Definition – Gamma distribution

Contents
1. Definitions
2. Derivation of the gammadistribution
3. Uses of the gamma distribution
4. Mean.
5. Variance.
6. Moment generating function

1. Definition

Let $X$ be a random variable with domain $[0,\infty) = \mathbb{R}^+$ . We say that $X$ follows a gamma distribution if,

$g(X=x) = \frac{\beta^\alpha}{\Gamma (\alpha)}x^{\alpha -1}e^{-\beta x}$

Where $e$ is Euler’s constant. We write that $X \sim Gamma(\alpha, \beta)$ , where $\lambda= \frac{1}{\beta}$ , and $\lambda$ is the rate parameter of the Poisson distribution.

2. Derivation of the gamma distribution

We can derive the gamma distribution as the distribution of waiting times for $k\in\mathbb{N}$ events