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Beta-binomial distribution
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In probability theory and statistics, the beta-binomial distribution is a family of discrete probability distributions on a finite support of non-negative integers arising when the probability of success in each of a fixed or known number of Bernoulli trials is either unknown or random. The beta-binomial distribution is the binomial distribution in which the probability of success at each of n trials is not fixed but randomly drawn from a beta distribution. It is frequently used in Bayesian statistics, empirical Bayes methods and classical statistics to capture overdispersion in binomial type distributed data.
Probability mass function
Probability mass function for the beta-binomial distribution
Cumulative distribution function
Cumulative probability distribution function for the beta-binomial distribution
Parameters
n ∈ N0 — number of trials
{\displaystyle \alpha >0}\alpha >0 (real)
{\displaystyle \beta >0}\beta >0 (real)
Support
k ∈ { 0, …, n }
PMF
{\displaystyle {\binom {n}{k}}{\frac {\mathrm {B} (k+\alpha ,n-k+\beta )}{\mathrm {B} (\alpha ,\beta )}}\!}{\displaystyle {\binom {n}{k}}{\frac {\mathrm {B} (k+\alpha ,n-k+\beta )}{\mathrm {B} (\alpha ,\beta )}}\!}
CDF
{\displaystyle {\begin{cases}0,&k<0\\{\binom {n}{k}}{\tfrac {\mathrm {B} (k+\alpha ,n-k+\beta )}{\mathrm {B} (\alpha ,\beta )}}{}_{3}\!F_{2}({\boldsymbol {a}},{\boldsymbol {b}},k),&0\leq k<n\\1,&k\geq n\end{cases}}}{\displaystyle {\begin{cases}0,&k<0\\{\binom {n}{k}}{\tfrac {\mathrm {B} (k+\alpha ,n-k+\beta )}{\mathrm {B} (\alpha ,\beta )}}{}_{3}\!F_{2}({\boldsymbol {a}},{\boldsymbol {b}},k),&0\leq k<n\\1,&k\geq n\end{cases}}}
where 3F2(a,b,k) is the generalized hypergeometric function
{\displaystyle {}_{3}\!F_{2}(1,-k,n\!-\!k\!+\!\beta ;n\!-\!k\!-\!1,1\!-\!k\!-\!\alpha ;1)\!}{\displaystyle {}_{3}\!F_{2}(1,-k,n\!-\!k\!+\!\beta ;n\!-\!k\!-\!1,1\!-\!k\!-\!\alpha ;1)\!}
Mean
{\displaystyle {\frac {n\alpha }{\alpha +\beta }}\!}{\frac {n\alpha }{\alpha +\beta }}\!
Variance
{\displaystyle {\frac {n\alpha \beta (\alpha +\beta +n)}{(\alpha +\beta )^{2}(\alpha +\beta +1)}}\!}{\frac {n\alpha \beta (\alpha +\beta +n)}{(\alpha +\beta )^{2}(\alpha +\beta +1)}}\!
Skewness
{\displaystyle {\tfrac {(\alpha +\beta +2n)(\beta -\alpha )}{(\alpha +\beta +2)}}{\sqrt {\tfrac {1+\alpha +\beta }{n\alpha \beta (n+\alpha +\beta )}}}\!}{\tfrac {(\alpha +\beta +2n)(\beta -\alpha )}{(\alpha +\beta +2)}}{\sqrt {{\tfrac {1+\alpha +\beta }{n\alpha \beta (n+\alpha +\beta )}}}}\!
Ex. kurtosis
See text
MGF
{\displaystyle _{2}F_{1}(-n,\alpha ;\alpha +\beta ;1-e^{t})\!}_{{2}}F_{{1}}(-n,\alpha ;\alpha +\beta ;1-e^{{t}})\!
{\displaystyle {\text{for }}t<\log _{e}(2)}{\text{for }}t<\log _{e}(2)
CF
{\displaystyle _{2}F_{1}(-n,\alpha ;\alpha +\beta ;1-e^{it})\!}_{{2}}F_{{1}}(-n,\alpha ;\alpha +\beta ;1-e^{{it}})\!
PGF
{\displaystyle {\frac {_{2}F_{1}(-n,\alpha ;-\beta -n+1;z)}{_{2}F_{1}(-n,\alpha ;-\beta -n+1;1)}}}{\displaystyle {\frac {_{2}F_{1}(-n,\alpha ;-\beta -n+1;z)}{_{2}F_{1}(-n,\alpha ;-\beta -n+1;1)}}}
It reduces to the Bernoulli distribution as a special case when n = 1. For α = β = 1, it is the discrete uniform distribution from 0 to n. It also approximates the binomial distribution arbitrarily well for large α and β. Similarly, it contains the negative binomial distribution in the limit with large β and n. The beta-binomial is a one-dimensional version of the Dirichlet-multinomial distribution as the binomial and beta distributions are univariate versions of the multinomial and Dirichlet distributions respectively.
Motivation and derivation
Moments and properties
Point estimates
Further Bayesian considerations
Shrinkage factors
Related distributions
See also
References
External links
Last edited 10 months ago by TripleShortOfACycle
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