Hypoexponential
Parameters	${\displaystyle {\lambda }_1,\dots ,{\lambda }_k>0}$ rates (real)
Support	${\displaystyle x\in [0;\mathrm{\infty })}$
PDF	Expressed as aphase-type distribution ${\displaystyle -\mathit{\alpha }{e}^{x\mathrm{\Theta }}\mathrm{\Theta }\mathbf{1}}$ Has no other simple form; see article for details
CDF	Expressed as a phase-type distribution ${\displaystyle 1-\mathit{\alpha }{e}^{x\mathrm{\Theta }}\mathbf{1}}$
Mean	${\displaystyle \sum _{i=1}^{k}1/{\lambda }_i}$
Median	General closed form does not exist[1]
Mode	${\displaystyle (k-1)/\lambda }$ if${\displaystyle {\lambda }_k=\lambda }$, for all k
Variance	${\displaystyle \sum _{i=1}^{k}1/{\lambda }_i^2}$
Skewness	${\displaystyle 2(\sum _{i=1}^{k}1/{\lambda }_i^3)/(\sum _{i=1}^{k}1/{\lambda }_i^2{)}^{3/2}}$
Excess kurtosis	no simple closed form
MGF	${\displaystyle \mathit{\alpha }(tI-\mathrm{\Theta }{)}^{-1}\mathrm{\Theta }\mathbf{1}}$
CF	${\displaystyle \mathit{\alpha }(itI-\mathrm{\Theta }{)}^{-1}\mathrm{\Theta }\mathbf{1}}$

Inprobability theory thehypoexponential distribution or thegeneralizedErlang distribution is acontinuous distribution, that has found use in the same fields as the Erlang distribution, such asqueueing theory,teletraffic engineering and more generally instochastic processes. It is called the hypoexponetial distribution as it has acoefficient of variation less than one, compared to thehyper-exponential distribution which has coefficient of variation greater than one and theexponential distribution which has coefficient of variation of one.

TheErlang distribution is a series ofk exponential distributions all with rate${\displaystyle \lambda }$. The hypoexponential is a series ofk exponential distributions each with their own rate${\displaystyle {\lambda }_i}$, the rate of the${\displaystyle i^{th}}$ exponential distribution. If we havek independently distributed exponential random variables${\displaystyle X_i}$, then the random variable,

${\displaystyle X=\sum _{i=1}^k{X}_i}$

is hypoexponentially distributed. The hypoexponential has a minimum coefficient of variation of${\displaystyle 1/k}$.

As a result of the definition it is easier to consider this distribution as a special case of thephase-type distribution.^[2] The phase-type distribution is the time to absorption of a finite stateMarkov process. If we have ak+1 state process, where the firstk states are transient and the statek+1 is an absorbing state, then the distribution of time from the start of the process until the absorbing state is reached is phase-type distributed. This becomes the hypoexponential if we start in the first 1 and move skip-free from statei toi+1 with rate${\displaystyle {\lambda }_i}$ until statek transitions with rate${\displaystyle {\lambda }_k}$ to the absorbing statek+1. This can be written in the form of a subgenerator matrix,

For simplicity denote the above matrix${\displaystyle \mathrm{\Theta }\equiv \mathrm{\Theta }({\lambda }_1,\dots ,{\lambda }_k)}$. If the probability of starting in each of thek states is

${\displaystyle \mathit{\alpha }=(1,0,\dots ,0)}$

then${\displaystyle Hypo({\lambda }_1,\dots ,{\lambda }_k)=PH(\mathit{\alpha },\mathrm{\Theta }).}$

Where the distribution has two parameters (${\displaystyle {\lambda }_1\ne {\lambda }_2}$) the explicit forms of the probability functions and the associated statistics are:^[3]

CDF:${\displaystyle F(x)=1-\frac{{\lambda }_2}{{\lambda }_2-{\lambda }_1}{e}^{-{\lambda }_{1}x}-\frac{{\lambda }_1}{{\lambda }_1-{\lambda }_2}{e}^{-{\lambda }_{2}x}}$

PDF:${\displaystyle f(x)=\frac{{\lambda }_1{\lambda }_2}{{\lambda }_1-{\lambda }_2}(e^{-x{\lambda }_2}-e^{-x{\lambda }_1})}$

Mean:${\displaystyle \frac{1}{{\lambda }_1}+\frac{1}{{\lambda }_2}}$

Variance:${\displaystyle \frac{1}{{\lambda }_1^2}+\frac{1}{{\lambda }_2^2}}$

Coefficient of variation:${\displaystyle \frac{\sqrt{{\lambda }_1^2+{\lambda }_2^2}}{{\lambda }_1+{\lambda }_2}}$

The coefficient of variation is always less than 1.

Given the sample mean (${\displaystyle \overline{x}}$) and sample coefficient of variation (${\displaystyle c}$), the parameters${\displaystyle {\lambda }_1}$ and${\displaystyle {\lambda }_2}$ can be estimated as follows:

${\displaystyle {\lambda }_1=\frac{2}{\overline{x}}{\left[1+\sqrt{1+2(c^2-1)}\right]}^{-1}}$

${\displaystyle {\lambda }_2=\frac{2}{\overline{x}}{\left[1-\sqrt{1+2(c^2-1)}\right]}^{-1}}$

These estimators can be derived from the methods of moments by setting${\displaystyle \frac{1}{{\lambda }_1}+\frac{1}{{\lambda }_2}=\overline{x}}$ and${\displaystyle \frac{\sqrt{{\lambda }_1^2+{\lambda }_2^2}}{{\lambda }_1+{\lambda }_2}=c}$.

The resulting parameters${\displaystyle {\lambda }_1}$ and${\displaystyle {\lambda }_2}$ are real values if${\displaystyle c^2\in [0.5,1]}$.

A random variable${\displaystyle X\sim Hypo({\lambda }_1,\dots ,{\lambda }_k)}$ hascumulative distribution function given by,

${\displaystyle F(x)=1-\mathit{\alpha }{e}^{x\mathrm{\Theta }}\mathbf{1}}$

anddensity function,

${\displaystyle f(x)=-\mathit{\alpha }{e}^{x\mathrm{\Theta }}\mathrm{\Theta }\mathbf{1},}$

where${\displaystyle \mathbf{1}}$ is acolumn vector of ones of the sizek and${\displaystyle e^A}$ is thematrix exponential ofA. When${\displaystyle {\lambda }_i\ne {\lambda }_j}$ for all${\displaystyle i\ne j}$, thedensity function can be written as

${\displaystyle f(x)=\sum _{i=1}^k{\lambda }_i{e}^{-x{\lambda }_i}\left(\prod _{j=1,j\ne i}^k\frac{{\lambda }_j}{{\lambda }_j-{\lambda }_i}\right)=\sum _{i=1}^k{\ell }_i(0){\lambda }_i{e}^{-x{\lambda }_i}}$

where${\displaystyle {\ell }_1(x),\dots ,{\ell }_k(x)}$ are theLagrange basis polynomials associated with the points${\displaystyle {\lambda }_1,\dots ,{\lambda }_k}$.

The distribution hasLaplace transform of

${\displaystyle \mathcal{L}\{f(x)\}=-\mathit{\alpha }(sI-\mathrm{\Theta }{)}^{-1}\mathrm{\Theta }\mathbf{1}}$

Which can be used to find moments,

${\displaystyle E[X^n]=(-1{)}^{n}n!\mathit{\alpha }{\mathrm{\Theta }}^{-n}\mathbf{1}.}$

In the general casewhere there are${\displaystyle a}$ distinct sums of exponential distributionswith rates${\displaystyle {\lambda }_1,{\lambda }_2,\cdots ,{\lambda }_a}$ and a number of terms in eachsum equals to${\displaystyle r_1,r_2,\cdots ,r_a}$ respectively. The cumulativedistribution function for${\displaystyle t\ge 0}$ is given by

${\displaystyle F(t)=1-\left(\prod _{j=1}^a{\lambda }_j^{r_j}\right)\sum _{k=1}^a\sum _{l=1}^{r_k}\frac{{\mathrm{\Psi }}_{k,l}(-{\lambda }_k)t^{r_k-l}\exp (-{\lambda }_{k}t)}{(r_k-l)!(l-1)!},}$

with

${\displaystyle {\mathrm{\Psi }}_{k,l}(x)=-\frac{{\mathrm{\partial }}^{l-1}}{\mathrm{\partial }{x}^{l-1}}\left(\prod _{j=0,j\ne k}^a{\left({\lambda }_j+x\right)}^{-r_j}\right).}$

with the additional convention${\displaystyle {\lambda }_0=0,r_0=1}$.^[4]

This distribution has been used in population genetics,^[5] cell biology,^[6][7] and queuing theory.^[8][9]

• Phase-type distribution
• Coxian distribution

• M. F. Neuts. (1981) Matrix-Geometric Solutions in Stochastic Models: an Algorithmic Approach, Chapter 2: Probability Distributions of Phase Type; Dover Publications Inc.
• G. Latouche, V. Ramaswami. (1999) Introduction to Matrix Analytic Methods in Stochastic Modelling, 1st edition. Chapter 2: PH Distributions; ASA SIAM,
• Colm A. O'Cinneide (1999).Phase-type distribution: open problems and a few properties, Communication in Statistic - Stochastic Models, 15(4), 731–757.
• L. Leemis and J. McQueston (2008).Univariate distribution relationships, The American Statistician, 62(1), 45—53.
• S. Ross. (2007) Introduction to Probability Models, 9th edition, New York: Academic Press

• Continuous distributions

• Articles with short description
• Short description matches Wikidata