| Beta distribution | CDF | I x ( α , β ) {\displaystyle I_{x}(\alpha ,\beta )\!} (the regularized incomplete beta function) | 1.00 | infobox |
| Beta distribution | CF | 1 F 1 ( α ; α + β ; i t ) {\displaystyle {}_{1}F_{1}(\alpha ;\alpha +\beta ;i\,t)\!} (see Confluent hypergeometric function) | 1.00 | infobox |
| Beta distribution | Entropy | ln B ( α , β ) − ( α − 1 ) ψ ( α ) − ( β − 1 ) ψ ( β ) + ( α + β − 2 ) ψ ( α + β ) {\displaystyle {\begin{matrix}\ln \mathrm {B} (\alpha ,\beta )-(\alpha -1)\psi (\alpha )-(\b… | 1.00 | infobox |
| Beta distribution | Excess kurtosis | 6 [ ( α − β ) 2 ( α + β + 1 ) − α β ( α + β + 2 ) ] α β ( α + β + 2 ) ( α + β + 3 ) {\displaystyle {\frac {6[(\alpha -\beta )^{2}(\alpha +\beta +1)-\alpha \beta (\alpha +\beta +… | 1.00 | infobox |
| Beta distribution | Fisher information | [ var [ ln X ] cov [ ln X , ln ( 1 − X ) ] cov [ ln X , ln ( 1 − X ) ] var [ ln ( 1 − X ) ] ] {\displaystyle {\begin{bmatrix}\operatorname {var} [\ln X]&\ope… | 1.00 | infobox |
| Beta distribution | Mean | E [ X ] = α α + β {\displaystyle \operatorname {E} [X]={\frac {\alpha }{\alpha +\beta }}\!} E [ ln X ] = ψ ( α ) − ψ ( α + β ) {\displaystyle \operatorname {E} [\ln X]=\ps… | 1.00 | infobox |
| Beta distribution | Median | I 1 2 [ − 1 ] ( α , β ) (in general) ≈ α − 1 3 α + β − 2 3 for α , β > 1 {\displaystyle {\begin{matrix}I_{\frac {1}{2}}^{[-1]}(\alpha ,\beta ){\text{ (in general) }}\\[0.5em]\ap… | 1.00 | infobox |
| Beta distribution | Method of moments | α = ( E [ X ] ( 1 − E [ X ] ) V [ X ] − 1 ) E [ X ] {\displaystyle \alpha =\left({\frac {E[X](1-E[X])}{V[X]}}-1\right)E[X]} β = ( E [ X ] ( 1 − E [ X ] ) V [ X ] − 1 ) ( 1 − E [… | 1.00 | infobox |
| Beta distribution | MGF | 1 + ∑ k = 1 ∞ ( ∏ r = 0 k − 1 α + r α + β + r ) t k k ! {\displaystyle 1+\sum _{k=1}^{\infty }\left(\prod _{r=0}^{k-1}{\frac {\alpha +r}{\alpha +\beta +r}}\right){\frac {t^{k}}{… | 1.00 | infobox |
| Beta distribution | Mode | α − 1 α + β − 2 {\displaystyle {\frac {\alpha -1}{\alpha +\beta -2}}\!} for α, β > 1 Any value in the domain for α = β = 1 No mode if α<1 or β<1. Density diverges at 0 for α ≤ 1… | 1.00 | infobox |
| Beta distribution | Notation | Beta(α, β) | 1.00 | infobox |
| Beta distribution | Parameters | α > 0 shape (real) β > 0 shape (real) | 1.00 | infobox |
| Beta distribution | PDF | x α − 1 ( 1 − x ) β − 1 B ( α , β ) {\displaystyle {\frac {x^{\alpha -1}(1-x)^{\beta -1}}{\mathrm {B} (\alpha ,\beta )}}\!} where B ( α , β ) = Γ ( α ) Γ ( β ) Γ ( α + β ) {\dis… | 1.00 | infobox |
| Beta distribution | Skewness | 2 ( β − α ) α + β + 1 ( α + β + 2 ) α β {\displaystyle {\frac {2\,(\beta -\alpha ){\sqrt {\alpha +\beta +1}}}{(\alpha +\beta +2){\sqrt {\alpha \beta }}}}} | 1.00 | infobox |
| Beta distribution | Support | x ∈ [ 0 , 1 ] {\displaystyle x\in [0,1]\!} or x ∈ ( 0 , 1 ) {\displaystyle x\in (0,1)\!} | 1.00 | infobox |
| Beta distribution | Variance | var [ X ] = α β ( α + β ) 2 ( α + β + 1 ) {\displaystyle \operatorname {var} [X]={\frac {\alpha \beta }{(\alpha +\beta )^{2}(\alpha +\beta +1)}}\!} var [ ln X ] = ψ 1 ( α… | 1.00 | infobox |
| Beta distribution | is a | family of continuous probability distributions defined on the interval | 0.90 | text |
| Beta distribution | is a | suitable model for the random behavior of percentages and proportions.In Bayesian inference | 0.90 | text |
| Beta distribution | is a | unique real number x | 0.90 | text |
| Beta distribution | is a | suitable model for the random behavior of percentages and it is particularly suitable to the statistical modelling of proportions | 0.90 | text |
| Beta distribution | is a | same as the uniform distribution | 0.90 | text |
| Beta distribution | is a | special case of the noncentral beta distribution where λ | 0.90 | text |
| Beta distribution | is a | rule of succession | 0.90 | text |
| Beta distribution | is a | 2-dimensional surface | 0.90 | text |
