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Gompertz distribution

Probability density function
Cumulative distribution function
Parameters	${\displaystyle \eta ,b>0,x>0}$
PDF	${\displaystyle b\eta e^{bx}e^{\eta }exp\left(-\eta e^{bx}\right)}$
CDF	${\displaystyle 1-exp\left(-\eta \left(e^{bx}-1\right)\right)}$
Mean	${\displaystyle (-1/b)e^{\eta }Ei\left(-\eta \right)}$${\displaystyle whereEi\left(x\right)=\int \limits _{-x}^{\infty }\left(e^{-u}/u\right)du}$
Mode	${\displaystyle \left(x_{\eta }^{*}\right)=\left(1/b\right)\ln \left(1/\eta \right)\ with0<F\left(x_{\eta }^{*}\right)<1-e^{-1}}$${\displaystyle =0.632121,0<\eta <1\ }$${\displaystyle \left(x_{\eta }^{*}\right)=0,\eta \geq 1}$
Variance	${\displaystyle (1/b^{2})(\mathrm {E} \{^{2}\}-(\mathrm {E} )^{2})\,}$ where ${\displaystyle X=\eta e^{-bx}\,}$ and ${\displaystyle {\begin{aligned}\mathrm {E} \{^{2}\}=&\!\!\int _{0}^{\eta }\!\!\!\!e^{-X}^{2}\,dX\\&{}-1/\eta \!\!\int _{0}^{\eta }\!\!\!\!Xe^{-X}^{2}\,dX\end{aligned}}}$

The Gompertz distribution is an extreme value (reverted Gumbel) distribution (distribution of -x) truncated at zero. It has been used as a model of customer lifetime.

Specification

Probability density function

The probability density function of the Gompertz distribution is:

${\displaystyle f\left(x|\eta ,b\right)=b\eta e^{bx}e^{\eta }exp\left(-\eta e^{bx}\right)}$

where ${\displaystyle b>0}$ is the scale parameter and ${\displaystyle \eta >0}$ is the shape parameter of the Gompertz distribution.

Cumulative distribution function

The cumulative distribution function of the Gompertz distribution is:

${\displaystyle F\left(x|\eta ,b\right)=1-exp\left(-\eta \left(e^{bx}-1\right)\right)}$

where ${\displaystyle \eta ,b>0,x>0}$

Properties

The Gompertz distribution is right-skewed for all values of ${\displaystyle \eta }$.

Shapes

The Gompertz density function can take on different shapes depending on the values of the shape parameter ${\displaystyle \eta }$:

• ${\displaystyle \eta \geq 1\,}$ the probability density function has its mode at 0.
• ${\displaystyle \eta <1\,}$ the probability density function has its mode at

${\displaystyle {\text{mode}}=-{\frac {\ln(z^{\star })}{b}}\,\qquad 0<z^{\star }<1}$

where ${\displaystyle z^{\star }\,}$ is the smallest root of

${\displaystyle \eta ^{2}z^{2}-\eta (3+\eta )z+\eta +1=0\,,}$

which is

${\displaystyle z^{\star }=/(2\eta ).}$

Related distributions

The Gompertz distribution is a natural conjugate to a gamma distribution. If ${\displaystyle \eta }$ varies according to a gamma distribution with shape parameter ${\displaystyle \alpha }$ and scale parameter ${\displaystyle \beta }$ (mean = ${\displaystyle \alpha \beta }$), the cumulative distribution function is Gamma/Gompertz (G/G).

See also

• Gompertz function
• Gompertz–Makeham law of mortality
• Customer lifetime value
• Gumbel distribution

• Continuous distributions