 2.021506
pdf(X, 2)#>  0.3907336log_pdf(X, 2)#>  -0.9397292
cdf(X, 4)#>  0.9003676quantile(X, 0.7)#>  2.945181
cdf(X, quantile(X, 0.7))#>  0.7quantile(X, cdf(X, 7))#>  7