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.
The probability density function of the Gamma/Gompertz distribution is:
f(x;b,s,\beta)=
| bsebx\betas | |
| \left(\beta-1+ebx\right)s+1 |
where
b>0
\beta,s>0
The cumulative distribution function of the Gamma/Gompertz distribution is:
\begin{align}F(x;b,s,\beta)&=1-
| \betas | |
| \left(\beta-1+ebx\right)s |
,{ }x>0,{ }b,s,\beta>0\\[6pt] &=1-e-bsx,{ }\beta=1\\\end{align}
The moment generating function is given by:
\begin{align} E(e-tx)= \begin{cases}\displaystyle \betas
| sb | |
| t+sb |
{ }{2F1}(s+1,(t/b)+s;(t/b)+s+1;1-\beta),&\beta\ne1;\\ \displaystyle
| sb | |
| t+sb |
,&\beta=1. \end{cases} \end{align}
{2F1}(a,b;c;z)=
| infty[(a) | |
| \sum | |
| k(b) |
k/(c)
| k/k! | |
| k]z |
The Gamma/Gompertz distribution is a flexible distribution that can be skewed to the right or to the left.
b.
η
\alpha
\beta
\alpha/\beta
x