| Beta-binomial distribution | CDF | { 0 , x < 0 ( n x ) B ( x + α , n − x + β ) B ( α , β ) 3 F 2 ( a ; b ; x ) , 0 ≤ x < n 1 , x ≥ n {\displaystyle {\begin{cases}0,&x<0\\{\binom {n}{x}}{\tfrac {\mathrm {B} (x+\al… | 1.00 | infobox |
| Beta-binomial distribution | CF | 2 F 1 ( − n , α ; α + β ; 1 − e i t ) {\displaystyle _{2}F_{1}(-n,\alpha ;\alpha +\beta ;1-e^{it})\!} | 1.00 | infobox |
| Beta-binomial distribution | Excess kurtosis | See text | 1.00 | infobox |
| Beta-binomial distribution | Mean | n α α + β {\displaystyle {\frac {n\alpha }{\alpha +\beta }}\!} | 1.00 | infobox |
| Beta-binomial distribution | MGF | 2 F 1 ( − n , α ; α + β ; 1 − e t ) {\displaystyle _{2}F_{1}(-n,\alpha ;\alpha +\beta ;1-e^{t})\!} where 2 F 1 {\displaystyle _{2}F_{1}} is the hypergeometric function | 1.00 | infobox |
| Beta-binomial distribution | Notation | B e t a B i n ( n , α , β ) {\displaystyle \mathrm {BetaBin} (n,\alpha ,\beta )} | 1.00 | infobox |
| Beta-binomial distribution | Parameters | n ∈ N0 — number of trials α > 0 {\displaystyle \alpha >0} (real) β > 0 {\displaystyle \beta >0} (real) | 1.00 | infobox |
| Beta-binomial distribution | PGF | 2 F 1 ( − n , α ; α + β ; 1 − z ) {\displaystyle _{2}F_{1}(-n,\alpha ;\alpha +\beta ;1-z)\!} | 1.00 | infobox |
| Beta-binomial distribution | PMF | ( n x ) B ( x + α , n − x + β ) B ( α , β ) {\displaystyle {\binom {n}{x}}{\frac {\mathrm {B} (x+\alpha ,n-x+\beta )}{\mathrm {B} (\alpha ,\beta )}}\!} where B ( x , y ) = Γ ( x… | 1.00 | infobox |
| Beta-binomial distribution | Skewness | ( α + β + 2 n ) ( β − α ) ( α + β + 2 ) 1 + α + β n α β ( n + α + β ) {\displaystyle {\tfrac {(\alpha +\beta +2n)(\beta -\alpha )}{(\alpha +\beta +2)}}{\sqrt {\tfrac {1+\alpha +… | 1.00 | infobox |
| Beta-binomial distribution | Support | x ∈ { 0, …, n } | 1.00 | infobox |
| Beta-binomial distribution | Variance | n α β ( α + β + n ) ( α + β ) 2 ( α + β + 1 ) {\displaystyle {\frac {n\alpha \beta (\alpha +\beta +n)}{(\alpha +\beta )^{2}(\alpha +\beta +1)}}\!} | 1.00 | infobox |
| 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… | 0.90 | text |
| Beta-binomial distribution | is a | binomial distribution in which the probability of success at each of n trials is not fixed but randomly drawn from a beta distribution | 0.90 | text |
