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

$f_X(x) = \sum_{i=1}^n f_{Y_i}(x)\;p_i,$

where $Y_i$ is an exponentially distributed random variable with rate parameter $\lambda\,_i$, and $p_i$ is the probability that X will take on the form of the exponential distribution with rate $\lambda\,_i$. 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 $\lambda\,_1$ and probability q of them using their internet connection with rate $\lambda\,_2.$

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

$E(X) = \int_{-\infty}^\infty x f(x) dx= p_1\int_0^\infty x\lambda\,_1e^{-\lambda\,_1x} dx+ p_2\int_0^\infty x\lambda\,_2e^{-\lambda\,_2x} dx+ \cdots + p_n\int_0^\infty x\lambda\,_ne^{-\lambda\,_nx} dx$

$= \sum_{i=1}^n \frac{p_i}{\lambda\,_i}$

and

$E(X^2) = \int_{-\infty}^\infty x^2 f(x) \, dx = p_1\int_0^\infty x^2\lambda\,_1e^{-\lambda\,_1x} \, dx + p_2\int_0^\infty x^2\lambda\,_2e^{-\lambda\,_2x} \, dx+ \cdots + p_n\int_0^\infty x^2\lambda\,_ne^{-\lambda\,_nx}\, dx,$

$= \sum_{i=1}^n \frac{2}{\lambda\,_i^2}p_i,$

from which we can derive the variance.

The moment-generating function is given by

$E(e^{tx}) = \int_{-\infty}^\infty e^{tx} f(x) dx= p_1\int_0^\infty e^{tx}\lambda\,_1e^{-\lambda\,_1x} dx+ p_2\int_0^\infty e^{tx}\lambda\,_2e^{-\lambda\,_2x} dx+ \cdots + p_n\int_0^\infty e^{tx}\lambda\,_ne^{-\lambda\,_nx} dx$

$= \sum_{i=1}^n \frac{\lambda\,_i}{\lambda_i - t}p_i.$

See also

• phase-type distribution