Gompertz distribution
Encyclopedia
The Gompertz distribution is an extreme value (reverted Gumbel
) distribution (i.e., the distribution of
) truncated at zero. It has been used as a model of customer lifetime.
of the Gompertz distribution is:
where
is the scale parameter
and
is the shape parameter
of the Gompertz distribution.
of the Gompertz distribution is:
where
Let us make the following change of variable: y=
. The moment generating function becomes:
With
:
varies according to a gamma distribution with shape parameter 
and scale parameter 
(mean = 
), the cumulative distribution function is Gamma/Gompertz (G/G).
Gumbel
Gumbel may refer to:* Gumbel distribution, named after Emil Julius Gumbel* Bryant Gumbel , American television sportscaster, brother of Greg* Emil Julius Gumbel, , German mathematician, pacifist and anti-Nazi campaigner...
) distribution (i.e., the distribution of
) truncated at zero. It has been used as a model of customer lifetime.Probability density function
The probability density functionProbability density function
In probability theory, a probability density function , or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the...
of the Gompertz distribution is:
where
is the scale parameterScale parameter
In probability theory and statistics, a scale parameter is a special kind of numerical parameter of a parametric family of probability distributions...
and
is the shape parameterShape parameter
In probability theory and statistics, a shape parameter is a kind of numerical parameter of a parametric family of probability distributions.- Definition :...
of the Gompertz distribution.
Cumulative distribution function
The cumulative distribution functionCumulative distribution function
In probability theory and statistics, the cumulative distribution function , or just distribution function, describes the probability that a real-valued random variable X with a given probability distribution will be found at a value less than or equal to x. Intuitively, it is the "area so far"...
of the Gompertz distribution is:
where
Moment generating function
The moment generating function is such as:Let us make the following change of variable: y=
. The moment generating function becomes:With
Shapes
The Gompertz density function can take on different shapes depending on the values of the shape parameter:
-
the probability density function has its mode at 0. -
the probability density function has its mode at
-
- where
is the smallest root of
- which is
Related distributions
The Gompertz distribution is a natural conjugate to a gamma distribution. If varies according to a gamma distribution with shape parameter and scale parameter (mean = ), the cumulative distribution function is Gamma/Gompertz (G/G).See also
- Gompertz function
- Gompertz–Makeham law of mortality
- Customer lifetime valueCustomer lifetime value-Definition of Customer Lifetime Value:In marketing, customer lifetime value , lifetime customer value , or lifetime value is the net present value of the cash flows attributed to the relationship with a customer...
- Gumbel distribution