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.
We recommend reading this documentation on
https://pkg.mitchelloharawild.com/distributional/, where the math
will render nicely.
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):
$$
E(e^{tX}) = \Big(\frac{\beta}{ \beta - t}\Big)^{\alpha}, \thinspace t < \beta
$$