Hyper-exponential Distribution

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

where Y_i is an exponentially distributed random variable with rate parameter λ_i, and p_i is the probability that X will take on the form of the exponential distribution with rate λ_i. 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.