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