Beta Scaled Distribution#

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

The Beta scaled distribution is a continuous probability distribution similar to the Beta distribution but instead of being bounded between 0 and1 it is bounded between \(lower\) and \(upper\). 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 scaled distribution can adopt a wide range of “shapes” including uniform, U-shape, normal-like, exponential-like, and many others, always restricted to a given interval. This flexibility makes it a versatile choice for modeling random variables that are known to be bounded like percentages, grades, some physical quantities like temperature of liquid water at a given pressure.

Key properties and parameters#

Support

\(x \in (lower, upper)\)

Mean

\(\dfrac{\alpha}{\alpha + \beta} (upper-lower) + lower\)

Variance

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

Parameters:

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

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

  • \(lower\) : (float) Lower bound of the distribution, \(lower < upper\).

  • \(upper\) : (float) Upper bound of the distribution, \(upper > lower\).

Probability Density Function (PDF)#

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

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

/home/docs/checkouts/readthedocs.org/user_builds/preliz/envs/stable/lib/python3.11/site-packages/pytensor/link/numba/dispatch/basic.py:211: UserWarning: Numba will use object mode to run XlogY0's perform method. Set `pytensor.config.compiler_verbose = True` to see more details.
  warnings.warn(
Beta Scaled Distribution PDF

Cumulative Distribution Function (CDF)#

\[ F(x \mid \alpha,\beta, lower, upper) = \frac{B(y;\alpha,\beta)}{B(\alpha,\beta)} = I_y(\alpha,\beta) \]

where \(y\) is the scaled variable \(y = \frac{(x - lower)}{(upper - lower)}\). \(B(x;\alpha,\beta)\) is the Incomplete beta function and \(I_x(\alpha,\beta)\) is the regularized incomplete beta function.

Beta Scaled Distribution CDF

See also

Related Distributions:

  • Beta - A Beta scaled distribution with \(lower=0\) and \(upper=1\).

  • Kumaraswamy - It is similar to the Beta scaled distribution, but restricted to the [0, 1] interval and with closed form expression for its probability density function, cumulative distribution function and quantile function.

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

References#