2.11.3 Erlang PDFA random variable Lk is said to be a kth-order Erlang random variable if its pdf is given by 
The mean, variance, and s-transforms are 
The Erlang pdf arises frequently in the application of Poisson processes. As we will show in Section 2.12, the kth-order Erlang pdf describes the probabilistic behavior of time until the kth arrival in a Poisson process. Note that for k = 1, the Erlang reduces to the familiar negative exponential pdf. In fact, Lk may be thought of as the sum of k independent, identically distributed negative exponential random variables, each with mean 1/  ; this provides a
convenient way to understand intuitively the Erlang pdf and to remember the mean, variance, and s-transform. |