where each Yi is an exponentially distributed random variable with rate parameter λi, and pi is the probability that X will take on the form of the exponential distribution with rate λi. 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 λ1 and probability q of them using their internet connection with rate λ2.
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
from which we can derive the variance:
The standard deviation exceeds the mean in general (except for the degenerate case of all the λs being equal), so the coefficient of variation is greater than 1.
The moment-generating function is given by
A given probability distribution, including a heavy-tailed distribution, can be approximated by a hyperexponential distribution by fitting recursively to different time scales using Prony's method.
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