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GompertzMakehamDistribution

 GompertzMakehamDistribution represents a Gompertz distribution with scale parameter and frailty parameter . GompertzMakehamDistributionrepresents a Gompertz-Makeham distribution with parameters , , , and .
• The hazard function for value in a Gompertz distribution is given by for , and is zero for .
• The hazard function for value in a Gompertz-Makeham distribution is given by for and is zero for .
Probability density function of Gompertz distribution:
Cumulative distribution function of Gompertz distribution:
Probability density function of Gompertz-Makeham distribution:
Cumulative distribution function of Gompertz-Makeham distribution:
Mean and variance of a Gompertz distribution:
Median of Gompertz distribution:
Probability density function of Gompertz distribution:
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Cumulative distribution function of Gompertz distribution:
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Probability density function of Gompertz-Makeham distribution:
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Cumulative distribution function of Gompertz-Makeham distribution:
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Mean and variance of a Gompertz distribution:
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Median of Gompertz distribution:
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 Scope   (9)
Generate a set of pseudorandom numbers that are Gompertz distributed:
Compare its histogram to the PDF:
Distribution parameters estimation:
Estimate the distribution parameters from sample data:
Compare the density histogram of the sample with the PDF of the estimated distribution:
Skewness of Gompertz distribution:
Kurtosis of Gompertz distribution:
Different moments of Gompertz distribution with closed forms as functions of parameters:
Hazard function of Gompertz distribution:
Hazard function of Gompertz-Makeham distribution:
Quantile function of Gompertz distribution:
Quantile function of Gompertz-Makeham distribution:
 Applications   (4)
The lifetime of a device has a Gompertz distribution. Find the reliability of the device:
The hazard function increasing in time:
Find the reliability of two such devices in series:
Find the reliability of two such devices in parallel:
Compare the reliability of both systems for and :
A steel pipe with thickness is exposed to corrosion and fails if any of the n microscopic pits penetrates the surface. Assume the time to penetration at each pit is proportional to the remaining thickness with factor k. If the depths of the pits are initially random and each follows a right-truncated exponential distribution with parameter , then time to failure of the pipe follows a Gompertz distribution. Find the reliability of the pipe:
Find the mean time to failure:
The female mortality in 1900 according to the Society of Actuaries is given by the table:
Create a sample population to use maximum likelihood estimation:
Fit a Gompertz-Makeham distribution into the data:
Plot probability density function:
Compare the mortality data with the survival function of the estimated distribution:
Find the average female life length in 1900:
Compare the mean residual lifetime from the data with estimated distribution:
Define expo-power distribution using Gompertz distribution:
Hazard function:
Survival function:
Probability density function:
WeibullDistribution is a limiting case:
Parameter influence on the CDF of a Gompertz distribution for each :
Parameter influence on the CDF of a Gompertz-Makeham distribution for each :
GompertzMakehamDistribution is closed under scaling by a positive factor:
The family of Gompertz distributions is closed under a minimum:
For different frailty parameters:
The family of Gompertz-Makeham distributions is closed under a minimum:
For different frailty parameters:
Relationships to other distributions:
Gompertz distribution is related to exponential distribution:
The Gompertz-Makeham distribution simplifies to the Gompertz distribution for and :
Gompertz distribution is a truncated GumbelDistribution:
Gompertz distribution is related to truncated WeibullDistribution:
WeibullDistribution is related to Gompertz distribution:
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