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.
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(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