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2 . 6
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4
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F r o m W i k i p e d i a , t h e f r e e e n c y c l o p e d i a
Heavy-tail probability distribution
The Lomax distribution , conditionally also called the Pareto Type II distribution , is a heavy-tail probability distribution used in business, economics, actuarial science, queueing theory and Internet traffic modeling.[1] [2] [3] It is named after K. S. Lomax. It is essentially a Pareto distribution that has been shifted so that its support begins at zero.[4]
Characterization [ edit ]
Probability density function [ edit ]
The probability density function (pdf) for the Lomax distribution is given by
p
(
x
)
=
α
λ
[
1
+
x
λ
]
−
(
α
+
1
)
,
x
≥
0
,
{\displaystyle p(x )={\alpha \over \lambda }\left[{1+{x \over \lambda }}\right]^{-(\alpha +1)},\qquad x\geq 0,}
with shape parameter
α
>
0
{\displaystyle \alpha >0}
and scale parameter
λ
>
0
{\displaystyle \lambda >0}
. The density can be rewritten in such a way that more clearly shows the relation to the Pareto Type I distribution . That is:
p
(
x
)
=
α
λ
α
(
x
+
λ
)
α
+
1
{\displaystyle p(x )={{\alpha \lambda ^{\alpha }} \over {(x+\lambda )^{\alpha +1}}}}
.
Non-central moments [ edit ]
The
ν
{\displaystyle \nu }
th non-central moment
E
[
X
ν
]
{\displaystyle E\left[X^{\nu }\right]}
exists only if the shape parameter
α
{\displaystyle \alpha }
strictly exceeds
ν
{\displaystyle \nu }
, when the moment has the value
E
(
X
ν
)
=
λ
ν
Γ
(
α
−
ν
)
Γ
(
1
+
ν
)
Γ
(
α
)
{\displaystyle E\left(X^{\nu }\right)={\frac {\lambda ^{\nu }\Gamma (\alpha -\nu )\Gamma (1+\nu )}{\Gamma (\alpha )}}}
Related distributions [ edit ]
Relation to the Pareto distribution [ edit ]
The Lomax distribution is a Pareto Type I distribution shifted so that its support begins at zero. Specifically:
If
Y
∼
Pareto
(
x
m
=
λ
,
α
)
,
then
Y
−
x
m
∼
Lomax
(
α
,
λ
)
.
{\displaystyle {\text{If }}Y\sim {\mbox{Pareto}}(x_{m}=\lambda ,\alpha ),{\text{ then }}Y-x_{m}\sim {\mbox{Lomax}}(\alpha ,\lambda ).}
The Lomax distribution is a Pareto Type II distribution with x m =λ and μ=0:[5]
If
X
∼
Lomax
(
α
,
λ
)
then
X
∼
P(II )
(
x
m
=
λ
,
α
,
μ
=
0
)
.
{\displaystyle {\text{If }}X\sim {\mbox{Lomax}}(\alpha ,\lambda ){\text{ then }}X\sim {\text{P(II )}}\left(x_{m}=\lambda ,\alpha ,\mu =0\right).}
Relation to the generalized Pareto distribution [ edit ]
The Lomax distribution is a special case of the generalized Pareto distribution . Specifically:
μ
=
0
,
ξ
=
1
α
,
σ
=
λ
α
.
{\displaystyle \mu =0,~\xi ={1 \over \alpha },~\sigma ={\lambda \over \alpha }.}
Relation to the beta prime distribution [ edit ]
The Lomax distribution with scale parameter λ = 1 is a special case of the beta prime distribution . If X has a Lomax distribution, then
X
λ
∼
β
′
(
1
,
α
)
{\displaystyle {\frac {X}{\lambda }}\sim \beta ^{\prime }(1,\alpha )}
.
Relation to the F distribution [ edit ]
The Lomax distribution with shape parameter α = 1 and scale parameter λ = 1 has density
f
(
x
)
=
1
(
1
+
x
)
2
{\displaystyle f(x )={\frac {1}{(1+x)^{2}}}}
, the same distribution as an F (2,2) distribution . This is the distribution of the ratio of two independent and identically distributed random variables with exponential distributions .
Relation to the q-exponential distribution [ edit ]
The Lomax distribution is a special case of the q-exponential distribution . The q-exponential extends this distribution to support on a bounded interval. The Lomax parameters are given by:
α
=
2
−
q
q
−
1
,
λ
=
1
λ
q
(
q
−
1
)
.
{\displaystyle \alpha ={{2-q} \over {q-1}},~\lambda ={1 \over \lambda _{q}(q-1)}.}
Relation to the (log-) logistic distribution [ edit ]
The logarithm of a Lomax(shape = 1.0, scale = λ)-distributed variable follows a logistic distribution with location log(λ) and scale 1.0.
This implies that a Lomax(shape = 1.0, scale = λ)-distribution equals a log-logistic distribution with shape β = 1.0 and scale α = log(λ).
Gamma-exponential (scale-) mixture connection [ edit ]
The Lomax distribution arises as a mixture of exponential distributions where the mixing distribution of the rate is a gamma distribution .
If λ|k,θ ~ Gamma(shape = k, scale = θ) and X |λ ~ Exponential(rate = λ) then the marginal distribution of X |k,θ is Lomax(shape = k, scale = 1/θ).
Since the rate parameter may equivalently be reparameterized to a scale parameter , the Lomax distribution constitutes a scale mixture of exponentials (with the exponential scale parameter following an inverse-gamma distribution ).
See also [ edit ]
References [ edit ]
^ Johnson, N. L.; Kotz, S.; Balakrishnan, N. (1994). "20 Pareto distributions ". Continuous univariate distributions . Vol. 1 (2nd ed.). New York: Wiley. p. 573.
^ J. Chen, J., Addie, R. G., Zukerman. M., Neame, T. D. (2015) "Performance Evaluation of a Queue Fed by a Poisson Lomax Burst Process", IEEE Communications Letters , 19, 3, 367-370.
^ Van Hauwermeiren M and Vose D (2009). A Compendium of Distributions [ebook]. Vose Software, Ghent, Belgium. Available at www.vosesoftware.com.
^ Kleiber, Christian; Kotz, Samuel (2003), Statistical Size Distributions in Economics and Actuarial Sciences , Wiley Series in Probability and Statistics, vol. 470, John Wiley & Sons, p. 60, ISBN 9780471457169 .
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=Lomax_distribution&oldid=1213538064 "
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H i d d e n c a t e g o r i e s :
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● S h o r t d e s c r i p t i o n i s d i f f e r e n t f r o m W i k i d a t a
● T h i s p a g e w a s l a s t e d i t e d o n 1 3 M a r c h 2 0 2 4 , a t 1 7 : 1 5 ( U T C ) .
● T e x t i s a v a i l a b l e u n d e r t h e C r e a t i v e C o m m o n s A t t r i b u t i o n - S h a r e A l i k e L i c e n s e 4 . 0 ;
a d d i t i o n a l t e r m s m a y a p p l y . B y u s i n g t h i s s i t e , y o u a g r e e t o t h e T e r m s o f U s e a n d P r i v a c y P o l i c y . W i k i p e d i a ® i s a r e g i s t e r e d t r a d e m a r k o f t h e W i k i m e d i a F o u n d a t i o n , I n c . , a n o n - p r o f i t o r g a n i z a t i o n .
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