What is the Erlang distribution?

The Erlang distribution is a probability distribution used to model the time between events in a Poisson process.

The Erlang distribution is named after the Danish mathematician Agner Krarup Erlang, who developed it in the early 1900s to model the arrival of telephone calls at a switchboard. It is a continuous probability distribution that is often used to model the time between events in a Poisson process, which is a stochastic process that models the occurrence of random events over time.

The Erlang distribution is a special case of the gamma distribution, which is a more general probability distribution that can also be used to model the time between events in a Poisson process. The Erlang distribution is characterized by two parameters: the shape parameter k, which determines the number of events that must occur before the distribution is considered to be complete, and the rate parameter λ, which determines the average rate at which events occur.

The probability density function (PDF) of the Erlang distribution is given by:

f(x;k,λ) = (λ^k * x^(k-1) * e^(-λx)) / (k-1)!

where x is the time between events, k is the shape parameter, λ is the rate parameter, and k! is the factorial of k.

The cumulative distribution function (CDF) of the Erlang distribution is given by:

F(x;k,λ) = 1 - ∑(i=0 to k-1) [(λx)^i / i! * e^(-λx)]

where ∑ denotes the sum from i=0 to k-1.

The mean and variance of the Erlang distribution are given by:

μ = k/λ

σ^2 = k/λ^2

where μ is the mean and σ^2 is the variance.