Gamma distribution

The Erlang distribution is a special case of a more general distribution called the Gamma distribution which has probability density function

\[ f(x) \;\;=\;\; \begin{cases} \dfrac {\beta^\alpha }{\Gamma(\alpha)} x^{\alpha - 1} e^{-x\beta}& \text{if }x \gt 0 \\ 0 & \text{otherwise} \end{cases} \]

The Erlang distribution is the same as this with \(\alpha = k\) and \(\beta = \lambda\), but the Gamma distribution also allows non-integer values of \(\alpha\) by replacing the factorial in the Erlang distribution's pdf by a gamma function, \((k-1)! \to \Gamma(\alpha)\).

Probabilities

Because of the strong relationship between Erlang and Gamma distributions, we will wait until the section about Gamma distributions in the next chapter before explaining how to find probabilities for Erlang random variables.