Bètaverdeling
Sign in to saveAlso known as beta distribution of the first kind
distributions defined on [0, 1] in terms of two positive parameters
Key facts
- Notation
- Beta( α , β )
- Parameters
- α > 0 shape ( real ), β > 0 shape ( real )
- Support
- x ∈ [ 0 , 1 ] {\displaystyle x\in [0,1]\!} or x ∈ ( 0 , 1 ) {\displaystyle x\in (0,1)\!}
- Cdf
- I x ( α , β ) {\displaystyle I_{x}(\alpha ,\beta )\!} (the regularized incomplete beta function )
- 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, and at 1 if β ≤ 1
- 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 }}}}}
- 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 +2)]}{\alpha \beta (\alpha +\beta +2)(\alpha +\beta +3)}}}
- 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 )-(\beta -1)\psi (\beta )\\[0.5em]{}+(\alpha +\beta -2)\psi (\alpha +\beta )\end{matrix}}}
- 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}}{k!}}}
- Cf
- 1 F 1 ( α ; α + β ; i t ) {\displaystyle {}_{1}F_{1}(\alpha ;\alpha +\beta ;i\,t)\!} (see Confluent hypergeometric function )
- 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 [ X ] ) {\displaystyle \beta =\left({\frac {E[X](1-E[X])}{V[X]}}-1\right)(1-E[X])}
via Wikipedia infobox
Wikidata facts
- Subclass of
- continuous probability distribution
Show 3 more facts
- different from
- beta function
- Commons category
- Beta distribution
- maintained by WikiProject
- WikiProject Mathematics
via Wikidata · CC0
Article · Nederlands
In de kansrekening en statistiek is de bètaverdeling een continue kansverdeling, met twee parameters. De bètaverdeling wordt gebruikt om de kansverdeling van gesorteerde grootheden te beschrijven. Tevens wordt de bètaverdeling uitgebreid gebruikt in de Bayesiaanse statistiek vanwege handige wiskundige eigenschappen van deze verdeling.
Abstract from DBpedia / Wikipedia · CC BY-SA