Gamma Distribution

Used to model the total waiting time until multiple events occur. Denoted by $\text{Gamma}(\alpha, \beta)$, where $\alpha > 0$, $\beta > 0$. Has the probably density function:

$$f(x;\alpha,\beta) = \begin{cases} \displaystyle \frac{x^{\alpha - 1}}{\beta^\alpha \Gamma(\alpha)} \exp\left( \frac{-x}{\beta} \right) &;\; \text{for } x > 0 \\ \displaystyle 0 &;\; \text{otherwise} \end{cases}$$

Here $\Gamma$ is the gamma function. A short revision on gamma function is included below.

Gamma function

$$\tau(x) = \int_0^\infty y^{x - 1} e^{-y} \, \text{d}y \;\;\; \text{for } x > 0$$

Recursive. Relates to the factorial function, for integer values.

$$\tau(x) = (x - 1)!$$

$\tau(1) = 1$ and $\tau(0.5) = \sqrt{\pi}$.

Properties

Mean

$$\mu = \alpha \beta$$

Variance

$$\sigma^2 = \alpha \beta^2$$

Relation with Exponential Distribution

$$\text{Gamma}(1, \frac{1}{\lambda}) \equiv \text{Exp}(\lambda)$$

Relation with Chi-Squared Distribution

$$\text{Gamma}\left(\frac{k}{2}, \frac{1}{2}\right) \equiv \chi^2(k)$$

Relation with Normal Distribution

If $X \sim N(\mu, \sigma^2)$ then: $$

$$\frac{Z^2}{2} \sim \text{Gamma}\left(\frac{1}{2}\right)$$