Poisson binomial distribution
The Generalized binomial distribution is a discrete probability distribution. It is defined as the sum of independent identically distributed random variables are not necessarily which are subject to a Bernoulli distribution.
The Generalized Binomial thus describes the success of a series of independent tests, which may take exactly two results. The difference to the binomial distribution is that any attempt another success probability can be assigned.
It is also possible to define the generalized binomial distribution as a sum of independent, non-identical, binomial distributed random variables, where the Bernoulli random variables are combined with identical probabilities of success for a binomial random variable.
In English, the generalized binomial distribution is often referred to as a Poisson binomial distribution.
- 2.1 Expectation value
- 2.2 variance
- 2.3 skewness
- 2.4 curvature
- 2.5 Characteristic Function
- 2.6 moment generating function
- 3.1 Special case of binomial distribution
- 3.2 approximation by Poisson distribution
- 3.3 approximation by the normal distribution
- 4.1 Radar Speed
- 4.2 Production process
Definition of Generalized Binomial
A discrete random variable follows a Generalized binomial distribution with parameter vector, if it has the following probability function
Where the vector of probabilities of success per trial and the total number of successes in trials referred.
Notation:
Is the set of all subsets that can be formed from the carrier. is the complement of, ie.
The corresponding cumulative distribution function is
Alternative parameterization
The Generalized binomial distribution can also be defined as the sum of binomial distributed random variables, by the Bernoulli random variables are combined with equal success probabilities binomial distributed random variables.
Where the parameter vector containing the probabilities of success of a binomial random variable and the parameter vector, respectively associated number of attempts.
It thus applies. Here is the one vector of length consisting of all ones.
Properties of the Generalized Binomial
Is a random variable that follows a binomial distribution Generalized below.
Expected value
The Generalized Binomial distribution has the expected value
Variance
The Generalized Binomial distribution has the variance
Skew
The Generalized binomial distribution has skewness
Curvature
The Generalized Binomial distribution has the curvature
Characteristic function
The characteristic function of the generalized binomial distribution is:
Moment generating function
The moment generating function of the Generalized binomial distribution is:
Relationship to other distributions
Special case of the binomial distribution
If all the success probabilities are equal, that is, is then obtained from the Generalized binomial binomial distribution.
Approximation by Poisson distribution
For a very large number of attempts and very small, but different success probabilities, the probability function of the Generalized binomial distribution can be approximated by the Poisson distribution.
The parameter is equal to the expected value of the Generalized binomial distribution.
Approximation by the normal distribution
The distribution function of the binomial distribution can be approximated by the generalized for a very large number of attempts by the normal distribution.
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 has to drive every working two different routes. The probabilities in a radar control should be advised for directions and for directions.
What are the chances to fall on a working day in controls?
The random number of speed cameras can be modeled as a sum of two Bernoulli random variable for Route and Route:, with
Since and have different probabilities of success, one can not be solved using the binomial distribution this example.
Follows a binomial distribution with parameters Generalized vector.
The desired probabilities can be calculated as follows:
- Controls:
- Control:
- Controls:
Manufacturing process
In a factory units are produced and then subjected to quality control. There may be various types of faults. The probabilities that a particular type of error are occurring for the error type and for each of the error types, and.
What are the probabilities that a unit is produced with errors?
The random number of errors can be written as a sum of three random variables Bernoulli - distributed, and: , with
Generalized has a binomial distribution with parameter vector.
Alternatively, the parameterization can be chosen by the identical Bernoulli random variables are combined into a binomially distributed random variable.
The desired probabilities can be calculated as follows:
- Error:
- Error:
- Error:
- Error:
Application & Calculation
The Generalized Binomial comes in many areas; e.g. Surveys, production processes, quality assurance. Often, however, an approximation is used because the exact calculation is very complex. Without appropriate software simple models with few Bernoulli random variables themselves are hard to calculate.
Random numbers
For the generation of random numbers, the inversion method can be used.