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Gompertz distribution

Probability density function Note: b=2.322
Cumulative distribution function Note: b=2.322
Parameters	${\displaystyle x,\eta ,b>0\,\!}$
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	${\displaystyle (-1/b)e^{\eta }{\text{Ei}}\left(-\eta \right)}$ ${\displaystyle {\text{where Ei}}\left(z\right)=\int \limits _{-z}^{\infty }\left(e^{-u}/u\right)du}$
Mode	${\displaystyle =\left(1/b\right)\ln \left(1/\eta \right)\ }$ ${\displaystyle {\text{with }}0<{\text{F}}\left(x_{\eta }^{*}\right)<\left(1-e^{-1}=0.632121\right),0<\eta <1}$ ${\displaystyle =0,\quad \eta \geq 1}$
Variance	${\displaystyle \left(1/b\right)^{2}e^{\eta }\{-2\eta {\ }_{3}{\text{F}}_{3}\left(1,1,1;2,2,2;-\eta \right)+\gamma ^{2}}$${\displaystyle +\left(\pi ^{2}/6\right)+2\gamma \ln \left(\eta \right)+^{2}-e^{\eta }^{2}\}}$ ${\displaystyle {\text{ where }}\gamma {\text{ is the Euler constant }}\,\!}$ ${\displaystyle \gamma =-\psi \left(1\right)={\text{0.5777215...}}}$${\displaystyle {\text{and }}{}_{3}{\text{F}}_{3}\left(1,1,1;2,2,2;-z\right)=}$ ${\displaystyle \sum _{k=0}^{\infty }\{1/\left(k+1\right)^{3}\}\left(-1\right)^{k}\left(z^{k}/k!\right)}$

The Gompertz distribution is an extreme value (reverted Gumbel distribution) distribution (i.e., the distribution of ${\displaystyle -x}$) 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 ${\displaystyle \eta ,b>0,x>0\,\!}$

Moment generating function

The moment generating function is such as: ${\displaystyle {\text{E}}\left(e^{-tx}|\eta ,b\right)=\eta e^{\eta }{\text{E}}_{t/b}\left(\eta \right)}$
With ${\displaystyle {\text{E}}_{t/b}\left(\eta \right)=\int _{1}^{\infty }e^{-\eta v}v^{-t/b}dv,\ t>0}$

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

References

• Bemmaor, Albert C. (2011). "Modeling Purchasing Behavior With Sudden 'Death': A Flexible Customer Lifetime Model". Management Science. Articles in Advance. doi:http://dx.doi.org/10.1287/mnsc.1110.1461. {{cite journal}}: Check |doi= value (help); External link in |doi= (help); Unknown parameter |coauthors= ignored (|author= suggested) (help)
• Bemmaor, Albert C.; Glady, Nicolas. "Implementing the Gamma/Gompertz/NBD Model in MATLAB" (PDF).
• Gompertz, B. (1825). "On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies". Philos. Trans. Roy. Soc. 115. London: 513–583. {{cite journal}}: Cite has empty unknown parameter: |1= (help)
• Johnson, N. L.; Kotz, S.; Balakrishnan, N. (1995). "Continuous Univariate Distributions". 2 (2nd ed.). New York: John Wiley & Sons. {{cite journal}}: Cite has empty unknown parameter: |1= (help); Cite journal requires |journal= (help)
• Sheikh, A. K. (1989). "Truncated extreme value model for pipeline reliability". Reliability, Engrg. System Safety. 25 (1): 1–14. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)

• Continuous distributions