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.
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
set.seed(27)
X <- Gamma(5, 2)
X
#> [1] "Gamma distribution (shape = 5, rate = 2)"
random(X, 10)
#> [1] 4.727510 3.628168 1.512156 4.771854 2.257310 3.645070 5.083710 2.509344
#> [9] 1.093361 2.021506
pdf(X, 2)
#> [1] 0.3907336
log_pdf(X, 2)
#> [1] -0.9397292
cdf(X, 4)
#> [1] 0.9003676
quantile(X, 0.7)
#> [1] 2.945181
cdf(X, quantile(X, 0.7))
#> [1] 0.7
quantile(X, cdf(X, 7))
#> [1] 7