Beta Distribution#

Univariate, Continuous, Bounded, Symmetric (when α = β)

The Beta distribution is a continuous probability distribution bounded between 0 and 1. It is usually defined by two positive shape parameters: (\(\alpha\)) and (\(\beta\)). But other parametrization like mean (\(\mu\)) and concentration (\(\nu\)) are also common.

The Beta distribution can adopt a wide range of “shapes” including uniform, U-shape, normal-like, exponential-like, and many others, always restricted to the unit interval. This flexibility makes it a versatile choice for modeling random variables that represent proportions, probabilities, or rates. For example, the Beta distribution is commonly used to model the uncertainty of the true proportion of successes in a series of Bernoulli trials, where \(\alpha\) and \(\beta\) represent the number of successes and failures, respectively.

Key properties and parameters#

Support

\(x \in (0, 1)\)

Mean

\(\dfrac{\alpha}{\alpha + \beta}\)

Variance

\(\dfrac{\alpha \beta}{(\alpha+\beta)^2(\alpha+\beta+1)}\)

Parameters:

  • \(\alpha\) : (float) Shape parameter, \(\alpha > 0\).

  • \(\beta\) : (float) Shape parameter, \(\beta > 0\).

  • \(\mu\) : (float) Mean of the distribution, \(0 < \mu < 1\).

  • \(\sigma\) : (float) Standard deviation of the distribution, \(\sigma < sqrt(\mu(1-\mu))\).

  • \(\nu\) : (float) Concentration parameter, \(\nu > 0\).

Alternative parameterization

The Beta distribution has 3 alternative parameterizations. In terms of \(\alpha\) and \(\beta\), \(\mu\) and \(\sigma\), and \(\mu\) and \(\nu\).

The link between the parameters is given by:

\[\begin{split} \alpha = \mu \nu \\ \beta = (1 - \mu) \nu \end{split}\]

Probability Density Function (PDF)#

\[ f(x \mid \alpha, \beta) = \frac{x^{\alpha - 1} (1 - x)^{\beta - 1}}{B(\alpha, \beta)} \]

where \(B(\alpha,\beta)\) is the Beta function

../../_images/beta_0_1.png
../../_images/beta_1_0.png
../../_images/beta_2_0.png

Cumulative Distribution Function (CDF)#

\[ F(x \mid \alpha,\beta) = \frac{B(x;\alpha,\beta)}{B(\alpha,\beta)} = I_x(\alpha,\beta) \]

where \(B(x;\alpha,\beta)\) is the Incomplete beta function and \(I_x(\alpha,\beta)\) is the regularized incomplete beta function.

../../_images/beta_3_0.png
../../_images/beta_4_0.png
../../_images/beta_5_0.png

See also

Common Alternatives:

  • Kumaraswamy - It is similar to the Beta distribution, but with closed form expression for its probability density function, cumulative distribution function and quantile function.

Related Distributions:

  • Beta Scaled - A Beta distribution defined on an arbitrary range.

  • Uniform - The Uniform distribution on the interval [0, 1] is a special case of the Beta distribution with \(\alpha = \beta = 1\).

  • Dirichlet - The Dirichlet distribution is the generalization of the Beta to higher dimensions.

References#