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: \(\frac{n}{\lambda^{2}}\)