distribuzione Beta
Sign in to saveAlso known as beta distribution of the first kind
distribuzione di probabilità
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 · Italiano
In teoria delle probabilità e in statistica la distribuzione (Beta) è una distribuzione di probabilità continua definita da due parametri e sull'intervallo unitario . Questa distribuzione trova particolare utilizzo nella inferenza bayesiana perché governa la probabilità di un processo di Bernoulli a posteriori dell'osservazione di "successi" e "fallimenti", quando è a priori distribuita uniformemente tra e .
Abstract from DBpedia / Wikipedia · CC BY-SA