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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 :
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↑
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).
