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| {{Probability distribution |
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| name =Hypoexponential|
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| type =density|
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| pdf_image =|
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| cdf_image =|
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| parameters =<math>\lambda_{1},\dots,\lambda_{k} > 0\,</math> rates ([[real number|real]])|
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| support =<math>x \in [0; \infty)\!</math>|
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| pdf =Expressed as a [[phase-type distribution]]<br /><math>-\boldsymbol{\alpha}e^{x\Theta}\Theta\boldsymbol{1}</math><br />Has no other simple form; see article for details|
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| cdf =Expressed as a phase-type distribution<br /><math>1-\boldsymbol{\alpha}e^{x\Theta}\boldsymbol{1}</math>|
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| mean =<math>\sum^{k}_{i=1}1/\lambda_{i}\,</math>|
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| mode =<math>(k-1)/\lambda</math> if <math>\lambda_{k} = \lambda</math>, for all k|
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| variance =<math>\sum^{k}_{i=1}1/\lambda^2_{i}</math>|
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| median =<math>\ln(2)\sum^{k}_{i=1}1/\lambda_{i}</math>|
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| skewness =<math>2(\sum^{k}_{i=1}1/\lambda_{i}^3)/(\sum^{k}_{i=1}1/\lambda_{i}^2)^{3/2}</math>|
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| kurtosis =no simple closed form|
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| entropy =|
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| mgf =<math>\boldsymbol{\alpha}(tI-\Theta)^{-1}\Theta\mathbf{1}</math>|
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| char =<math>\boldsymbol{\alpha}(itI-\Theta)^{-1}\Theta\mathbf{1}</math>|
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| }}
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| In [[probability theory]] the '''hypoexponential distribution''' or the '''generalized [[Erlang distribution]]''' is a [[continuous distribution]], that has found use in the same fields as the Erlang distribution, such as [[queueing theory]], [[teletraffic engineering]] and more generally in [[stochastic processes]]. It is called the hypoexponetial distribution as it has a [[coefficient of variation]] less than one, compared to the [[hyper-exponential distribution]] which has coefficient of variation greater than one and the [[exponential distribution]] which has coefficient of variation of one.
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| ==Overview==
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| The Erlang distribution is a series of ''k'' exponential distributions all with rate <math>\lambda</math>. The hypoexponential is a series of ''k'' exponential distributions each with their own rate <math>\lambda_{i}</math>, the rate of the <math>i^{th}</math> exponential distribution. If we have ''k'' independently distributed exponential random variables <math>\boldsymbol{X}_{i}</math>, then the random variable,
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| :<math>
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| \boldsymbol{X}=\sum^{k}_{i=1}\boldsymbol{X}_{i}
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| </math>
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| is hypoexponentially distributed. The hypoexponential has a minimum coefficient of variation of <math>1/k</math>.
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| ===Relation to the phase-type distribution===
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| As a result of the definition it is easier to consider this distribution as a special case of the [[phase-type distribution]]. The phase-type distribution is the time to absorption of a finite state [[Markov process]]. If we have a ''k+1'' state process, where the first ''k'' states are transient and the state ''k+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 state ''i'' to ''i+1'' with rate <math>\lambda_{i}</math> until state ''k'' transitions with rate <math>\lambda_{k}</math> to the absorbing state ''k+1''. This can be written in the form of a subgenerator matrix,
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| :<math> | |
| \left[\begin{matrix}-\lambda_{1}&\lambda_{1}&0&\dots&0&0\\
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| 0&-\lambda_{2}&\lambda_{2}&\ddots&0&0\\
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| \vdots&\ddots&\ddots&\ddots&\ddots&\vdots\\
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| 0&0&\ddots&-\lambda_{k-2}&\lambda_{k-2}&0\\
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| 0&0&\dots&0&-\lambda_{k-1}&\lambda_{k-1}\\
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| 0&0&\dots&0&0&-\lambda_{k}
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| \end{matrix}\right]\; .
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| </math>
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| For simplicity denote the above matrix <math>\Theta\equiv\Theta(\lambda_{1},\dots,\lambda_{k})</math>. If the probability of starting in each of the ''k'' states is
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| :<math>
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| \boldsymbol{\alpha}=(1,0,\dots,0)
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| </math>
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| then <math>Hypo(\lambda_{1},\dots,\lambda_{k})=PH(\boldsymbol{\alpha},\Theta).</math>
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| ==Two parameter case==
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| Where the distribution has two parameters (<math>\mu_1 \neq \mu_2</math>) the explicit forms of the probability functions and the associated statistics are<ref>{{cite doi|10.1002/0471200581.ch1}}</ref>
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| CDF: <math>F(x) = 1 - \frac{\mu_2}{\mu_2-\mu_1}e^{-\mu_1x} + \frac{\mu_1}{\mu_2-\mu_1}e^{-\mu_2x}</math>
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| PDF: <math>f(x) = \frac{\mu_1\mu_2}{\mu_1-\mu_2}( e^{-x \mu_2} - e^{-x \mu_1} )</math>
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| Mean: <math>\frac{1}{\mu_1}+\frac{1}{\mu_2}</math>
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| Variance: <math>\frac{1}{\mu_1^2}+\frac{1}{\mu_2^2}</math>
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| Coefficient of variation: <math>\frac{\sqrt{\mu_1 + \mu_2}}{ \mu_1 + \mu_2 }</math>
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| The coefficient of variation is always < 1.
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| Given the sample mean (<math>\bar{x}</math>) and sample coefficient of variation (<math>c</math>) the parameters <math>\mu_1</math> and <math>\mu_2</math> can be estimated:
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| <math>\mu_1= \frac{ 2}{ \bar{x} } \left[ 1 + \sqrt{ 1 + 2 ( c^2 - 1 ) } \right]^{-1}</math>
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| <math>\mu_2 = \frac{ 2 }{ \bar{x} } \left[ 1 - \sqrt{ 1 + 2 ( c^2 - 1 ) } \right]^{-1}</math>
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| ==Characterization==
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| A random variable <math>\boldsymbol{X}\sim Hypo(\lambda_{1},\dots,\lambda_{k})</math> has [[cumulative distribution function]] given by,
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| :<math>
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| F(x)=1-\boldsymbol{\alpha}e^{x\Theta}\boldsymbol{1}
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| </math>
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| and [[density function]],
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| :<math>
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| f(x)=-\boldsymbol{\alpha}e^{x\Theta}\Theta\boldsymbol{1}\; ,
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| </math>
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| where <math>\boldsymbol{1}</math> is a [[column vector]] of ones of the size ''k'' and <math>e^{A}</math> is the [[matrix exponential]] of ''A''. When <math>\lambda_{i} \ne \lambda_{j}</math> for all <math>i \ne j</math>, the [[density function]] can be written as
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| :<math>
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| 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}
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| </math>
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| where <math>\ell_1(x), \dots, \ell_k(x)</math> are the [[Lagrange polynomial|Lagrange basis polynomials]] associated with the points <math>\lambda_1,\dots,\lambda_k</math>.
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| The distribution has [[Laplace transform]] of
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| :<math>
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| \mathcal{L}\{f(x)\}=-\boldsymbol{\alpha}(sI-\Theta)^{-1}\Theta\boldsymbol{1}
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| </math>
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| Which can be used to find moments,
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| :<math>
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| E[X^{n}]=(-1)^{n}n!\boldsymbol{\alpha}\Theta^{-n}\boldsymbol{1}\; .
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| </math>
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| ==General case==
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| In the general case
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| where there are <math>a</math> distinct sums of exponential distributions
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| with rates <math>\lambda_1,\lambda_2,\cdots,\lambda_a</math> and a number of terms in each
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| sum equals to <math>r_1,r_2,\cdots,r_a</math> respectively. The cumulative
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| distribution function for <math>t\geq0</math> is given by
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| :<math>F(t)
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| = 1 - \left(\prod_{j=1}^a \lambda_j^{r_j} \right)
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| \sum_{k=1}^a \sum_{l=1}^{r_k}
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| \frac{\Psi_{k,l}(-\lambda_k) t^{r_k-l} \exp(-\lambda_k t)}
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| {(r_k-l)!(l-1)!} ,
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| </math>
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| with
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| :<math>\Psi_{k,l}(x)
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| = -\frac{\partial^{l-1}}{\partial x^{l-1}}
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| \left(\prod_{j=0,j\neq k}^a \left(\lambda_j+x\right)^{-r_j} \right) .
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| </math>
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| with the additional convention <math>\lambda_0 = 0, r_0 = 1</math>.
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| ==Uses==
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| This distribution has been used in population genetics<ref name=Strimmer2001>Strimmer K, Pybus OG (2001) "Exploring the demographic history of DNA sequences using the generalized skyline plot", ''Mol Biol Evol'' 18(12):2298-305</ref> and queuing theory<ref name=Calinescu2009>http://www.few.vu.nl/en/Images/stageverslag-calinescu_tcm39-105827.pdf</ref><ref name=Bekker2011>Bekker R, Koeleman PM (2011) "Scheduling admissions and reducing variability in bed demand". ''Health Care Manag Sci'', 14(3):237-249</ref>
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| ==See also==
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| * [[Phase-type distribution]]
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| * [[Phase-type distribution#Coxian distribution|Coxian distribution]]
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| ==References==
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| {{reflist}}
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| ===Additional material===
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| * M. F. Neuts. (1981) Matrix-Geometric Solutions in Stochastic Models: an Algorthmic Approach, Chapter 2: Probability Distributions of Phase Type; Dover Publications Inc.
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| * G. Latouche, V. Ramaswami. (1999) Introduction to Matrix Analytic Methods in Stochastic Modelling, 1st edition. Chapter 2: PH Distributions; ASA SIAM,
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| * Colm A. O'Cinneide (1999). ''Phase-type distribution: open problems and a few properties'', Communication in Statistic - Stochastic Models, 15(4), 731–757.
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| * L. Leemis and J. McQueston (2008). ''Univariate distribution relationships'', The American Statistician, 62(1), 45—53.
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| * S. Ross. (2007) Introduction to Probability Models, 9th edition, New York: Academic Press
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| * S.V. Amari and R.B. Misra (1997) ''Closed-form expressions for distribution of sum of exponential random variables'',IEEE Trans. Reliab. 46, 519–522
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| {{ProbDistributions|continuous-semi-infinite}}
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| {{DEFAULTSORT:Hypoexponential Distribution}}
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| [[Category:Continuous distributions]]
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| [[Category:Probability distributions]]
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| [[zh:Erlang分布]]
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