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Probability density function | |||
Cumulative distribution function | |||
Parameters |
shape rate alt.: scale | ||
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Support | |||
CDF | |||
Mean | |||
Median | No simple closed form | ||
Mode | |||
Variance | |||
Skewness | |||
Excess kurtosis | |||
Entropy | |||
MGF | for | ||
CF |
The Erlang distribution is a two-parameter family of continuous probability distributions with support . The two parameters are:
The Erlang distribution is the distribution of a sum of independent exponential variables with mean each. Equivalently, it is the distribution of the time until the kth event of a Poisson process with a rate of . The Erlang and Poisson distributions are complementary, in that while the Poisson distribution counts the events that occur in a fixed amount of time, the Erlang distribution counts the amount of time until the occurrence of a fixed number of events. When , the distribution simplifies to the exponential distribution. The Erlang distribution is a special case of the gamma distribution in which the shape of the distribution is discretized.
The Erlang distribution was developed by A. K. Erlang to examine the number of telephone calls that 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.