Generalized binomial distribution
Generalized binomial distribution  
Probability distribution 

Distribution function 

parameter   Probability of success for each of the n attempts 

carrier  
Density function  
Distribution function  
Expected value  
Variance  
Crookedness  
Bulge  
Moment generating function  
Characteristic function 
The generalized binomial distribution (sometimes also called Poisson's generalization of the binomial distribution , or Poisson binomial distribution ) is a probability distribution and can therefore be assigned to the mathematical subarea of stochastics . It is a univariate discrete probability distribution . It is defined as the sum of independent , not necessarily identically distributed random variables that are subject to a Bernoulli distribution .
The generalized binomial distribution describes the success of a series of independent experiments, which can each assume exactly two results. The difference to the binomial distribution is that each attempt can be assigned a different probability of success.
It is also possible to define the generalized binomial distribution as the sum of independent, nonidentical, binomially distributed random variables, whereby the Bernoulli random variables with identical success probabilities are combined to binomially distributed random variables.
Definition of the generalized binomial distribution
A discrete random variable follows a Generalized binomial distribution with parameter vector when the following probability function has
 ,
where denotes the vector of the probabilities of success per attempt and the total number of successes in attempts .
Notation:
is the set of all elementary subsets that can be formed from the carrier . is the complement of , that is .
The associated distribution function is
Alternative parameterization
The generalized binomial distribution can also be defined as the sum of binomially distributed random variables by combining the Bernoulli random variables with the same success probabilities to binomially distributed random variables.
 ,
whereby the parameter vector contains the success probabilities of binomially distributed random variables and the parameter vector the respective number of attempts.
It is therefore true . Here is the one vector of the length , consisting of all ones.
Properties of the generalized binomial distribution
in the following be a random variable that follows a generalized binomial distribution .
Expected value
The generalized binomial distribution has the expectation value
Variance
The generalized binomial distribution has the variance
Crookedness
The generalized binomial distribution is skewed
Bulge and excess
The generalized binomial distribution has the curvature
and with it the excess
Accumulators
The cumulative generating function is
 .
Therefore, the kth cumulant is exactly the sum of the kth cumulants of the n Bernoullidistributed random variables from which the generalized binomial distribution is composed:
The recursion equation of the cumulants of the Bernoulli distribution then also applies to these cumulants
Probability generating function
The probability generating function of the generalized binomial distribution is
Characteristic function
The characteristic function of the generalized binomial distribution is:
Moment generating function
The momentgenerating function of the generalized binomial distribution is:
Sum of generalized binomial random variables
Is and two independent generalized binomial random variables, then is generalized binomial distribution: . Hence the generalized binomial distribution is reproductive .
Relationship to other distributions
Relationship to the binomial distribution
The sum of mutually independent binomially distributed random variables is generally binomially distributed. If all success probabilities are equal, that is , then the generalized binomial distribution results in the binomial distribution. In fact, the binomial distribution for fixed expectation and fixed order is the generalized binomial distribution with maximum entropy . That is, under the condition that the parameter vector of the length has maximizes the entropy .
Relationship to the Bernoulli distribution
The sum of mutually independent Bernoullidistributed random variables , which all have a different parameter , is generally binomially distributed.
Approximation by the Poisson distribution
For a very large number of attempts and very small but different probabilities of success , the probability function of the generalized binomial distribution can be approximated by the Poisson distribution .
The parameter is equal to the expectation value of the generalized binomial distribution .
Approximation by the normal distribution
The distribution function of the generalized binomial distribution can be approximated by the normal distribution for a very large number of experiments .
The parameter corresponds to the expected value and the standard deviation of the generalized binomial distribution . is the distribution function of the standard normal distribution .
Examples
Radar control
An employee must drive to work on the motorway and through the local area every working day. The probabilities of getting into a radar control are on the motorway and in the local area.
How high are the chances of getting into controls on a working day ?
The random number of radar controls can be modeled as the sum of two Bernoullidistributed random variables for the motorway and for the local area:, with
Since and have different probabilities of success, this example cannot be solved using the binomial distribution.
follows a generalized binomial distribution with a parameter vector .
The probabilities we are looking for can be calculated as follows:
 Controls:
 Control:
 Controls:
Manufacturing process
Devices are produced in a factory and then subjected to quality control. It can different types of faults. The probabilities that a particular error type will occur are for the error of the type and for the error types and, respectively .
What are the chances that a device will be produced with errors?
The random number of errors can be written as the sum of three Bernoullidistributed random variables , and :, with
has a generalized binomial distribution with a parameter vector .
Alternatively, the parameterization can be selected by combining the identical Bernoulli random variables into a binomially distributed random variable.
The probabilities we are looking for can be calculated as follows:
 Error:
 Error:
 Error:
 Error:
Application & calculation
The generalized binomial distribution is used in many areas; z. B. surveys, manufacturing processes, quality assurance. However, an approximation is often used because the exact calculation is very complex. Without the appropriate software, even simple models with a few Bernoulli random variables can hardly be calculated.
Random numbers
The inversion method can be used to generate random numbers . Alternatively, you can also generate Bernoullidistributed random numbers for the parameters and add them up. The result is then generalized binomially distributed.
literature
 M.Fisz, probability calculation and mathematical statistics, VEB Deutscher Verlag der Wissenschaften, 1973, p. 164 ff.
 KJ Klauer, CriterionOriented Tests, Verlag für Psychologie, Hogrefe, 1987, Göttingen, p. 208 ff.
Web links
 GenBinomApps  R Package . R Package for the computation of Clopper Pearson confidence intervals and the generalized binomial distribution. Retrieved July 30, 2015.
Individual evidence
 ^ On the Number of Successes in Independent Trials . (PDF; 1.6 MB) YHWang, Statistica Sinica, Vol. 3, 1993, p. 295312. Retrieved September 23, 2013.
 ↑ ^{a } ^{b } ^{c } On Computing the Distribution Function for the Sum of Independent and Nonidentical Random Indicators . ( Memento of the original from October 23, 2015 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. (PDF; 110 kB) Y. Hong, Blacksburg, USA, April 5, 2011. Retrieved September 23, 2013.
 ↑ Peter Harremoës: Binomial and Poisson Distributions as Maximum Entropy Distributions . In: IEEE Information Theory Society (Ed.): IEEE Transactions on Information Theory . 47, 2001, pp. 20392041. doi : 10.1109 / 18.930936 .