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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 actuaries. Related fields of science such as biology and gerontology also considered the Gompertz distribution for the analysis of survival. More recently, computer scientists have also started to model the failure rates of computer codes by the Gompertz distribution. In Marketing Science, it has been used as an individual-level simulation for customer lifetime value modeling. 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.
Probability density function
Cumulative distribution function
Probability density functionEdit
The probability density function of the Gompertz distribution is:
where is the scale parameter and is the shape parameter of the Gompertz distribution. In the actuarial and biological sciences and in demography, the Gompertz distribution is parametrized slightly differently (Gompertz–Makeham law of mortality).
Cumulative distribution functionEdit
The cumulative distribution function of the Gompertz distribution is:
Moment generating functionEdit
The moment generating function is:
The Gompertz distribution is a flexible distribution that can be skewed to the right and to the left. Its hazard function is a convex function of . The model can be fitted into the innovation-imitation paradigm with as the coefficient of innovation and as the coefficient of imitation. When becomes large, approaches . The model can also belong to the propensity-to-adopt paradigm with as the propensity to adopt and as the overall appeal of the new offering.
The Gompertz density function can take on different shapes depending on the values of the shape parameter :
- When the probability density function has its mode at 0.
- When the probability density function has its mode at
If and are the probability density functions of two Gompertz distributions, then their Kullback-Leibler divergence is given by
- If X is defined to be the result of sampling from a Gumbel distribution until a negative value Y is produced, and setting X=−Y, then X has a Gompertz distribution.
- The gamma distribution is a natural conjugate prior to a Gompertz likelihood with known scale parameter 
- When varies according to a gamma distribution with shape parameter and scale parameter (mean = ), the distribution of is Gamma/Gompertz.
- In hydrology the Gompertz distribution is applied to extreme events such as annual maximum one-day rainfalls and river discharges. The blue picture illustrates an example of fitting the Gompertz distribution to ranked annually maximum one-day rainfalls showing also the 90% confidence belt based on the binomial distribution. The rainfall data are represented by plotting positions as part of the cumulative frequency analysis.
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- Calculator for probability distribution fitting 
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