Browse history interactively ← Previous edit Next edit → Content deleted Content addedVisual Wikitext Inline Revision as of 13:19, 28 November 2011
Gompertz distribution
Probability density function
Cumulative distribution function Parameters
η
,
b
>
0
,
x
>
0
{\displaystyle \eta ,b>0,x>0}
PDF
b
η
e
b
x
e
η
e
x
p
(
−
η
e
b
x
)
{\displaystyle b\eta e^{bx}e^{\eta }exp\left(-\eta e^{bx}\right)}
CDF
1
−
e
x
p
(
−
η
(
e
b
x
−
1
)
)
{\displaystyle 1-exp\left(-\eta \left(e^{bx}-1\right)\right)}
Mean
(
−
1
/
b
)
e
η
E
i
(
−
η
)
{\displaystyle (-1/b)e^{\eta }Ei\left(-\eta \right)}
w
h
e
r
e
E
i
(
x
)
=
∫
−
x
∞
(
e
−
u
/
u
)
d
u
{\displaystyle whereEi\left(x\right)=\int \limits _{-x}^{\infty }\left(e^{-u}/u\right)du}
Mode
(
x
η
∗
)
=
(
1
/
b
)
ln
(
1
/
η
)
w
i
t
h
0
<
F
(
x
η
∗
)
<
1
−
e
−
1
{\displaystyle \left(x_{\eta }^{*}\right)=\left(1/b\right)\ln \left(1/\eta \right)\ with0<F\left(x_{\eta }^{*}\right)<1-e^{-1}}
=
0.632121
,
0
<
η
<
1
{\displaystyle =0.632121,0<\eta <1\ }
(
x
η
∗
)
=
0
,
η
≥
1
{\displaystyle \left(x_{\eta }^{*}\right)=0,\eta \geq 1}
Variance
(
1
/
b
2
)
(
E
{
[
ln
(
X
)
]
2
}
−
(
E
[
ln
(
X
)
]
)
2
)
{\displaystyle (1/b^{2})(\mathrm {E} \{^{2}\}-(\mathrm {E} )^{2})\,}
where
X
=
η
e
−
b
x
{\displaystyle X=\eta e^{-bx}\,}
E
{
[
ln
(
X
)
]
2
}
=
[
1
+
1
/
η
]
∫
0
η
e
−
X
[
ln
(
X
)
]
2
d
X
−
1
/
η
∫
0
η
X
e
−
X
[
ln
(
X
)
]
2
d
X
{\displaystyle {\begin{aligned}\mathrm {E} \{^{2}\}=&\!\!\int _{0}^{\eta }\!\!\!\!e^{-X}^{2}\,dX\\&{}-1/\eta \!\!\int _{0}^{\eta }\!\!\!\!Xe^{-X}^{2}\,dX\end{aligned}}}
The Gompertz distribution is an extreme value (reverted Gumbel ) distribution (i.e., the distribution of
−
x
)
t
r
u
n
c
a
t
e
d
a
t
z
e
r
o
.
I
t
h
a
s
b
e
e
n
u
s
e
d
a
s
a
m
o
d
e
l
o
f
c
u
s
t
o
m
e
r
l
i
f
e
t
i
m
e
.
==
S
p
e
c
i
f
i
c
a
t
i
o
n
=====
P
r
o
b
a
b
i
l
i
t
y
d
e
n
s
i
t
y
f
u
n
c
t
i
o
n
===
T
h
e
[
[
p
r
o
b
a
b
i
l
i
t
y
d
e
n
s
i
t
y
f
u
n
c
t
i
o
n
]
]
o
f
t
h
e
G
o
m
p
e
r
t
z
d
i
s
t
r
i
b
u
t
i
o
n
i
s
::<
m
a
t
h
>
f
(
x
|
η
,
b
)
=
b
η
e
b
x
e
η
e
x
p
(
−
η
e
b
x
)
{\displaystyle -x)truncatedatzero.Ithasbeenusedasamodelofcustomerlifetime.==Specification=====Probabilitydensityfunction===The]oftheGompertzdistributionis::<math>f\left(x|\eta ,b\right)=b\eta e^{bx}e^{\eta }exp\left(-\eta e^{bx}\right)}
where
b
>
0
{\displaystyle b>0}
scale parameter and
η
>
0
{\displaystyle \eta >0}
shape parameter of the Gompertz distribution.
Cumulative distribution function
The cumulative distribution function of the Gompertz distribution is:
F
(
x
|
η
,
b
)
=
1
−
e
x
p
(
−
η
(
e
b
x
−
1
)
)
{\displaystyle F\left(x|\eta ,b\right)=1-exp\left(-\eta \left(e^{bx}-1\right)\right)}
where
η
,
b
>
0
,
x
>
0
{\displaystyle \eta ,b>0,x>0}
Properties
The Gompertz distribution is right-skewed for all values of
η
{\displaystyle \eta }
Shapes
The Gompertz density function can take on different shapes depending on the values of the shape parameter
η
{\displaystyle \eta }
η
≥
1
{\displaystyle \eta \geq 1\,}
η
<
1
{\displaystyle \eta <1\,}
mode
=
−
ln
(
z
⋆
)
b
0
<
z
⋆
<
1
{\displaystyle {\text{mode}}=-{\frac {\ln(z^{\star })}{b}}\,\qquad 0<z^{\star }<1}
where
z
⋆
{\displaystyle z^{\star }\,}
η
2
z
2
−
η
(
3
+
η
)
z
+
η
+
1
=
0
,
{\displaystyle \eta ^{2}z^{2}-\eta (3+\eta )z+\eta +1=0\,,}
which is
z
⋆
=
[
3
+
η
−
(
η
2
+
2
η
+
5
)
1
/
2
]
/
(
2
η
)
.
{\displaystyle z^{\star }=/(2\eta ).}
Related distributions
The Gompertz distribution is a natural conjugate to a gamma distribution. If
η
{\displaystyle \eta }
gamma distribution with shape parameter
α
{\displaystyle \alpha }
β
{\displaystyle \beta }
α
β
{\displaystyle \alpha \beta }
See also
Category :
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
**DISCLAIMER** We are not affiliated with Wikipedia, and Cloudflare.
The information presented on this site is for general informational purposes only and does not constitute medical advice.
You should always have a personal consultation with a healthcare professional before making changes to your diet, medication, or exercise routine.
AI helps with the correspondence in our chat.
We participate in an affiliate program. If you buy something through a link, we may earn a commission 💕
↑
and 
![{\displaystyle -x)truncatedatzero.Ithasbeenusedasamodelofcustomerlifetime.==Specification=====Probabilitydensityfunction===The]oftheGompertzdistributionis::<math>f\left(x|\eta ,b\right)=b\eta e^{bx}e^{\eta }exp\left(-\eta e^{bx}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3bbacabe0339f1d879f7d7b326ab04a9f2f6c3a9)
is the 
is the 
.
the probability density function has its mode at 0.
the probability density function has its mode at
is the smallest root of
and scale parameter 
(mean = 
), the cumulative distribution function is Gamma/Gompertz (G/G).
