| Dirichlet-multinomial distribution | CF | E ( ∏ k = 1 K e i t k ⋅ x k ) = Γ ( α 0 ) Γ ( n + 1 ) Γ ( n + α 0 ) ⋅ D n ( α , ( e i t 1 , . . . , e i t K ) ) {\displaystyle \operatorname {E} (\prod \limits _{k=1}^{K}{e}^{… | 1.00 | infobox |
| Dirichlet-multinomial distribution | Mean | E ( X i ) = n α i α 0 {\displaystyle \operatorname {E} (X_{i})=n{\frac {\alpha _{i}}{\alpha _{0}}}} | 1.00 | infobox |
| Dirichlet-multinomial distribution | MGF | E ( ∏ k = 1 K e t k ⋅ x k ) = Γ ( α 0 ) Γ ( n + 1 ) Γ ( n + α 0 ) ⋅ D n ( α , ( e t 1 , . . . , e t K ) ) {\displaystyle \operatorname {E} (\prod \limits _{k=1}^{K}{e}^{t_{k}\… | 1.00 | infobox |
| Dirichlet-multinomial distribution | Notation | D i r M u l t ( n , α ) {\displaystyle \mathrm {DirMult} (n,{\boldsymbol {\alpha }})} | 1.00 | infobox |
| Dirichlet-multinomial distribution | Parameters | n ∈ { 0 , 1 , 2 , … } {\displaystyle n\in \{0,1,2,\ldots \}} number of trials α 1 , … , α K > 0 , α 0 = ∑ α k {\displaystyle \alpha _{1},\ldots ,\alpha _{K}>0,\alpha _{0}=\sum \… | 1.00 | infobox |
| Dirichlet-multinomial distribution | PGF | E ( ∏ k = 1 K z k x k ) = Γ ( α 0 ) Γ ( n + 1 ) Γ ( n + α 0 ) ⋅ D n ( α , z ) {\displaystyle \operatorname {E} (\prod \limits _{k=1}^{K}{z_{k}}^{x_{k}})={\frac {\Gamma (\alpha… | 1.00 | infobox |
| Dirichlet-multinomial distribution | PMF | Γ ( α 0 ) Γ ( n + 1 ) Γ ( n + α 0 ) ∏ k = 1 K Γ ( x k + α k ) Γ ( α k ) Γ ( x k + 1 ) {\displaystyle {\frac {\Gamma \left(\alpha _{0}\right)\Gamma \left(n+1\right)}{\Gamma \left… | 1.00 | infobox |
| Dirichlet-multinomial distribution | Support | x i ∈ { 0 , … , n } {\displaystyle x_{i}\in \{0,\dots ,n\}} Σ x i = n , 1 ≤ i ≤ K {\displaystyle \Sigma x_{i}=n\!,1\leq i\leq K} | 1.00 | infobox |
| Dirichlet-multinomial distribution | Variance | Var ( X i ) = n α i α 0 ( 1 − α i α 0 ) ( n + α 0 1 + α 0 ) {\displaystyle \operatorname {Var} (X_{i})=n{\frac {\alpha _{i}}{\alpha _{0}}}\left(1-{\frac {\alpha _{i}}{\alpha _… | 1.00 | infobox |
| Dirichlet-multinomial distribution | is a | family of discrete multivariate probability distributions on a finite support of non-negative integers | 0.90 | text |
| Dirichlet-multinomial distribution | is a | set | 0.90 | text |
| the one in this model | instance of | the same simplification would apply in a larger joint probability expression | 0.80 | text |
