Gompertz distribution information

Gompertz distribution
	Probability density function
	Cumulative distribution function
Parameters	shape , scale
Support
PDF
CDF
Quantile
Mean	;
Median
Mode	; ;
Variance	;
MGF	;

In probability and statistics, the Gompertz distribution is a continuous probability distribution, named after Benjamin Gompertz. The Gompertz distribution is often applied to describe the distribution of adult lifespans by demographers^[1]^[2] and actuaries.^[3]^[4] Related fields of science such as biology^[5] and gerontology^[6] also considered the Gompertz distribution for the analysis of survival. More recently, computer scientists have also started to model the failure rates of computer code by the Gompertz distribution.^[7] In Marketing Science, it has been used as an individual-level simulation for customer lifetime value modeling.^[8] In network theory, particularly the Erdős–Rényi model, the walk length of a random self-avoiding walk (SAW) is distributed according to the Gompertz distribution.^[9]

^ Cite error: The named reference Vaupel1986 was invoked but never defined (see the help page).
^ Cite error: The named reference Preston2001 was invoked but never defined (see the help page).
^ Cite error: The named reference Benjamin1980 was invoked but never defined (see the help page).
^ Cite error: The named reference Willemse2000 was invoked but never defined (see the help page).
^ Cite error: The named reference Economos1982 was invoked but never defined (see the help page).
^ Cite error: The named reference Brown1974 was invoked but never defined (see the help page).
^ Cite error: The named reference Ohishi2009 was invoked but never defined (see the help page).
^ Cite error: The named reference BG was invoked but never defined (see the help page).
^ Tishby, Biham, Katzav (2016), The distribution of path lengths of self avoiding walks on Erdős-Rényi networks, arXiv:1603.06613.

[Vaupel1986-1] Cite error: The named reference Vaupel1986 was invoked but never defined (see the help page).

[Preston2001-2] Cite error: The named reference Preston2001 was invoked but never defined (see the help page).

[Benjamin1980-3] Cite error: The named reference Benjamin1980 was invoked but never defined (see the help page).

[Willemse2000-4] Cite error: The named reference Willemse2000 was invoked but never defined (see the help page).

[Economos1982-5] Cite error: The named reference Economos1982 was invoked but never defined (see the help page).

[Brown1974-6] Cite error: The named reference Brown1974 was invoked but never defined (see the help page).

[Ohishi2009-7] Cite error: The named reference Ohishi2009 was invoked but never defined (see the help page).

[BG-8] Cite error: The named reference BG was invoked but never defined (see the help page).

[9] Tishby, Biham, Katzav (2016), The distribution of path lengths of self avoiding walks on Erdős-Rényi networks, arXiv:1603.06613.

Gompertz distribution information

and 24 Related for: Gompertz distribution information

Gompertz distribution

Shifted Gompertz distribution

Gumbel distribution

List of probability distributions

Gompertz function

Benjamin Gompertz

Logistic function

Probability distribution fitting

List of statistics articles

Bass diffusion model

Accelerated failure time model

Churn rate

Multivariate normal distribution

Gamma distribution

Customer lifetime value

CumFreq

Elliptical distribution

Sigmoid function

Generalized beta distribution

Multivariate Laplace distribution

Bathtub curve

Logit

Celadon

Demography

Gompertz distribution
Probability density function
Cumulative distribution function
Parameters	shape $\eta >0\,\!$ , scale $b>0\,\!$
Support	$x\in [0,\infty )\!$
PDF	$b\eta \exp \left(\eta +bx-\eta e^{bx}\right)$
CDF	$1-\exp \left(-\eta \left(e^{bx}-1\right)\right)$
Quantile	${\frac {1}{b}}\ln \left(1-{\frac {1}{\eta }}\ln(1-u)\right)$
Mean	$(1/b)e^{\eta }{\text{Ei}}\left(-\eta \right)$ ${\text{where Ei}}\left(z\right)=\int \limits _{-z}^{\infty }\left(e^{-v}/v\right)dv$
Median	$\left(1/b\right)\ln \left[\left(1/\eta \right)\ln \left(1/2\right)+1\right]$
Mode	$=\left(1/b\right)\ln \left(1/\eta \right)\$ ${\text{with }}0<{\text{F}}\left(x^{*}\right)<1-e^{-1}=0.632121,0<\eta <1$ $=0,\quad \eta \geq 1$
Variance	$\left(1/b\right)^{2}e^{\eta }\{-2\eta {\ }_{3}{\text{F}}_{3}\left(1,1,1;2,2,2;\eta \right)+\gamma ^{2}$ $+\left(\pi ^{2}/6\right)+2\gamma \ln \left(\eta \right)+[\ln \left(\eta \right)]^{2}-e^{\eta }[{\text{Ei}}\left(-\eta \right)]^{2}\}$ ${\begin{aligned}{\text{ where }}&\gamma {\text{ is the Euler constant: }}\,\!\\&\gamma =-\psi \left(1\right)={\text{0.577215... }}\end{aligned}}$ ${\begin{aligned}{\text{ and }}{}_{3}{\text{F}}_{3}&\left(1,1,1;2,2,2;-z\right)=\\&\sum _{k=0}^{\infty }\left[1/\left(k+1\right)^{3}\right]\left(-1\right)^{k}\left(z^{k}/k!\right)\end{aligned}}$
MGF	${\text{E}}\left(e^{-tx}\right)=\eta e^{\eta }{\text{E}}_{t/b}\left(\eta \right)$ ${\text{with E}}_{t/b}\left(\eta \right)=\int _{1}^{\infty }e^{-\eta v}v^{-t/b}dv,\ t>0$