{(k-1)!\,}| cdf =\scriptstyle \frac{\gamma(k,\, \lambda x)}{(k \,-\, 1)!} \;=\; 1 \,-\, \sum_{n=0}^{k-1}\frac{1}{n!}e^{-\lambda x}(\lambda x)^{n}| mean =\scriptstyle \frac{k}{\lambda}\,| mode =\scriptstyle \frac{1}{\lambda}(k \,-\, 1)\, for \scriptstyle k \;\geq\; 1\, | variance =\scriptstyle \frac{k}{\lambda^2}\,| median =no simple closed form| skewness =\scriptstyle \frac{2}{\sqrt{k}}| scv =\scriptstyle \frac{1}{k}| kurtosis =\scriptstyle \frac{6}{k}| entropy =\scriptstyle (1 \,-\, k)\psi(k) \,+\, \ln\left[\frac{\gamma(k)}{\lambda}\right] \,+\, k| mgf =\scriptstyle \left(1 \,-\, \frac{t}{\lambda}\right)^{-k}\, for \scriptstyle t \;<\; \lambda\,| char =\scriptstyle \left(1 \,-\, \frac{it}{\lambda}\right)^{-k}\,| }} the erlang distribution is a two parameter family of continuous probability distributions with support x \;\in\; (0,\, \infty). the two parameters are: * a positive integer 'shape' k * a positive real 'rate' \lambda; sometimes the scale \mu, the inverse of the rate is used. the erlang distribution with shape parameter k equals to 1 simplifies to the exponential distribution. it is a special case of the gamma distribution. it is the distribution of a sum of k independent exponential variables with mean \mu. 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 now used in the fields of stochastic processes and of biomathematics. == characterization == === probability density function === the probability density function of the erlang distribution is :f(x; k,\lambda)={\lambda^k x^{k-1} e^{-\lambda x} \over (k-1)!}\quad\mbox{for }x, \lambda \geq 0, the parameter k is called the shape parameter, and the parameter \lambda is called the rate parameter. an alternative, but equivalent, parametrization uses the scale parameter \mu, which is the reciprocal of the rate parameter (i.e., \mu = 1/\lambda): :f(x; k,\mu)=\frac{ x^{k-1} e^{-\frac{x}{\mu}} }{\mu^k (k-1)!}\quad\mbox{for }x, \mu \geq 0. when the sc
Continuous_distributions Exponential_family_distributions Infinitely_divisible_probability_distributions Probability_distributions