Erlang distribution

The Erlang distribution is a two-parameter family of continuous probability distributions with support x ∈ [ 0 , ∞ ) {displaystyle xin [0,infty )} . The two parameters are: The Erlang distribution is a two-parameter family of continuous probability distributions with support x ∈ [ 0 , ∞ ) {displaystyle xin [0,infty )} . The two parameters are: The Erlang distribution with shape parameter k = 1 {displaystyle k=1} simplifies to the exponential distribution. It is a special case of the gamma distribution. It is the distribution of a sum of k {displaystyle k} independent exponential variables with mean 1 / λ {displaystyle 1/lambda } each. The Erlang distribution was developed by A. K. Erlang to examine the number of telephone calls which might be made at the same time to the operators of the switching stations. This work on telephone traffic engineering has been expanded to consider waiting times in queueing systems in general. The distribution is also used in the field of stochastic processes. The probability density function of the Erlang distribution is The parameter k is called the shape parameter, and the parameter λ {displaystyle lambda } is called the rate parameter. An alternative, but equivalent, parametrization uses the scale parameter μ {displaystyle mu } , which is the reciprocal of the rate parameter (i.e., μ = 1 / λ {displaystyle mu =1/lambda } ): When the scale parameter μ {displaystyle mu } equals 2, the distribution simplifies to the chi-squared distribution with 2k degrees of freedom. It can therefore be regarded as a generalized chi-squared distribution for even numbers of degrees of freedom. The cumulative distribution function of the Erlang distribution is where γ {displaystyle gamma } is the lower incomplete gamma function.The CDF may also be expressed as