Beta Distribution

Table of contents

Density Function

The density function of the Beta distribution:

\[f(x; a,b) = \frac{1}{\mathcal{B}(a,b)} x^{a-1} (1-x)^{b-1} \times \mathbf{1}[0 \leq x \leq 1]\]

where \(\mathcal{B}(a,b)\) denotes the Beta function.

Methods for scalar input, as well as for list input, are listed below.

Scalar Input

pystats.dbeta(x: float, shape1: float, shape2: float, log: bool = False) → float

Density function of the Beta distribution.

Example

>>> pystats.dbeta(0.5, 3.0, 2.0)
1.5
Parameters

  • x (float) – A real-valued input.

  • shape1 (float) – A real-valued shape parameter.

  • shape2 (float) – A real-valued shape parameter.

  • log (bool) – Return the log-density or the true form.

Returns

The density function evaluated at x.

List Input

pystats.dbeta(x: List[float], shape1: float, shape2: float, log: bool = False) → List[float]

Density function of the Beta distribution.

Example

>>> pystats.dbeta([0.3, 0.5, 0.9], 3.0, 2.0)
[0.756, 1.5, 0.972]
Parameters

  • x (List[float]) – A standard list input.

  • shape1 (float) – A real-valued shape parameter.

  • shape2 (float) – A real-valued shape parameter.

  • log (bool) – Return the log-density or the true form.

Returns

A list of density values corresponding to the elements of x.

Cumulative Distribution Function

The cumulative distribution function (CDF) of the Beta distribution:

\[F(x; a, b) = \int_0^x f(z; a,b) dz = I_x (a,b)\]

where \(I_x (a,b)\) denotes the regularized incomplete Beta function.

Methods for scalar input, as well as for list input, are listed below.

Scalar Input

pystats.pbeta(p: float, shape1: float, shape2: float, log: bool = False) → float

Distribution function of the Beta distribution.

Example

>>> pystats.pbeta(0.5, 3.0, 2.0)
0.3125
Parameters

  • p (float) – A real-valued input.

  • shape1 (float) – A real-valued shape parameter.

  • shape2 (float) – A real-valued shape parameter.

  • log (bool) – Return the log-density or the true form.

Returns

The cumulative distribution function evaluated at p.

List Input

pystats.pbeta(p: List[float], shape1: float, shape2: float, log: bool = False) → List[float]

Distribution function of the Beta distribution.

Example

>>> pystats.pbeta([0.3, 0.5, 0.9], 3.0, 2.0)
[0.0837, 0.3125, 0.9477]
Parameters

  • p (List[float]) – A standard list input.

  • shape1 (float) – A real-valued shape parameter.

  • shape2 (float) – A real-valued shape parameter.

  • log (bool) – Return the log-density or the true form.

Returns

A list of CDF values corresponding to the elements of p.

Quantile Function

The quantile function of the Beta distribution:

\[q(p; a,b) = \inf \left\{ x : p \leq I_x (a,b) \right\}\]

Methods for scalar input, as well as for list input, are listed below.

Scalar Input

pystats.qbeta(q: float, shape1: float, shape2: float) → float

Quantile function of the Beta distribution.

Example

>>> pystats.qbeta(0.5, 3.0, 2.0)
0.61427243186
Parameters

  • q (float) – A real-valued input.

  • shape1 (float) – A real-valued shape parameter.

  • shape2 (float) – A real-valued shape parameter.

Returns

The quantile function evaluated at q.

List Input

pystats.qbeta(q: List[float], shape1: float, shape2: float) → List[float]

Quantile function of the Beta distribution.

Example

>>> pystats.qbeta([0.3, 0.5, 0.9], 3.0, 2.0)
[0.4915952451274149, 0.6142724318676113, 0.8574406832899696]
Parameters

  • q (List[float]) – A standard list input.

  • shape1 (float) – A real-valued shape parameter.

  • shape2 (float) – A real-valued shape parameter.

Returns

A list of quantiles values corresponding to the elements of q.

Random Sampling

Random sampling for the Beta distribution is achieved by simulating two independent gamma-distributed random variables, \(X \sim G(a,1), Y \sim G(a,1)\), then returning:

\[Z = \frac{X}{X+Y} \sim B(a,b)\]

Scalar Output

pystats.rbeta(shape1: float, shape2: float) → float

Random sampling function for the Beta distribution.

Example

>>> pystats.rbeta(3.0, 2.0)
0.3102933967927699
Parameters

  • shape1 (float) – A real-valued shape parameter.

  • shape2 (float) – A real-valued shape parameter.

Returns

A pseudo-random draw from the Beta distribution.

List Output

pystats.rbeta(n: int, shape1: float, shape2: float) → List[float]

Random sampling function for the Beta distribution.

Example

>>> pystats.rbeta(5, 3.0, 2.0)
[0.2098009340766354, 0.483502928842804, 0.5101449459492882, 0.46683636135288137, 0.791214142230094]
Parameters

  • n (int) – The number of output values.

  • shape1 (float) – A real-valued shape parameter.

  • shape2 (float) – A real-valued shape parameter.

Returns

A list of pseudo-random draws from the Beta distribution.