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)

shape | The shape parameter. Can be any positive number. |
---|---|

rate | The rate parameter. Can be any positive number. Defaults
to |

A `Gamma`

object.

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`

#> Gamma distribution (shape = 5, rate = 2)#> [1] 4.727510 3.628168 1.512156 4.771854 2.257310 3.645070 5.083710 2.509344 #> [9] 1.093361 2.021506#> [1] 0.3907336#> [1] -0.9397292#> [1] 0.9003676#> [1] 2.945181#> [1] 0.7#> [1] 7