# Hyper-exponential distribution

In probability theory, a **hyper-exponential distribution** is a continuous distribution such that the probability density function of the random variable is given by

where is an exponentially distributed random variable with rate parameter , and is the probability that *X* will take on the form of the exponential distribution with rate .^{[1]} It is named the *hyper*-exponential distribution since its coefficient of variation is greater than that of the exponential distribution, whose coefficient of variation is 1, and the hypoexponential distribution, which has a coefficient of variation less than one. While the exponential distribution is the continuous analogue of the geometric distribution, the hyper-exponential distribution is not analogous to the hypergeometric distribution. The hyper-exponential distribution is an example of a mixture density.

An example of a hyper-exponential random variable can be seen in the context of telephony, where, if someone has a modem and a phone, their phone line usage could be modeled as a hyper-exponential distribution where there is probability p of them talking on the phone with rate and probability *q* of them using their internet connection with rate

## Properties of the hyper-exponential distribution

Since the expected value of a sum is the sum of the expected values, the expected value of a hyper-exponential random variable can be shown as

and

from which we can derive the variance.

The moment-generating function is given by

## See also

## References

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