probability Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...

and

statistics Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...

, 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 In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can ...

of the Gamma/Gompertz distribution is: :

f(x;b,s,\beta) = \frac

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 In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Ev ...

of the Gamma/Gompertz distribution is: :

& = 1-e^, \beta=1\\\end

Moment generating function

The moment generating function is given by: :

\begin
\text(e^)=
\begin\displaystyle
\beta^s \frac (s+1,(t/b)+s;(t/b)+s+1;1-\beta), & \beta \ne 1; \\
\displaystyle
\frac,& \beta =1.
\end
\end

where

(a,b;c;z) = \sum_^\infty a)_k(b)_k/(c)_k^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 In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average ...

with parameter ''sb''. *The

gamma distribution In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions. The exponential distribution, Erlang distribution, and chi-square distribution are special cases of the gamma distri ...

is a natural

conjugate prior In Bayesian probability theory, if the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posterior are then called conjugate distributions, and th ...

to a Gompertz likelihood with known, scale parameter

b \,\!.

* When the shape parameter

\eta\,\!

of a

Gompertz distribution 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 and actu ...

varies according to a

with shape parameter

\alpha\,\!

and scale parameter

\beta\,\!

(mean =

\alpha/\beta\,\!

), the distribution of

x

is Gamma/Gompertz.

Notes

References

* * * * * {{DEFAULTSORT:Gamma Gompertz distribution Continuous distributions hu:Gompertz-eloszlás