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Beta distribution

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The Beta distribution is the reference model for proportions and unknown probabilities. In Phitter it can live on [A,B]; its two shape parameters separately control proximity to the lower and upper boundary.

Shape Qualitative reading
alpha = beta = 1 uniform
alpha = beta > 1 symmetric, concentrated at the centre
alpha < beta skewed toward the lower endpoint
alpha > beta skewed toward the upper endpoint

Origin and terminology

Its development is tied to Euler’s Beta function and Pearson’s distributions. The family became central to Bayesian inference because Beta is conjugate to Bernoulli and Binomial likelihoods.

In real models

  • uncertainty about a probability of success
  • rates, percentages, composition fractions, and normalized times

Comparisons

Do not automatically read alpha and beta as observations or probabilities: they are positive shape parameters, and a mean near a boundary can arise through different mechanisms.

One family, many geometries between two bounds

Beta can be flat, symmetric, tilted, U-shaped, or concentrated around an interior point. This variety comes from only two shape parameters. It also arises naturally in order statistics: the observation occupying a chosen rank in a sorted Uniform sample has a Beta distribution.

Its Bayesian role is equally familiar. Combining a Beta probability with Bernoulli or Binomial results lets success and failure counts update its parameters directly. That convenience does not make Beta an automatic answer for every percentage. Exact zeros and ones, inflated endpoints, or measurements with a changing denominator may need different models. Rescaling to [A,B] does not remove the need for those endpoints to have substantive meaning.

Decision guide

A good candidate when: a continuous proportion lies between known bounds and behaviour near each endpoint must be controlled separately.

Compare it with: Kumaraswamy and zero- or one-inflated models. Exact boundary values do not belong to an ordinary Beta model and should be explained rather than silently shifted.

References

  • SciPy reference: scipy.stats.beta — definition and parameterization
  • Johnson, N. L., Kotz, S. & Balakrishnan, N. (1994). Continuous Univariate Distributions, 2nd ed., Vol. 1. Wiley.
  • Evans, M., Hastings, N. & Peacock, B. (2000). Statistical Distributions, 3rd ed. Wiley.
  • Feller, W. (1968). An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd ed. Wiley.
  • Abramowitz, M. & Stegun, I. A., eds. (1972). Handbook of Mathematical Functions. Dover.

Beta Distribution: equations and calculator

Distribution defintion

X∼Beta(α,β,A,B)X\sim\mathrm{Beta}\left(\alpha,\beta,A,B\right)

Distribution domain

x∈(A,B)x\in\left(A,B\right)

Parameters domain and parameters constraints

α∈R+,β∈R+,A∈R,B∈R,A<B\alpha\in\mathbb{R}^{+},\beta\in\mathbb{R}^{+},A\in\mathbb{R},B\in\mathbb{R},A < B

Cumulative distribution function

FX(x)=I(z(x),α,β)F_{X}\left(x\right)=I\left(z(x),\alpha,\beta\right)

Probability density function

fX(x)=z(x)α−1(1−z(x))β−1Beta(α,β)(B−A)f_{X}\left(x\right)=\frac{z(x)^{\alpha-1}\left(1-z(x)\right)^{\beta-1}}{\text{Beta}(\alpha,\beta)(B-A)}

Percent point function/Sample

FX−1(u)=A+(B−A)×I−1(u,α,β)F^{-1}_{X}\left(u\right)=A+(B-A)\times I^{-1}\left(u,\alpha,\beta\right)

Non-central parametric moments

μ~k′=E[X~k]=∫01xkfX~(x)dx\tilde{\mu}'_{k}=E[\tilde{X}^k]=\int_{0}^{1}x^{k}f_{\tilde{X}}\left(x\right)dx

Parametric mean

Mean(X)=A+(B−A)⋅μ~1′=A+α(B−A)α+β\mathrm{Mean}(X)=A+\left(B-A\right)\cdot\tilde{\mu}'_{1}=A+\frac{\alpha\left(B-A\right)}{\alpha+\beta}

Parametric variance

Variance(X)=(B−A)2⋅(μ~2′−μ~1′2)=αβ(B−A)2(α+β)2(α+β+1)\mathrm{Variance}(X)=\left(B-A\right)^{2}\cdot(\tilde{\mu}'_{2}-\tilde{\mu}'^{2}_{1})=\frac{\alpha\beta\left(B-A\right)^{2}}{(\alpha+\beta)^2(\alpha+\beta+1)}

Parametric skewness

Skewness(X)=μ~3′−3μ~2′μ~1′+2μ~1′3(μ~2′−μ~1′2)1.5=2 (β−α)α+β+1(α+β+2)αβ\mathrm{Skewness}(X)=\frac{\tilde{\mu}'_{3}-3\tilde{\mu}'_{2}\tilde{\mu}'_{1}+2\tilde{\mu}'^{3}_{1}}{(\tilde{\mu}'_{2}-\tilde{\mu}'^{2}_{1})^{1.5}}=\frac{2\,(\beta-\alpha)\sqrt{\alpha+\beta+1}}{(\alpha+\beta+2)\sqrt{\alpha\beta}}

Parametric kurtosis

Kurtosis(X)=μ~4′−4μ~1′μ~3′+6μ~1′2μ~2′−3μ~1′4(μ~2′−μ~1′2)2=3+6[(α−β)2(α+β+1)−αβ(α+β+2)]αβ(α+β+2)(α+β+3)\mathrm{Kurtosis}(X)=\frac{\tilde{\mu}'_{4}-4\tilde{\mu}'_{1}\tilde{\mu}'_{3}+6\tilde{\mu}'^{2}_{1}\tilde{\mu}'_{2}-3\tilde{\mu}'^{4}_{1}}{(\tilde{\mu}'_{2}-\tilde{\mu}'^{2}_{1})^{2}}=3+\frac{6[(\alpha-\beta)^2 (\alpha +\beta+1)-\alpha \beta (\alpha+\beta+2)]}{\alpha \beta (\alpha+\beta+2) (\alpha+\beta+3)}

Parametric median

Median(X)=A+(B−A)×I−1(12,α,β)if α,β>1\mathrm{Median}(X)=A+(B-A)\times I^{-1}\left(\frac{1}{2},\alpha,\beta\right) \quad \text{if }\alpha,\beta>1

Parametric mode

Mode(X)=A+(B−A)α−1α+β−2if α,β>1\mathrm{Mode}(X)=A+(B-A)\frac{\alpha-1}{\alpha+\beta-2} \quad \text{if }\alpha,\beta>1

Additional information and definitions

X~∼Beta(α,β,0,1)\tilde{X}\sim\mathrm{Beta}\left(\alpha,\beta,0,1\right)
z(x)=(x−A)/(B−A)z\left(x\right)=\left(x-A\right)/\left(B-A\right)
u:Uniform[0,1] random varibleu:\text{Uniform[0,1] random varible}
I(x,a,b):Regularized incomplete beta functionI\left(x,a,b\right):\text{Regularized incomplete beta function}
I−1(x,a,b):Inverse of regularized incomplete beta functionI^{-1}\left(x,a,b\right):\text{Inverse of regularized incomplete beta function}
Beta(x,y):Beta function\text{Beta}\left(x,y\right):\text{Beta function}