Exponential vs Erlang Distribution

exponential distributionpoisson distributionprobability distributionsstatistics

I am confused as to what the primary difference is between an Exponential and an Erlang Random Variable since both are also related with a Poisson Distribution

From what I've read, Erlang refers to the xth occurences within a set time interval while Exponential also refers to the intervals.

Suppose we have the following examples:

  1. The distribution of the arrivals of a shipment is Poisson Distributed. Let A be the time between two shipment arrivals at the company. Does A refer to an Erlang RV referring to the time it takes until the second occurence happens?

  2. If for example we have the variable B which refers to the time between each arrival? Is this now an exponential random variable?

Best Answer

The relationship between the exponential and Erlang distributions is exactly analogous to the relationship between a geometric and negative binomial random variable.

A geometric random variable counts the number of Bernoulli trials needed to obtain the first success. A negative binomial random variable generalizes this idea to count the number of trials needed to obtain $r$ successes. Thus a geometric distribution is a special case of the negative binomial with $r = 1$. Another way to conceptualize a negative binomial random variable is that it is the sum of $r$ independent and identically distributed geometric random variables.

Similarly, an exponential random variable counts the time until the first event of interest occurs. An Erlang random variable counts the time it takes to observe $r$ events. And so it is also the sum of $r$ IID exponential random variables.

The following claim is incorrect:

Erlang refers to the $x^{\rm th}$ occurrences within a set time interval while Exponential also refers to the intervals.

The incorrect part of this claim is "within a set time interval." There is no such set time interval; we are only interested in the time it takes for the $x^{\rm th}$ occurrence, no matter how long that may be.

As for your example questions:

  1. Not quite. If arrivals are Poisson distributed with rate $\lambda$ per unit time, then the time between two consecutive shipment arrivals is simply exponential with rate $\lambda$. Since the exponential distribution is a special case of the Erlang with $r = 1$, it is "Erlang" with shape parameter $r = 1$ and rate $\lambda$. What is genuinely (nontrivially) Erlang, however, is the total time until we observe two arrivals, which includes the time it takes to wait for the first arrival, plus the time between the first arrival and the second. In this case, the total waiting time is Erlang with shape $r = 2$ and rate $\lambda$.

  2. As already implied above, the interrarival time is exponential for a (homogeneous) Poisson process.

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