Probability density function |
Cumulative distribution function |
Notation |
Beta(α, β) |
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Parameters |
α > 0 shape (real) β > 0 shape (real) |
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Support |
or ![{\displaystyle x\in (0,1)\!}](//wikimedia.org/api/rest_v1/media/math/render/svg/6c4bd4921b023da2cf81472604e1583c7526af1d) |
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PDF |
![{\displaystyle {\frac {x^{\alpha -1}(1-x)^{\beta -1}}{\mathrm {B} (\alpha ,\beta )}}\!}](//wikimedia.org/api/rest_v1/media/math/render/svg/125fdaa41844a8703d1a8610ac00fbf3edacc8e7) where and is the Gamma function. |
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CDF |
![{\displaystyle I_{x}(\alpha ,\beta )\!}](//wikimedia.org/api/rest_v1/media/math/render/svg/630767808887e1bd81c51a75934e8a196907bb93)
(the regularized incomplete beta function) |
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Mean |
![{\displaystyle \operatorname {E} [X]={\frac {\alpha }{\alpha +\beta }}\!}](//wikimedia.org/api/rest_v1/media/math/render/svg/3905662ceed484cba5580951e29eda96f4d2605e)
![{\displaystyle \operatorname {E} [\ln X]=\psi (\alpha )-\psi (\alpha +\beta )\!}](//wikimedia.org/api/rest_v1/media/math/render/svg/de67df996fa33237ab7f415e7edc9fa8e71997a0)
![{\displaystyle \operatorname {E} [X\,\ln X]={\frac {\alpha }{\alpha +\beta }}\,\left[\psi (\alpha +1)-\psi (\alpha +\beta +1)\right]\!}](//wikimedia.org/api/rest_v1/media/math/render/svg/50106a787db7d72ce3066a5a3238813cffebcc2e) (see section: Geometric mean)
where is the digamma function |
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Median |
![{\displaystyle {\begin{matrix}I_{\frac {1}{2}}^{[-1]}(\alpha ,\beta ){\text{ (in general) }}\\[0.5em]\approx {\frac {\alpha -{\tfrac {1}{3}}}{\alpha +\beta -{\tfrac {2}{3}}}}{\text{ for }}\alpha ,\beta >1\end{matrix}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/af887ef0331cde970dad14ad670cf3592334f845) |
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Mode |
for α, β > 1
any value in for α, β = 1
{0, 1} (bimodal) for α, β < 1
0 for α ≤ 1, β ≥ 1, α ≠ β
1 for α ≥ 1, β ≤ 1, α ≠ β |
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Variance |
![{\displaystyle \operatorname {var} [X]={\frac {\alpha \beta }{(\alpha +\beta )^{2}(\alpha +\beta +1)}}\!}](//wikimedia.org/api/rest_v1/media/math/render/svg/f90a6ad61b4b436749ca37a6c2a1aa077b032ce3)
![{\displaystyle \operatorname {var} [\ln X]=\psi _{1}(\alpha )-\psi _{1}(\alpha +\beta )\!}](//wikimedia.org/api/rest_v1/media/math/render/svg/b4941f45412823abd34d3befea7f8fbf544135e4) (see trigamma function and see section: Geometric variance) |
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Skewness |
![{\displaystyle {\frac {2\,(\beta -\alpha ){\sqrt {\alpha +\beta +1}}}{(\alpha +\beta +2){\sqrt {\alpha \beta }}}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/43ec71817c032c8eb21b5feadd0ec9b91c747530) |
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Excess kurtosis |
![{\displaystyle {\frac {6[(\alpha -\beta )^{2}(\alpha +\beta +1)-\alpha \beta (\alpha +\beta +2)]}{\alpha \beta (\alpha +\beta +2)(\alpha +\beta +3)}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/eea65a8d7c9e00ba6299b727eab679117776f41e) |
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Entropy |
![{\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}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/ff4b6cc1848fe96318adb734393b701cb816f88a) |
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MGF |
![{\displaystyle 1+\sum _{k=1}^{\infty }\left(\prod _{r=0}^{k-1}{\frac {\alpha +r}{\alpha +\beta +r}}\right){\frac {t^{k}}{k!}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/b97b0e33f3134c2fc5c484016ab8e03e18d85481) |
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CF |
;\alpha +\beta ;i\,t)\!}
(see Confluent hypergeometric function) |
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Fisher information |
see section: Fisher information matrix |
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Method of moments |
![{\displaystyle \alpha =\left({\frac {E[X](1-E[X])}{V[X]}}-1\right)E[X]}](//wikimedia.org/api/rest_v1/media/math/render/svg/d2b596a180ef813a0baa1d6f2063950e20da1f62)
![{\displaystyle \beta =\left({\frac {E[X](1-E[X])}{V[X]}}-1\right)(1-E[X])}](//wikimedia.org/api/rest_v1/media/math/render/svg/05ace15e23f6ac9be43eea861f44c018fd3d00de) |
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