# The Erlang Distribution

Posted by Beetle B. on Tue 06 June 2017

If the time between successive events is independent each with an exponential distribution with $$\lambda$$, then the total time $$X$$ that elapses before all of the next $$n$$ events occur is given by the Erlang distribution.

## Probability Density Function

Let $$\alpha=n,\beta=\frac{1}{\lambda}$$ in the Gamma distribution. $$n$$ is a positive integer.

\begin{equation*} f(x;\lambda,n)=\frac{\lambda\left(x\lambda\right)^{n-1}e^{-\lambda x}}{(n-1)!} \end{equation*}

for $$x\ge0$$

## Mean and Variance

Mean: $$\frac{n}{\lambda}$$

Variance: $$\frac{n}{\lambda^{2}}$$