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Beta Distribution

NIST/SEMATECH Section 1.3.6.6.17 Beta Distribution

PDF

Beta(2, 5) PDF 0 0.2 0.4 0.6 0.8 1 x 0 0.5 1 1.5 2 2.5 f(x)

CDF

Beta(2, 5) CDF 0 0.2 0.4 0.6 0.8 1 x 0 0.2 0.4 0.6 0.8 1 F(x)

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CDF

Formulas

Probability Density Function

f(x)=xα−1(1−x)β−1B(α,β),0≤x≤1f(x) = \frac{x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha,\beta)}, \quad 0 \le x \le 1

Cumulative Distribution Function

F(x)=Ix(α,β)F(x) = I_x(\alpha,\beta)

Properties

Mean

αα+β\frac{\alpha}{\alpha+\beta}

Variance

αβ(α+β)2(α+β+1)\frac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)}

Overview

The beta distribution is a continuous distribution defined on the interval [0, 1][0,\,1] with shape parameters α\alpha and β\beta. It is commonly used to model proportions, probabilities, and random variables with bounded support.

Related Distributions

Uniform Distribution Gamma Distribution Binomial Distribution