Beta binomial distribution
The beta binomial distribution is a special probability distribution in stochastics . It is one of the discrete probability distributions and is univariate . It can be seen as a kind of generalization of the binomial distribution , since it specifies the probability of successes given a given probability of a single success, while in the beta binomial distribution the probability of success is only imprecisely known and is described by a beta distribution B (a, b) becomes. It is therefore a mixed distribution .
The beta binomial distribution has three parameters: n , a , b
definition
A random variable has a beta binomial distribution with the parameters , and , in characters , if they are for all out of the carrier the probability function
has, where is the beta function .
construction
If the probability function is the binomial distribution and the density is the beta distribution, then the probability function of the mixed distribution is calculated as
- .
The integral corresponds exactly to the above probability function.
Alternative representation
Alternatively, the probability function can also be represented as
The constant C is a normalization constant and is calculated as follows:
Here is the gamma function .
properties
Expected value
The expected value depends on all three parameters:
Variance
The variance is:
Crookedness
The skew is indicated with
Probability generating function
The probability generating function of the beta binomial distribution is
- .
Here is the Gaussian hypergeometric function .
Characteristic function
The characteristic function follows through substitution :
- .
Moment generating function
This is the moment generating function
- .
Special cases
If and , then it is a discrete uniform distribution with , since the carrier contains values.
Areas of application
The beta binomial distribution is typically used in cases in which a binomial distribution would normally be used, but cannot assume that all individual events have the same probability of occurring, but that these probabilities are more or less bell-shaped around a value.
For example, if you want to know how many light bulbs will fail within the next 12 months, but assume that the probability of a light bulb failing varies between different delivery boxes, then a beta binomial distribution is appropriate.
Empirically one can assume to have to do with a beta binomial distribution, although one would rather think of a binomial model if the data spreads more than intended by the binomial distribution.
example
Model in Bayesian statistics
An urn contains an unknown number of balls, of which we know from other samples that the proportion of red balls is described by a beta distribution.
There are n are pulled times balls (with replacement). The probability that a red ball will be drawn x times is in the beta binomial distribution .
Numerical example
Based on a complete ignorance of the a priori distribution, which is described with a (alternatives are, for example ), a "preliminary study" is organized with a drawing (with repetition) of 15 balls. One of these balls is red. Thus the a posteriori distribution is described with the .
The actual "study" provides for a drawing of 40 balls. What is asked is the probability that a red ball will be drawn exactly twice.
Since in this second drawing the probability is that one , it can be calculated as follows:
- ,
in which
and there and besides is general , one obtains
This result differs significantly from that which would have been calculated with a "simple" binomial distribution . In that case the result would be .
The graph shows that the “simple” binomial distribution “allows” fewer results than that . This happens because one does not neglect in the Bayesian model that the “true” proportion of red balls is basically unknown, and thus the results are more widely spread.
literature
- Leonhard Held: Methods of statistical inference. Likelihood and Bayes , with the assistance of Daniel Sabanés Bové, Spektrum Akademischer Verlag Heidelberg 2008, ISBN 978-3-8274-1939-2 .
- Jim Albert: Bayesian Computation With R , Springer New York, 2009, ISBN 978-0-387-92297-3 , doi : 10.1007 / 978-0-387-92298-0 .
See also
Web links
- http://www.vosesoftware.com/ModelRiskHelp/Distributions/Discrete_distributions/Beta-Binomial_distribution.htm
- Eric W. Weisstein : Beta-Binomial Distribution . In: MathWorld (English).