Gamma/Gompertz distribution
# Gamma/Gompertz distribution

In probability and statistics, the Gamma/Gompertz distribution is a continuous probability distribution. It has been used as an aggregate-level model of customer lifetime and a model of mortality risks.

## Specification

### Probability density function

The probability density function of the Gamma/Gompertz distribution is:

    $f(x;b,s,\beta )={\frac {bse^{bx}\beta ^{s}}{\left(\beta -1+e^{bx}\right)^{s+1}}}$

where $b>0$ is the scale parameter and $\beta ,s>0\,\!$ are the shape parameters of the Gamma/Gompertz distribution.

### Cumulative distribution function

The cumulative distribution function of the Gamma/Gompertz distribution is:

    ${\begin{aligned}F(x;b,s,\beta )&=1-{\frac {\beta ^{s}}{\left(\beta -1+e^{bx}\right)^{s}}},{\ }x>0,{\ }b,s,\beta >0\\[6pt]&=1-e^{-bsx},{\ }\beta =1\\\end{aligned}}$

### Moment generating function

The moment generating function is given by:

    ${\begin{aligned}{\text{E}}(e^{-tx})={\begin{cases}\displaystyle \beta ^{s}{\frac {sb}{t+sb}}{\ }{_{2}{\text{F}}_{1}}(s+1,(t/b)+s;(t/b)+s+1;1-\beta ),&\beta \neq 1;\\\displaystyle {\frac {sb}{t+sb}},&\beta =1.\end{cases}}\end{aligned}}$

where ${_{2}{\text{F}}_{1}}(a,b;c;z)=\sum _{k=0}^{\infty }[(a)_{k}(b)_{k}/(c)_{k}]z^{k}/k!$ is a Hypergeometric function.

## Properties

The Gamma/Gompertz distribution is a flexible distribution that can be skewed to the right or to the left.

## Related distributions

* When β = 1, this reduces to an Exponential distribution with parameter sb.
* The gamma distribution is a natural conjugate prior to a Gompertz likelihood with known, scale parameter $b\,\!.$
* When the shape parameter $\eta \,\!$ of a Gompertz distribution varies according to a gamma distribution with shape parameter $\alpha \,\!$ and scale parameter $\beta \,\!$ (mean = $\alpha /\beta \,\!$), the distribution of $x$ is Gamma/Gompertz.
