In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, ...
Exponentialdistribution
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Probabilitydistribution
Nottobeconfusedwiththeexponentialfamilyofprobabilitydistributions.
Exponential
Probabilitydensityfunction
CumulativedistributionfunctionParameters
λ
>
0
,
{\displaystyle\lambda>0,}
rate,orinversescaleSupport
x
∈
[
0
,
∞
)
{\displaystylex\in[0,\infty)}
PDF
λ
e
−
λ
x
{\displaystyle\lambdae^{-\lambdax}}
CDF
1
−
e
−
λ
x
{\displaystyle1-e^{-\lambdax}}
Quantile
−
ln
(
1
−
p
)
λ
{\displaystyle-{\frac{\ln(1-p)}{\lambda}}}
Mean
1
λ
{\displaystyle{\frac{1}{\lambda}}}
Median
ln
2
λ
{\displaystyle{\frac{\ln2}{\lambda}}}
Mode
0
{\displaystyle0}
Variance
1
λ
2
{\displaystyle{\frac{1}{\lambda^{2}}}}
Skewness
2
{\displaystyle2}
Ex.kurtosis
6
{\displaystyle6}
Entropy
1
−
ln
λ
{\displaystyle1-\ln\lambda}
MGF
λ
λ
−
t
,
for
t
<
λ
{\displaystyle{\frac{\lambda}{\lambda-t}},{\text{for}}t0istheparameterofthedistribution,oftencalledtherateparameter.Thedistributionissupportedontheinterval [0,∞).IfarandomvariableXhasthisdistribution,wewrite X~Exp(λ).
Theexponentialdistributionexhibitsinfinitedivisibility.
Cumulativedistributionfunction
Thecumulativedistributionfunctionisgivenby
F
(
x
;
λ
)
=
{
1
−
e
−
λ
x
x
≥
0
,
0
x
<
0.
{\displaystyleF(x;\lambda)={\begin{cases}1-e^{-\lambdax}&x\geq0,\\0&x<0.\end{cases}}}
Alternativeparametrization
Theexponentialdistributionissometimesparametrizedintermsofthescaleparameterβ=1/λ,whichisalsothemean:
f
(
x
;
β
)
=
{
1
β
e
−
x
/
β
x
≥
0
,
0
x
<
0.
F
(
x
;
β
)
=
{
1
−
e
−
x
/
β
x
≥
0
,
0
x
<
0.
{\displaystylef(x;\beta)={\begin{cases}{\frac{1}{\beta}}e^{-x/\beta}&x\geq0,\\0&x<0.\end{cases}}\qquad\qquadF(x;\beta)={\begin{cases}1-e^{-x/\beta}&x\geq0,\\0&x<0.\end{cases}}}
Properties
Mean,variance,moments,andmedian
Themeanistheprobabilitymasscentre,thatis,thefirstmoment.
ThemedianisthepreimageF−1(1/2).
ThemeanorexpectedvalueofanexponentiallydistributedrandomvariableXwithrateparameterλisgivenby
E
[
X
]
=
1
λ
.
{\displaystyle\operatorname{E}[X]={\frac{1}{\lambda}}.}
Inlightoftheexamplesgivenbelow,thismakessense:ifyoureceivephonecallsatanaveragerateof2perhour,thenyoucanexpecttowaithalfanhourforeverycall.
ThevarianceofXisgivenby
Var
[
X
]
=
1
λ
2
,
{\displaystyle\operatorname{Var}[X]={\frac{1}{\lambda^{2}}},}
sothestandarddeviationisequaltothemean.
ThemomentsofX,for
n
∈
N
{\displaystylen\in\mathbb{N}}
aregivenby
E
[
X
n
]
=
n
!
λ
n
.
{\displaystyle\operatorname{E}\left[X^{n}\right]={\frac{n!}{\lambda^{n}}}.}
ThecentralmomentsofX,for
n
∈
N
{\displaystylen\in\mathbb{N}}
aregivenby
μ
n
=
!
n
λ
n
=
n
!
λ
n
∑
k
=
0
n
(
−
1
)
k
k
!
.
{\displaystyle\mu_{n}={\frac{!n}{\lambda^{n}}}={\frac{n!}{\lambda^{n}}}\sum_{k=0}^{n}{\frac{(-1)^{k}}{k!}}.}
where !nisthesubfactorialofn
ThemedianofXisgivenby
m
[
X
]
=
ln
(
2
)
λ
<
E
[
X
]
,
{\displaystyle\operatorname{m}[X]={\frac{\ln(2)}{\lambda}}
s
+
t
∣
T
>
s
)
=
Pr
(
T
>
t
)
,
∀
s
,
t
≥
0.
{\displaystyle\Pr\left(T>s+t\midT>s\right)=\Pr(T>t),\qquad\foralls,t\geq0.}
Thiscanbeseenbyconsideringthecomplementarycumulativedistributionfunction:
Pr
(
T
>
s
+
t
∣
T
>
s
)
=
Pr
(
T
>
s
+
t
∩
T
>
s
)
Pr
(
T
>
s
)
=
Pr
(
T
>
s
+
t
)
Pr
(
T
>
s
)
=
e
−
λ
(
s
+
t
)
e
−
λ
s
=
e
−
λ
t
=
Pr
(
T
>
t
)
.
{\displaystyle{\begin{aligned}\Pr\left(T>s+t\midT>s\right)&={\frac{\Pr\left(T>s+t\capT>s\right)}{\Pr\left(T>s\right)}}\\[4pt]&={\frac{\Pr\left(T>s+t\right)}{\Pr\left(T>s\right)}}\\[4pt]&={\frac{e^{-\lambda(s+t)}}{e^{-\lambdas}}}\\[4pt]&=e^{-\lambdat}\\[4pt]&=\Pr(T>t).\end{aligned}}}
WhenTisinterpretedasthewaitingtimeforaneventtooccurrelativetosomeinitialtime,thisrelationimpliesthat,ifTisconditionedonafailuretoobservetheeventoversomeinitialperiodoftimes,thedistributionoftheremainingwaitingtimeisthesameastheoriginalunconditionaldistribution.Forexample,ifaneventhasnotoccurredafter30seconds,theconditionalprobabilitythatoccurrencewilltakeatleast10moresecondsisequaltotheunconditionalprobabilityofobservingtheeventmorethan10secondsaftertheinitialtime.
Theexponentialdistributionandthegeometricdistributionaretheonlymemorylessprobabilitydistributions.
Theexponentialdistributionisconsequentlyalsonecessarilytheonlycontinuousprobabilitydistributionthathasaconstantfailurerate.
Quantiles
Tukeycriteriaforanomalies.[citationneeded]
Thequantilefunction(inversecumulativedistributionfunction)forExp(λ)is
F
−
1
(
p
;
λ
)
=
−
ln
(
1
−
p
)
λ
,
0
≤
p
<
1
{\displaystyleF^{-1}(p;\lambda)={\frac{-\ln(1-p)}{\lambda}},\qquad0\leqp<1}
Thequartilesaretherefore:
firstquartile:ln(4/3)/λ
median:ln(2)/λ
thirdquartile:ln(4)/λ
Andasaconsequencetheinterquartilerangeisln(3)/λ.
Kullback–Leiblerdivergence
ThedirectedKullback–Leiblerdivergenceinnatsof
e
λ
{\displaystylee^{\lambda}}
("approximating"distribution)from
e
λ
0
{\displaystylee^{\lambda_{0}}}
('true'distribution)isgivenby
Δ
(
λ
0
∥
λ
)
=
E
λ
0
(
log
p
λ
0
(
x
)
p
λ
(
x
)
)
=
E
λ
0
(
log
λ
0
e
λ
0
x
λ
e
λ
x
)
=
log
(
λ
0
)
−
log
(
λ
)
−
(
λ
0
−
λ
)
E
λ
0
(
x
)
=
log
(
λ
0
)
−
log
(
λ
)
+
λ
λ
0
−
1.
{\displaystyle{\begin{aligned}\Delta(\lambda_{0}\parallel\lambda)&=\mathbb{E}_{\lambda_{0}}\left(\log{\frac{p_{\lambda_{0}}(x)}{p_{\lambda}(x)}}\right)\\&=\mathbb{E}_{\lambda_{0}}\left(\log{\frac{\lambda_{0}e^{\lambda_{0}x}}{\lambdae^{\lambdax}}}\right)\\&=\log(\lambda_{0})-\log(\lambda)-(\lambda_{0}-\lambda)E_{\lambda_{0}}(x)\\&=\log(\lambda_{0})-\log(\lambda)+{\frac{\lambda}{\lambda_{0}}}-1.\end{aligned}}}
Maximumentropydistribution
Amongallcontinuousprobabilitydistributionswithsupport[0,∞)andmeanμ,theexponentialdistributionwithλ=1/μhasthelargestdifferentialentropy.Inotherwords,itisthemaximumentropyprobabilitydistributionforarandomvariateXwhichisgreaterthanorequaltozeroandforwhichE[X]isfixed.[1]
Distributionoftheminimumofexponentialrandomvariables
LetX1,…,Xnbeindependentexponentiallydistributedrandomvariableswithrateparametersλ1,…,λn.Then
min
{
X
1
,
…
,
X
n
}
{\displaystyle\min\left\{X_{1},\dotsc,X_{n}\right\}}
isalsoexponentiallydistributed,withparameter
λ
=
λ
1
+
⋯
+
λ
n
.
{\displaystyle\lambda=\lambda_{1}+\dotsb+\lambda_{n}.}
Thiscanbeseenbyconsideringthecomplementarycumulativedistributionfunction:
Pr
(
min
{
X
1
,
…
,
X
n
}
>
x
)
=
Pr
(
X
1
>
x
,
…
,
X
n
>
x
)
=
∏
i
=
1
n
Pr
(
X
i
>
x
)
=
∏
i
=
1
n
exp
(
−
x
λ
i
)
=
exp
(
−
x
∑
i
=
1
n
λ
i
)
.
{\displaystyle{\begin{aligned}&\Pr\left(\min\{X_{1},\dotsc,X_{n}\}>x\right)\\={}&\Pr\left(X_{1}>x,\dotsc,X_{n}>x\right)\\={}&\prod_{i=1}^{n}\Pr\left(X_{i}>x\right)\\={}&\prod_{i=1}^{n}\exp\left(-x\lambda_{i}\right)=\exp\left(-x\sum_{i=1}^{n}\lambda_{i}\right).\end{aligned}}}
Theindexofthevariablewhichachievestheminimumisdistributedaccordingtothecategoricaldistribution
Pr
(
X
k
=
min
{
X
1
,
…
,
X
n
}
)
=
λ
k
λ
1
+
⋯
+
λ
n
.
{\displaystyle\Pr\left(X_{k}=\min\{X_{1},\dotsc,X_{n}\}\right)={\frac{\lambda_{k}}{\lambda_{1}+\dotsb+\lambda_{n}}}.}
Aproofcanbeseenbyletting
I
=
argmin
i
∈
{
1
,
⋯
,
n
}
{
X
1
,
…
,
X
n
}
{\displaystyleI=\operatorname{argmin}_{i\in\{1,\dotsb,n\}}\{X_{1},\dotsc,X_{n}\}}
.Then,
Pr
(
I
=
k
)
=
∫
0
∞
Pr
(
X
k
=
x
)
Pr
(
∀
i
≠
k
X
i
>
x
)
d
x
=
∫
0
∞
λ
k
e
−
λ
k
x
(
∏
i
=
1
,
i
≠
k
n
e
−
λ
i
x
)
d
x
=
λ
k
∫
0
∞
e
−
(
λ
1
+
⋯
+
λ
n
)
x
d
x
=
λ
k
λ
1
+
⋯
+
λ
n
.
{\displaystyle{\begin{aligned}\Pr(I=k)&=\int_{0}^{\infty}\Pr(X_{k}=x)\Pr(\forall_{i\neqk}X_{i}>x)\,dx\\&=\int_{0}^{\infty}\lambda_{k}e^{-\lambda_{k}x}\left(\prod_{i=1,i\neqk}^{n}e^{-\lambda_{i}x}\right)dx\\&=\lambda_{k}\int_{0}^{\infty}e^{-\left(\lambda_{1}+\dotsb+\lambda_{n}\right)x}dx\\&={\frac{\lambda_{k}}{\lambda_{1}+\dotsb+\lambda_{n}}}.\end{aligned}}}
Notethat
max
{
X
1
,
…
,
X
n
}
{\displaystyle\max\{X_{1},\dotsc,X_{n}\}}
isnotexponentiallydistributed,ifX1,…,Xndonotallhaveparameter0.[2]
Jointmomentsofi.i.d.exponentialorderstatistics
Let
X
1
,
…
,
X
n
{\displaystyleX_{1},\dotsc,X_{n}}
be
n
{\displaystylen}
independentandidenticallydistributedexponentialrandomvariableswithrateparameterλ.
Let
X
(
1
)
,
…
,
X
(
n
)
{\displaystyleX_{(1)},\dotsc,X_{(n)}}
denotethecorrespondingorderstatistics.
For
i
<
j
{\displaystylei
λ
2
{\displaystyle\lambda_{1}>\lambda_{2}}
(withoutlossofgenerality),then
H
(
Z
)
=
1
+
γ
+
ln
(
λ
1
−
λ
2
λ
1
λ
2
)
+
ψ
(
λ
1
λ
1
−
λ
2
)
,
{\displaystyle{\begin{aligned}H(Z)&=1+\gamma+\ln\left({\frac{\lambda_{1}-\lambda_{2}}{\lambda_{1}\lambda_{2}}}\right)+\psi\left({\frac{\lambda_{1}}{\lambda_{1}-\lambda_{2}}}\right),\end{aligned}}}
where
γ
{\displaystyle\gamma}
istheEuler-Mascheroniconstant,and
ψ
(
⋅
)
{\displaystyle\psi(\cdot)}
isthedigammafunction.[3]
Inthecaseofequalrateparameters,theresultisanErlangdistributionwithshape2andparameter
λ
,
{\displaystyle\lambda,}
whichinturnisaspecialcaseofgammadistribution.
Relateddistributions
Thissectionincludesalistofgeneralreferences,butitlackssufficientcorrespondinginlinecitations.Pleasehelptoimprovethissectionbyintroducingmoreprecisecitations.(March2011)(Learnhowandwhentoremovethistemplatemessage)
If
X
∼
Laplace
(
μ
,
β
−
1
)
{\displaystyleX\sim\operatorname{Laplace}\left(\mu,\beta^{-1}\right)}
then|X−μ|~Exp(β).
IfX~Pareto(1,λ)thenlog(X)~Exp(λ).
IfX~SkewLogistic(θ),then
log
(
1
+
e
−
X
)
∼
Exp
(
θ
)
{\displaystyle\log\left(1+e^{-X}\right)\sim\operatorname{Exp}(\theta)}
.
IfXi~U(0,1)then
lim
n
→
∞
n
min
(
X
1
,
…
,
X
n
)
∼
Exp
(
1
)
{\displaystyle\lim_{n\to\infty}n\min\left(X_{1},\ldots,X_{n}\right)\sim\operatorname{Exp}(1)}
Theexponentialdistributionisalimitofascaledbetadistribution:
lim
n
→
∞
n
Beta
(
1
,
n
)
=
Exp
(
1
)
.
{\displaystyle\lim_{n\to\infty}n\operatorname{Beta}(1,n)=\operatorname{Exp}(1).}
Exponentialdistributionisaspecialcaseoftype3Pearsondistribution.
IfX~Exp(λ)andXi~Exp(λi)then:
k
X
∼
Exp
(
λ
k
)
{\displaystylekX\sim\operatorname{Exp}\left({\frac{\lambda}{k}}\right)}
,closureunderscalingbyapositivefactor.
1 + X~BenktanderWeibull(λ,1),whichreducestoatruncatedexponentialdistribution.
keX~Pareto(k,λ).
e−X~Beta(λ,1).
1/keX~PowerLaw(k,λ)
X
∼
Rayleigh
(
1
2
λ
)
{\displaystyle{\sqrt{X}}\sim\operatorname{Rayleigh}\left({\frac{1}{\sqrt{2\lambda}}}\right)}
,theRayleighdistribution
X
∼
Weibull
(
1
λ
,
1
)
{\displaystyleX\sim\operatorname{Weibull}\left({\frac{1}{\lambda}},1\right)}
,theWeibulldistribution
X
2
∼
Weibull
(
1
λ
2
,
1
2
)
{\displaystyleX^{2}\sim\operatorname{Weibull}\left({\frac{1}{\lambda^{2}}},{\frac{1}{2}}\right)}
μ−βlog(λX)∼Gumbel(μ,β).
⌊
X
⌋
∼
Geometric
(
1
−
e
−
λ
)
{\displaystyle\lfloorX\rfloor\sim\operatorname{Geometric}\left(1-e^{-\lambda}\right)}
,ageometricdistributionon0,1,2,3,...
⌈
X
⌉
∼
Geometric
(
1
−
e
−
λ
)
{\displaystyle\lceilX\rceil\sim\operatorname{Geometric}\left(1-e^{-\lambda}\right)}
,ageometricdistributionon1,2,3,4,...
IfalsoY~Erlang(n,λ)or
Y
∼
Γ
(
n
,
1
λ
)
{\displaystyleY\sim\Gamma\left(n,{\frac{1}{\lambda}}\right)}
then
X
Y
+
1
∼
Pareto
(
1
,
n
)
{\displaystyle{\frac{X}{Y}}+1\sim\operatorname{Pareto}(1,n)}
Ifalsoλ~Gamma(k,θ)(shape,scaleparametrisation)thenthemarginaldistributionofXisLomax(k,1/θ),thegammamixture
λ1X1−λ2Y2~Laplace(0,1).
min{X1,...,Xn}~Exp(λ1+...+λn).
Ifalsoλi=λthen:
X
1
+
⋯
+
X
k
=
∑
i
X
i
∼
{\displaystyleX_{1}+\cdots+X_{k}=\sum_{i}X_{i}\sim}
Erlang(k,λ)=Gamma(k,λ−1)=Gamma(k,λ)(in(k,θ)and(α,β)parametrization,respectively)withanintegershapeparameterk.[4]
Xi−Xj~Laplace(0,λ−1).
IfalsoXiareindependent,then:
X
i
X
i
+
X
j
{\displaystyle{\frac{X_{i}}{X_{i}+X_{j}}}}
~U(0,1)
Z
=
λ
i
X
i
λ
j
X
j
{\displaystyleZ={\frac{\lambda_{i}X_{i}}{\lambda_{j}X_{j}}}}
hasprobabilitydensityfunction
f
Z
(
z
)
=
1
(
z
+
1
)
2
{\displaystylef_{Z}(z)={\frac{1}{(z+1)^{2}}}}
.Thiscanbeusedtoobtainaconfidenceintervalfor
λ
i
λ
j
{\displaystyle{\frac{\lambda_{i}}{\lambda_{j}}}}
.
Ifalsoλ=1:
μ
−
β
log
(
e
−
X
1
−
e
−
X
)
∼
Logistic
(
μ
,
β
)
{\displaystyle\mu-\beta\log\left({\frac{e^{-X}}{1-e^{-X}}}\right)\sim\operatorname{Logistic}(\mu,\beta)}
,thelogisticdistribution
μ
−
β
log
(
X
i
X
j
)
∼
Logistic
(
μ
,
β
)
{\displaystyle\mu-\beta\log\left({\frac{X_{i}}{X_{j}}}\right)\sim\operatorname{Logistic}(\mu,\beta)}
μ−σlog(X)~GEV(μ,σ,0).
Furtherif
Y
∼
Γ
(
α
,
β
α
)
{\displaystyleY\sim\Gamma\left(\alpha,{\frac{\beta}{\alpha}}\right)}
then
X
Y
∼
K
(
α
,
β
)
{\displaystyle{\sqrt{XY}}\sim\operatorname{K}(\alpha,\beta)}
(K-distribution)
Ifalsoλ=1/2thenX∼χ22;i.e.,Xhasachi-squareddistributionwith2degreesoffreedom.Hence:
Exp
(
λ
)
=
1
2
λ
Exp
(
1
2
)
∼
1
2
λ
χ
2
2
⇒
∑
i
=
1
n
Exp
(
λ
)
∼
1
2
λ
χ
2
n
2
{\displaystyle\operatorname{Exp}(\lambda)={\frac{1}{2\lambda}}\operatorname{Exp}\left({\frac{1}{2}}\right)\sim{\frac{1}{2\lambda}}\chi_{2}^{2}\Rightarrow\sum_{i=1}^{n}\operatorname{Exp}(\lambda)\sim{\frac{1}{2\lambda}}\chi_{2n}^{2}}
If
X
∼
Exp
(
1
λ
)
{\displaystyleX\sim\operatorname{Exp}\left({\frac{1}{\lambda}}\right)}
and
Y
∣
X
{\displaystyleY\midX}
~Poisson(X)then
Y
∼
Geometric
(
1
1
+
λ
)
{\displaystyleY\sim\operatorname{Geometric}\left({\frac{1}{1+\lambda}}\right)}
(geometricdistribution)
TheHoytdistributioncanbeobtainedfromexponentialdistributionandarcsinedistribution
Otherrelateddistributions:
Hyper-exponentialdistribution–thedistributionwhosedensityisaweightedsumofexponentialdensities.
Hypoexponentialdistribution–thedistributionofageneralsumofexponentialrandomvariables.
exGaussiandistribution–thesumofanexponentialdistributionandanormaldistribution.
Statisticalinference
Below,supposerandomvariableXisexponentiallydistributedwithrateparameterλ,and
x
1
,
…
,
x
n
{\displaystylex_{1},\dotsc,x_{n}}
arenindependentsamplesfromX,withsamplemean
x
¯
{\displaystyle{\bar{x}}}
.
Parameterestimation
Themaximumlikelihoodestimatorforλisconstructedasfollows:
Thelikelihoodfunctionforλ,givenanindependentandidenticallydistributedsamplex=(x1,…,xn)drawnfromthevariable,is:
L
(
λ
)
=
∏
i
=
1
n
λ
exp
(
−
λ
x
i
)
=
λ
n
exp
(
−
λ
∑
i
=
1
n
x
i
)
=
λ
n
exp
(
−
λ
n
x
¯
)
,
{\displaystyleL(\lambda)=\prod_{i=1}^{n}\lambda\exp(-\lambdax_{i})=\lambda^{n}\exp\left(-\lambda\sum_{i=1}^{n}x_{i}\right)=\lambda^{n}\exp\left(-\lambdan{\overline{x}}\right),}
where:
x
¯
=
1
n
∑
i
=
1
n
x
i
{\displaystyle{\overline{x}}={\frac{1}{n}}\sum_{i=1}^{n}x_{i}}
isthesamplemean.
Thederivativeofthelikelihoodfunction'slogarithmis:
d
d
λ
ln
L
(
λ
)
=
d
d
λ
(
n
ln
λ
−
λ
n
x
¯
)
=
n
λ
−
n
x
¯
{
>
0
,
0
<
λ
<
1
x
¯
,
=
0
,
λ
=
1
x
¯
,
<
0
,
λ
>
1
x
¯
.
{\displaystyle{\frac{d}{d\lambda}}\lnL(\lambda)={\frac{d}{d\lambda}}\left(n\ln\lambda-\lambdan{\overline{x}}\right)={\frac{n}{\lambda}}-n{\overline{x}}\{\begin{cases}>0,&0{\frac{1}{\overline{x}}}.\end{cases}}}
Consequently,themaximumlikelihoodestimatefortherateparameteris:
λ
^
mle
=
1
x
¯
=
n
∑
i
x
i
{\displaystyle{\widehat{\lambda}}_{\text{mle}}={\frac{1}{\overline{x}}}={\frac{n}{\sum_{i}x_{i}}}}
Thisisnotanunbiasedestimatorof
λ
,
{\displaystyle\lambda,}
although
x
¯
{\displaystyle{\overline{x}}}
isanunbiased[5]MLE[6]estimatorof
1
/
λ
{\displaystyle1/\lambda}
andthedistributionmean.
Thebiasof
λ
^
mle
{\displaystyle{\widehat{\lambda}}_{\text{mle}}}
isequalto
b
≡
E
[
(
λ
^
mle
−
λ
)
]
=
λ
n
−
1
{\displaystyleb\equiv\operatorname{E}\left[\left({\widehat{\lambda}}_{\text{mle}}-\lambda\right)\right]={\frac{\lambda}{n-1}}}
whichyieldsthebias-correctedmaximumlikelihoodestimator
λ
^
mle
∗
=
λ
^
mle
−
b
^
.
{\displaystyle{\widehat{\lambda}}_{\text{mle}}^{*}={\widehat{\lambda}}_{\text{mle}}-{\widehat{b}}.}
Approximateminimizerofexpectedsquarederror
Assumeyouhaveatleastthreesamples.Ifweseekaminimizerofexpectedmeansquarederror(seealso:Bias–variancetradeoff)thatissimilartothemaximumlikelihoodestimate(i.e.amultiplicativecorrectiontothelikelihoodestimate)wehave:
λ
^
=
(
n
−
2
n
)
(
1
x
¯
)
=
n
−
2
∑
i
x
i
{\displaystyle{\widehat{\lambda}}=\left({\frac{n-2}{n}}\right)\left({\frac{1}{\bar{x}}}\right)={\frac{n-2}{\sum_{i}x_{i}}}}
Thisisderivedfromthemeanandvarianceoftheinverse-gammadistribution:
Inv-Gamma
(
n
,
λ
)
{\textstyle{\mbox{Inv-Gamma}}(n,\lambda)}
.[7]
Fisherinformation
TheFisherinformation,denoted
I
(
λ
)
{\displaystyle{\mathcal{I}}(\lambda)}
,foranestimatoroftherateparameter
λ
{\displaystyle\lambda}
isgivenas:
I
(
λ
)
=
E
[
(
∂
∂
λ
log
f
(
x
;
λ
)
)
2
|
λ
]
=
∫
(
∂
∂
λ
log
f
(
x
;
λ
)
)
2
f
(
x
;
λ
)
d
x
{\displaystyle{\mathcal{I}}(\lambda)=\operatorname{E}\left[\left.\left({\frac{\partial}{\partial\lambda}}\logf(x;\lambda)\right)^{2}\right|\lambda\right]=\int\left({\frac{\partial}{\partial\lambda}}\logf(x;\lambda)\right)^{2}f(x;\lambda)\,dx}
Plugginginthedistributionandsolvinggives:
I
(
λ
)
=
∫
0
∞
(
∂
∂
λ
log
λ
e
−
λ
x
)
2
λ
e
−
λ
x
d
x
=
∫
0
∞
(
1
λ
−
x
)
2
λ
e
−
λ
x
d
x
=
λ
−
2
.
{\displaystyle{\mathcal{I}}(\lambda)=\int_{0}^{\infty}\left({\frac{\partial}{\partial\lambda}}\log\lambdae^{-\lambdax}\right)^{2}\lambdae^{-\lambdax}\,dx=\int_{0}^{\infty}\left({\frac{1}{\lambda}}-x\right)^{2}\lambdae^{-\lambdax}\,dx=\lambda^{-2}.}
Thisdeterminestheamountofinformationeachindependentsampleofanexponentialdistributioncarriesabouttheunknownrateparameter
λ
{\displaystyle\lambda}
.
Confidenceintervals
The100(1−α)%confidenceintervalfortherateparameterofanexponentialdistributionisgivenby:[8]
2
n
λ
^
χ
1
−
α
2
,
2
n
2
<
1
λ
<
2
n
λ
^
χ
α
2
,
2
n
2
{\displaystyle{\frac{2n}{{\widehat{\lambda}}\chi_{1-{\frac{\alpha}{2}},2n}^{2}}}0.
Computationalmethods
Generatingexponentialvariates
Aconceptuallyverysimplemethodforgeneratingexponentialvariatesisbasedoninversetransformsampling:GivenarandomvariateUdrawnfromtheuniformdistributionontheunitinterval(0,1),thevariate
T
=
F
−
1
(
U
)
{\displaystyleT=F^{-1}(U)}
hasanexponentialdistribution,whereF−1isthequantilefunction,definedby
F
−
1
(
p
)
=
−
ln
(
1
−
p
)
λ
.
{\displaystyleF^{-1}(p)={\frac{-\ln(1-p)}{\lambda}}.}
Moreover,ifUisuniformon(0,1),thensois1−U.Thismeansonecangenerateexponentialvariatesasfollows:
T
=
−
ln
(
U
)
λ
.
{\displaystyleT={\frac{-\ln(U)}{\lambda}}.}
OthermethodsforgeneratingexponentialvariatesarediscussedbyKnuth[15]andDevroye.[16]
Afastmethodforgeneratingasetofready-orderedexponentialvariateswithoutusingasortingroutineisalsoavailable.[16]
Seealso
Deadtime–anapplicationofexponentialdistributiontoparticledetectoranalysis.
Laplacedistribution,orthe"doubleexponentialdistribution".
Relationshipsamongprobabilitydistributions
Marshall–Olkinexponentialdistribution
References
^Park,SungY.;Bera,AnilK.(2009)."Maximumentropyautoregressiveconditionalheteroskedasticitymodel"(PDF).JournalofEconometrics.Elsevier.150(2):219–230.doi:10.1016/j.jeconom.2008.12.014.Archivedfromtheoriginal(PDF)on2016-03-07.Retrieved2011-06-02.
^Michael,Lugo."Theexpectationofthemaximumofexponentials"(PDF).Archivedfromtheoriginal(PDF)on20December2016.Retrieved13December2016.
^Eckford,AndrewW.;Thomas,PeterJ.(2016)."Entropyofthesumoftwoindependent,non-identically-distributedexponentialrandomvariables".arXiv:1609.02911[cs.IT].
^Ibe,OliverC.(2014).FundamentalsofAppliedProbabilityandRandomProcesses(2nd ed.).AcademicPress.p. 128.ISBN 9780128010358.
^RichardArnoldJohnson;DeanW.Wichern(2007).AppliedMultivariateStatisticalAnalysis.PearsonPrenticeHall.ISBN 978-0-13-187715-3.Retrieved10August2012.
^NIST/SEMATECHe-HandbookofStatisticalMethods
^Elfessi,Abdulaziz;Reineke,DavidM.(2001)."ABayesianLookatClassicalEstimation:TheExponentialDistribution".JournalofStatisticsEducation.9(1).doi:10.1080/10691898.2001.11910648.
^Ross,SheldonM.(2009).Introductiontoprobabilityandstatisticsforengineersandscientists(4th ed.).AssociatedPress.p. 267.ISBN 978-0-12-370483-2.
^Guerriero,V.(2012)."PowerLawDistribution:MethodofMulti-scaleInferentialStatistics".JournalofModernMathematicsFrontier.1:21–28.
^"Cumfreq,afreecomputerprogramforcumulativefrequencyanalysis".
^Ritzema,H.P.,ed.(1994).FrequencyandRegressionAnalysis.Chapter6in:DrainagePrinciplesandApplications,Publication16,InternationalInstituteforLandReclamationandImprovement(ILRI),Wageningen,TheNetherlands.pp. 175–224.ISBN 90-70754-33-9.
^Lawless,J.F.;Fredette,M.(2005)."Frequentistpredictionsintervalsandpredictivedistributions".Biometrika.92(3):529–542.doi:10.1093/biomet/92.3.529.
^Bjornstad,J.F.(1990)."PredictiveLikelihood:AReview".Statist.Sci.5(2):242–254.doi:10.1214/ss/1177012175.
^D.F.SchmidtandE.Makalic,"UniversalModelsfortheExponentialDistribution",IEEETransactionsonInformationTheory,Volume55,Number7,pp.3087–3090,2009doi:10.1109/TIT.2009.2018331
^DonaldE.Knuth(1998).TheArtofComputerProgramming,volume2:SeminumericalAlgorithms,3rdedn.Boston:Addison–Wesley.ISBN 0-201-89684-2.Seesection3.4.1,p.133.
^abLucDevroye(1986).Non-UniformRandomVariateGeneration.NewYork:Springer-Verlag.ISBN 0-387-96305-7.SeechapterIX,section2,pp.392–401.
Externallinks
"Exponentialdistribution",EncyclopediaofMathematics,EMSPress,2001[1994]
OnlinecalculatorofExponentialDistribution
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Benford
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beta-binomial
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hypergeometric
negative
Poissonbinomial
Rademacher
soliton
discreteuniform
Zipf
Zipf–Mandelbrot
withinfinitesupport
betanegativebinomial
Borel
Conway–Maxwell–Poisson
discretephase-type
Delaporte
extendednegativebinomial
Flory–Schulz
Gauss–Kuzmin
geometric
logarithmic
mixedPoisson
negativebinomial
Panjer
parabolicfractal
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Skellam
Yule–Simon
zeta
Continuousunivariatesupportedonaboundedinterval
arcsine
ARGUS
Balding–Nichols
Bates
beta
betarectangular
continuousBernoulli
Irwin–Hall
Kumaraswamy
logit-normal
noncentralbeta
PERT
raisedcosine
reciprocal
triangular
U-quadratic
uniform
Wignersemicircle
supportedonasemi-infiniteinterval
Benini
Benktander1stkind
Benktander2ndkind
betaprime
Burr
chi
chi-squared
noncentral
inverse
scaled
Dagum
Davis
Erlang
hyper
exponential
hyperexponential
hypoexponential
logarithmic
F
noncentral
foldednormal
Fréchet
gamma
generalized
inverse
gamma/Gompertz
Gompertz
shifted
half-logistic
half-normal
Hotelling'sT-squared
inverseGaussian
generalized
Kolmogorov
Lévy
log-Cauchy
log-Laplace
log-logistic
log-normal
log-t
Lomax
matrix-exponential
Maxwell–Boltzmann
Maxwell–Jüttner
Mittag-Leffler
Nakagami
Pareto
phase-type
Poly-Weibull
Rayleigh
relativisticBreit–Wigner
Rice
truncatednormal
type-2Gumbel
Weibull
discrete
Wilks'slambda
supportedonthewholerealline
Cauchy
exponentialpower
Fisher'sz
Gaussianq
generalizednormal
generalizedhyperbolic
geometricstable
Gumbel
Holtsmark
hyperbolicsecant
Johnson'sSU
Landau
Laplace
asymmetric
logistic
noncentralt
normal(Gaussian)
normal-inverseGaussian
skewnormal
slash
stable
Student'st
type-1Gumbel
Tracy–Widom
variance-gamma
Voigt
withsupportwhosetypevaries
generalizedchi-squared
generalizedextremevalue
generalizedPareto
Marchenko–Pastur
q-exponential
q-Gaussian
q-Weibull
shiftedlog-logistic
Tukeylambda
Mixedunivariatecontinuous-discrete
RectifiedGaussian
Multivariate(joint)
Discrete:
Ewens
multinomial
Dirichlet
negative
Continuous:
Dirichlet
generalized
multivariateLaplace
multivariatenormal
multivariatestable
multivariatet
normal-gamma
inverse
Matrix-valued:
LKJ
matrixnormal
matrixt
matrixgamma
inversematrixgamma
Wishart
normal
inverse
normal-inverse
Directional
Univariate(circular)directional
Circularuniform
univariatevonMises
wrappednormal
wrappedCauchy
wrappedexponential
wrappedasymmetricLaplace
wrappedLévy
Bivariate(spherical)
Kent
Bivariate(toroidal)
bivariatevonMises
Multivariate
vonMises–Fisher
Bingham
Degenerateandsingular
Degenerate
Diracdeltafunction
Singular
Cantor
Families
Circular
compoundPoisson
elliptical
exponential
naturalexponential
location–scale
maximumentropy
mixture
Pearson
Tweedie
wrapped
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Exponential distribution mean