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

:$f\_X(x)\; =\; sum\_\{i=1\}^n\; f\_\{Y\_i\}(y)\; p\_i,$

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

:$E(X)\; =\; int\_\{-infty\}^infty\; x\; f(x)\; dx=\; p\_1int\_0^infty\; xlambda,\_1e^\{-lambda,\_1x\}\; dx+\; p\_2int\_0^infty\; xlambda,\_2e^\{-lambda,\_2x\}\; dx+\; cdots\; +\; p\_nint\_0^infty\; xlambda,\_ne^\{-lambda,\_nx\}\; dx$

::$=\; sum\_\{i=1\}^n\; frac\{p\_i\}\{lambda,\_i\}$

and

:$E(X^2)\; =\; int\_\{-infty\}^infty\; x^2\; f(x)\; ,\; dx\; =\; p\_1int\_0^infty\; x^2lambda,\_1e^\{-lambda,\_1x\}\; ,\; dx\; +\; p\_2int\_0^infty\; x^2lambda,\_2e^\{-lambda,\_2x\}\; ,\; dx+\; cdots\; +\; p\_nint\_0^infty\; x^2lambda,\_ne^\{-lambda,\_nx\},\; dx,$

::$=\; sum\_\{i=1\}^n\; frac\{2\}\{lambda,\_i^2\}p\_i,$

from which we can derive the variance.

The moment-generating function is given by

:$E(e^\{tx\})\; =\; int\_\{-infty\}^infty\; e^\{tx\}\; f(x)\; dx=\; p\_1int\_0^infty\; e^\{tx\}lambda,\_1e^\{-lambda,\_1x\}\; dx+\; p\_2int\_0^infty\; e^\{tx\}lambda,\_2e^\{-lambda,\_2x\}\; dx+\; cdots\; +\; p\_nint\_0^infty\; e^\{tx\}lambda,\_ne^\{-lambda,\_nx\}\; dx$

::$=\; sum\_\{i=1\}^n\; frac\{lambda,\_i\}\{lambda\_i\; -\; t\}p\_i.$

ee also

* phase-type distribution