Hypoexponential distribution

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Characterization

Cumulative distribution function Density function Column vector Matrix exponential Lagrange basis polynomials

6 related topics

Overview

Probability theory Erlang distribution Continuous distribution Queueing theory Teletraffic engineering

11 related topics

Key facts & relationships

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CDF: Expressed as a phase-type distribution 1 − α e x Θ 1 {\displaystyle 1-{\boldsymbol {\alpha }}e^{x\Theta }{\boldsymbol {1}}}
CF: α ( i t I − Θ ) − 1 Θ 1 {\displaystyle {\boldsymbol {\alpha }}(itI-\Theta )^{-1}\Theta \mathbf {1} }
Excess kurtosis: no simple closed form
Mean: ∑ i = 1 k 1 / λ i {\displaystyle \sum _{i=1}^{k}1/\lambda _{i}\,}
Median: General closed form does not exist
MGF: α ( t I − Θ ) − 1 Θ 1 {\displaystyle {\boldsymbol {\alpha }}(tI-\Theta )^{-1}\Theta \mathbf {1} }

Topics to explore

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Overview

Characterization

Cumulative distribution function
Density function
Column vector
Matrix exponential
Lagrange basis polynomials Lagrange polynomial
Laplace transform

Advanced semantic analysis

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Map overview Semantic statistics

Nodes20

Edges19

Triples12

Avg. degree1.9

Density0.1

Components1

How this topic connects Entity context

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Hypoexponential distribution

Top relations

CDF · 1

Hypoexponential distribution → Expressed as a phase-type distribution 1 − α e x Θ 1 {\displaystyle 1-{\boldsymbol {\alpha }}e^{x\Theta }{\boldsymbol {1}}}

CF · 1

Hypoexponential distribution → α ( i t I − Θ ) − 1 Θ 1 {\displaystyle {\boldsymbol {\alpha }}(itI-\Theta )^{-1}\Theta \mathbf {1} }

Excess kurtosis · 1

Hypoexponential distribution → no simple closed form

Mean · 1

Hypoexponential distribution → ∑ i = 1 k 1 / λ i {\displaystyle \sum _{i=1}^{k}1/\lambda _{i}\,}

Median · 1

Hypoexponential distribution → General closed form does not exist

MGF · 1

Hypoexponential distribution → α ( t I − Θ ) − 1 Θ 1 {\displaystyle {\boldsymbol {\alpha }}(tI-\Theta )^{-1}\Theta \mathbf {1} }

Mode · 1

Hypoexponential distribution → ( k − 1 ) / λ {\displaystyle (k-1)/\lambda } if λ k = λ {\displaystyle \lambda _{k}=\lambda } , for all k

Parameters · 1

Hypoexponential distribution → λ 1 , … , λ k 0 {\displaystyle \lambda _{1},\dots ,\lambda _{k}0\,} rates (real)

PDF · 1

Hypoexponential distribution → Expressed as a phase-type distribution − α e x Θ Θ 1 {\displaystyle -{\boldsymbol {\alpha }}e^{x\Theta }\Theta {\boldsymbol {1}}} Has no other simple form; see article for details

Skewness · 1

Hypoexponential distribution → 2 ( ∑ i = 1 k 1 / λ i 3 ) / ( ∑ i = 1 k 1 / λ i 2 ) 3 / 2 {\displaystyle 2(\sum _{i=1}^{k}1/\lambda _{i}^{3})/(\sum _{i=1}^{k}1/\lambda _{i}^{2})^{3/2}}

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Important terminology

distribution displaystyle lambda phase-type exponential hypoexponential probability dots distributions stochastic coefficient variation parameters general case boldsymbol theta 1- mean see

Entity relationships Subject–Predicate–Object triples

Subject	Predicate	Object	Confidence	Src
Hypoexponential distribution	CDF	Expressed as a phase-type distribution 1 − α e x Θ 1 {\displaystyle 1-{\boldsymbol {\alpha }}e^{x\Theta }{\boldsymbol {1}}}	1.00	infobox
Hypoexponential distribution	CF	α ( i t I − Θ ) − 1 Θ 1 {\displaystyle {\boldsymbol {\alpha }}(itI-\Theta )^{-1}\Theta \mathbf {1} }	1.00	infobox
Hypoexponential distribution	Excess kurtosis	no simple closed form	1.00	infobox
Hypoexponential distribution	Mean	∑ i = 1 k 1 / λ i {\displaystyle \sum _{i=1}^{k}1/\lambda _{i}\,}	1.00	infobox
Hypoexponential distribution	Median	General closed form does not exist	1.00	infobox
Hypoexponential distribution	MGF	α ( t I − Θ ) − 1 Θ 1 {\displaystyle {\boldsymbol {\alpha }}(tI-\Theta )^{-1}\Theta \mathbf {1} }	1.00	infobox
Hypoexponential distribution	Mode	( k − 1 ) / λ {\displaystyle (k-1)/\lambda } if λ k = λ {\displaystyle \lambda _{k}=\lambda } , for all k	1.00	infobox
Hypoexponential distribution	Parameters	λ 1 , … , λ k > 0 {\displaystyle \lambda _{1},\dots ,\lambda _{k}>0\,} rates (real)	1.00	infobox
Hypoexponential distribution	PDF	Expressed as a phase-type distribution − α e x Θ Θ 1 {\displaystyle -{\boldsymbol {\alpha }}e^{x\Theta }\Theta {\boldsymbol {1}}} Has no other simple form; see article for details	1.00	infobox
Hypoexponential distribution	Skewness	2 ( ∑ i = 1 k 1 / λ i 3 ) / ( ∑ i = 1 k 1 / λ i 2 ) 3 / 2 {\displaystyle 2(\sum _{i=1}^{k}1/\lambda _{i}^{3})/(\sum _{i=1}^{k}1/\lambda _{i}^{2})^{3/2}}	1.00	infobox
Hypoexponential distribution	Support	x ∈ [ 0 ; ∞ ) {\displaystyle x\in [0;\infty )\!}	1.00	infobox
Hypoexponential distribution	Variance	∑ i = 1 k 1 / λ i 2 {\displaystyle \sum _{i=1}^{k}1/\lambda _{i}^{2}}	1.00	infobox

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