Inprobability theory, ahyperexponential distribution is acontinuous probability distribution whoseprobability density function of therandom variableX is given by

${\displaystyle f_X(x)=\sum _{i=1}^n{f}_{Y_i}(x)p_i,}$

where eachY_i is anexponentially distributed random variable with rate parameterλ_i, andp_i is the probability thatX will take on the form of the exponential distribution with rateλ_i.^[1] It is named thehyperexponential distribution since itscoefficient of variation is greater than that of the exponential distribution, whose coefficient of variation is 1, and thehypoexponential distribution, which has a coefficient of variation smaller than one. While theexponential distribution is the continuous analogue of thegeometric distribution, the hyperexponential distribution is not analogous to thehypergeometric distribution. The hyperexponential distribution is an example of amixture density.

An example of a hyperexponential random variable can be seen in the context oftelephony, where, if someone has a modem and a phone, their phone line usage could be modeled as a hyperexponential distribution where there is probabilityp of them talking on the phone with rateλ₁ and probabilityq of them using their internet connection with rate λ₂.

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

${\displaystyle E[X]={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}xf(x)dx=\sum _{i=1}^n{p}_i{\int }_0^{\mathrm{\infty }}x{\lambda }_i{e}^{-{\lambda }_{i}x}dx=\sum _{i=1}^n\frac{p_i}{{\lambda }_i}}$

and

${\displaystyle E\left[X^2\right]={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{x}^{2}f(x)dx=\sum _{i=1}^n{p}_i{\int }_0^{\mathrm{\infty }}{x}^2{\lambda }_i{e}^{-{\lambda }_{i}x}dx=\sum _{i=1}^n\frac{2}{{\lambda }_i^2}{p}_i,}$

from which we can derive the variance:^[2]

${\displaystyle \mathrm{Var}[X]=E\left[X^2\right]-E{\left[X\right]}^2=\sum _{i=1}^n\frac{2}{{\lambda }_i^2}{p}_i-{\left[\sum _{i=1}^n\frac{p_i}{{\lambda }_i}\right]}^2={\left[\sum _{i=1}^n\frac{p_i}{{\lambda }_i}\right]}^2+\sum _{i=1}^n\sum _{j=1}^n{p}_i{p}_j{\left(\frac{1}{{\lambda }_i}-\frac{1}{{\lambda }_j}\right)}^2.}$

The standard deviation exceeds the mean in general (except for the degenerate case of all theλs being equal), so thecoefficient of variation is greater than 1.

Themoment-generating function is given by

${\displaystyle E\left[e^{tx}\right]={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{tx}f(x)dx=\sum _{i=1}^n{p}_i{\int }_0^{\mathrm{\infty }}{e}^{tx}{\lambda }_i{e}^{-{\lambda }_{i}x}dx=\sum _{i=1}^n\frac{{\lambda }_i}{{\lambda }_i-t}{p}_i.}$

A givenprobability distribution, including aheavy-tailed distribution, can be approximated by a hyperexponential distribution by fitting recursively to different time scales usingProny's method.^[3]

• Phase-type distribution
• Hyper-Erlang distribution
• Lomax distribution (continuous mixture of exponentials)

• Continuous distributions

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