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Erlang distribution

The Erlang distribution is a two-parameter family of continuous probability distributions with support x ∈ [ 0 , ∞ ) {\displaystyle x\in [0,\infty )} . The two parameters are:

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CDF
P ( k , λ x ) = γ ( k , λ x ) ( k − 1 ) ! = 1 − ∑ n = 0 k − 1 1 n ! e − λ x ( λ x ) n {\displaystyle P(k,\lambda x)={\frac {\gamma (k,\lambda x)}{(k-1)!}}=1-\sum _{n=0}^{k-1}{\f…
CF
( 1 − i t λ ) − k {\displaystyle \left(1-{\frac {it}{\lambda }}\right)^{-k}}
Entropy
( 1 − k ) ψ ( k ) + ln ⁡ [ Γ ( k ) λ ] + k {\displaystyle (1-k)\psi (k)+\ln \left[{\frac {\Gamma (k)}{\lambda }}\right]+k}
Excess kurtosis
6 k {\displaystyle {\frac {6}{k}}}
Mean
k λ {\displaystyle {\frac {k}{\lambda }}}
Median
No simple closed form

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Erlang distribution

Nodes33
Edges32
Triples70
Avg. degree1.94
Density0.060606
Components1

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Erlang distribution

Top relations

related to Related distributions · 19
Erlang distribution → Because, CDF, Erlang, Erlang-2, Erlang-k, Exponential, Gamma, If, III, In, Pareto, PDF, Pearson, Poisson, Pr, Taking, That, The, The Erlang
related to Waiting times · 7
Erlang distribution → Erlang, Events, Poisson, The, The Erlang, The Erlang-B, This
is a · 5
Erlang distribution → distribution of a sum of k, distribution of the sum of k independent and identically distributed random variables, special case of the gamma distribution in which the shape of the distribution is discretized.The Erlang distribution was developed by A, special case of the Pearson type III distribution, two-parameter family of continuous probability distributions with support x
related to Erlang-k · 5
Erlang distribution → Erlang, Erlang-2, For, PDF, The Erlang-k
has application · 4
Erlang distribution → Erlang, More, The, When
related to Cumulative distribution function (CDF) · 3
Erlang distribution → Erlang, The, The CDF
related to External links · 3
Erlang distribution → Erlang DistributionResource Dimensioning Using, Erlang-B, Erlang-C
related to Median · 2
Erlang distribution → An, Erlang
related to Probability density function · 2
Erlang distribution → Erlang, The
CDF · 1
Erlang distribution → P ( k , λ x ) = γ ( k , λ x ) ( k − 1 ) ! = 1 − ∑ n = 0 k − 1 1 n ! e − λ x ( λ x ) n {\displaystyle P(k,\lambda x)={\frac {\gamma (k,\lambda x)}{(k-1)!}}=1-\sum _{n=0}^{k-1}{\f…

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Important terminology

distribution erlang displaystyle lambda rate time number probability frac exponential poisson shape sum function distributions used integer operatorname events beta

Entity relationships Subject–Predicate–Object triples

SubjectPredicateObjectConfidenceSrc
Erlang distributionCDFP ( k , λ x ) = γ ( k , λ x ) ( k − 1 ) ! = 1 − ∑ n = 0 k − 1 1 n ! e − λ x ( λ x ) n {\displaystyle P(k,\lambda x)={\frac {\gamma (k,\lambda x)}{(k-1)!}}=1-\sum _{n=0}^{k-1}{\f…1.00infobox
Erlang distributionCF( 1 − i t λ ) − k {\displaystyle \left(1-{\frac {it}{\lambda }}\right)^{-k}}1.00infobox
Erlang distributionEntropy( 1 − k ) ψ ( k ) + ln ⁡ [ Γ ( k ) λ ] + k {\displaystyle (1-k)\psi (k)+\ln \left[{\frac {\Gamma (k)}{\lambda }}\right]+k}1.00infobox
Erlang distributionExcess kurtosis6 k {\displaystyle {\frac {6}{k}}}1.00infobox
Erlang distributionMeank λ {\displaystyle {\frac {k}{\lambda }}}1.00infobox
Erlang distributionMedianNo simple closed form1.00infobox
Erlang distributionMGF( 1 − t λ ) − k {\displaystyle \left(1-{\frac {t}{\lambda }}\right)^{-k}} for t < λ {\displaystyle t<\lambda }1.00infobox
Erlang distributionMode1 λ ( k − 1 ) {\displaystyle {\frac {1}{\lambda }}(k-1)}1.00infobox
Erlang distributionParametersk ∈ { 1 , 2 , 3 , … } , {\displaystyle k\in \{1,2,3,\ldots \},} shape λ ∈ ( 0 , ∞ ) , {\displaystyle \lambda \in (0,\infty ),} rate alt.: β = 1 / λ , {\displaystyle \beta =1/\la…1.00infobox
Erlang distributionPDFλ k x k − 1 e − λ x ( k − 1 ) ! {\displaystyle {\frac {\lambda ^{k}x^{k-1}e^{-\lambda x}}{(k-1)!}}}1.00infobox
Erlang distributionSkewness2 k {\displaystyle {\frac {2}{\sqrt {k}}}}1.00infobox
Erlang distributionSupportx ∈ [ 0 , ∞ ) {\displaystyle x\in [0,\infty )}1.00infobox
Erlang distributionVariancek λ 2 {\displaystyle {\frac {k}{\lambda ^{2}}}}1.00infobox
Erlang distributionis atwo-parameter family of continuous probability distributions with support x0.90text
Erlang distributionis adistribution of a sum of k0.90text
Erlang distributionis aspecial case of the gamma distribution in which the shape of the distribution is discretized.The Erlang distribution was developed by A0.90text
Erlang distributionis adistribution of the sum of k independent and identically distributed random variables0.90text
Erlang distributionis aspecial case of the Pearson type III distribution0.90text

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