Beta Distribution (Beta₍α,β₎) ○◂｜appendage｜___

Beta	…	…	…
Probability density function	…	…	…
Cumulative distribution function	…	…	…
Notation	Beta(α, β)	…	…
Parameters	α > 0 shape (real) β > 0 shape (real)	…	…
Support	$x \in [0, 1]$ or $x \in (0, 1)$	…	…
PDF	$\frac{x ^{α - 1} ( 1 - x ) ^{β - 1}}{B ( α , β )}$ where $B (α, β) = \frac{Γ ( α ) Γ ( β )}{Γ ( α + β )}$ and $Γ$ is the Gamma function.	…	…
CDF	$I_{x} (α, β)$ (the regularized incomplete beta function)	…	…
Mean	$E [X] = \frac{α}{α + β}$ $E [ln X] = ψ (α) - ψ (α + β)$ $E [X ln X] = \frac{α}{α + β} [ψ (α + 1) - ψ (α + β + 1)]$ (see section: Geometric mean) where $ψ$ is the digamma function	…	…
Median	$I_{\frac{1}{2}}^{[- 1]} (α, β) (in general) \approx \frac{α - \frac{1}{3}}{α + β - \frac{2}{3}} for α, β > 1$	…	…
Mode	$\frac{α - 1}{α + β - 2}$ 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	…	…
Variance	$var [X] = \frac{α β}{( α + β ) ^{2} ( α + β + 1 )}$ $var [ln X] = ψ_{1} (α) - ψ_{1} (α + β)$ (see trigamma function and see section: Geometric variance)	…	…
Skewness	$\frac{2 ( β - α ) α + β + 1}{( α + β + 2 ) α β}$	…	…
Excess kurtosis	$\frac{6 [( α - β ) ^{2} ( α + β + 1 ) - α β ( α + β + 2 )]}{α β ( α + β + 2 ) ( α + β + 3 )}$	…	…
Entropy	$ln B (α, β) - (α - 1) ψ (α) - (β - 1) ψ (β) + (α + β - 2) ψ (α + β)$	…	…
MGF	$1 + k = 1 \sum \infty (r = 0 \prod k - 1 \frac{α + r}{α + β + r}) \frac{t ^{k}}{k !}$	…	…
CF	$_{1} F_{1} (α; α + β; i t)$ (see Confluent hypergeometric function)	…	…
Fisher information	$[var [ln X] cov [ln X, ln (1 - X)] cov [ln X, ln (1 - X)] var [ln (1 - X)]]$ see section: Fisher information matrix	…	…
Method of moments	$α = (\frac{E [ X ] ( 1 - E [ X ])}{V [ X ]} - 1) E [X]$ $β = (\frac{E [ X ] ( 1 - E [ X ])}{V [ X ]} - 1) (1 - E [X])$	…	…

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⏾ Concept Map

Beta Distribution (Beta₍α,β₎) ○◂｜appendage｜___｜20260628182425-01-•