MathPHP\Probability\Distribution\Continuous\Beta Class Reference

Beta distribution https://en.wikipedia.org/wiki/Beta_distribution. More...

Inheritance diagram for MathPHP\Probability\Distribution\Continuous\Beta:

Collaboration diagram for MathPHP\Probability\Distribution\Continuous\Beta:

Public Member Functions
	__construct (float $α, float $β)
	Constructor. More...
	pdf (float $x)
	Probability density function. More...
	cdf (float $x)
	Cumulative distribution function. More...
	inverse (float $x, float $tolerance=1.0e-15, int $max_iterations=200)
	Inverse cumulative distribution function (quantile function) Iterative method. More...
	mean ()
	Mean of the distribution. More...
	median ()
	Median of the distribution. More...
	mode ()
	Mode of the distribution. More...
	variance ()
	Variance of the distribution. More...
Public Member Functions inherited from MathPHP\Probability\Distribution\Continuous\Continuous
	inverse (float $target)
	The Inverse CDF of the distribution. More...
	between (float $x₁, float $x₂)
	CDF between - probability of being between two points, x₁ and x₂ The area under a continuous distribution, that lies between two specified points. More...
	outside (float $x₁, float $x₂)
	CDF outside - Probability of being below x₁ and above x₂. More...
	above (float $x)
	CDF above - Probability of being above x to ∞ Area under a continuous distribution, that lies above a specified point. More...
	rand ()
	Produce a random number with a particular distribution. More...
Public Member Functions inherited from MathPHP\Probability\Distribution\Distribution
	__construct (... $params)
	Constructor. More...

Public Attributes
const	PARAMETER_LIMITS
const	SUPPORT_LIMITS
Public Attributes inherited from MathPHP\Probability\Distribution\Continuous\Continuous
const	GUESS_THRESHOLD = 10
const	GUESS_ALLOWANCE = 8

Protected Attributes
	$α
	$β

Detailed Description

Beta distribution https://en.wikipedia.org/wiki/Beta_distribution.

Constructor & Destructor Documentation

__construct()

MathPHP\Probability\Distribution\Continuous\Beta::__construct	(	float	$α,
		float	$β
	)

Constructor.

Parameters

float	$α	shape parameter α > 0
float	$β	shape parameter β > 0

Member Function Documentation

cdf()

MathPHP\Probability\Distribution\Continuous\Beta::cdf

(

float

$x

)

Cumulative distribution function.

cdf = Iₓ(α,β)

Parameters

float

$x

x ∈ (0,1)

Returns

float

Implements MathPHP\Probability\Distribution\Continuous\ContinuousDistribution.

inverse()

MathPHP\Probability\Distribution\Continuous\Beta::inverse	(	float	$x,
		float	$tolerance = 1.0e-15,
		int	$max_iterations = 200
	)

Inverse cumulative distribution function (quantile function) Iterative method.

Parameters

float	$x
float	$tolerance	(optional)
int	$max_iterations	(optional)

Returns

float

Exceptions

MathException

if it fails to converge on a guess within the tolerance

mean()

MathPHP\Probability\Distribution\Continuous\Beta::mean

(

)

Mean of the distribution.

α μ = --— α + β

Returns

float

Implements MathPHP\Probability\Distribution\Continuous\ContinuousDistribution.

median()

MathPHP\Probability\Distribution\Continuous\Beta::median

(

)

Median of the distribution.

Closed forms

• For symmetric cases α = β, median = 1/2
• For α = 1 and β > 0, median = 1 - 2^(-1/β)
• For α > 0 and β = 1, median = 2^(-1/α)
• For α = 3 and β = 2, median = 0.6142724318676105
• For α = 2 and β = 3, median = 0.38572756813238945

Approximation α - ⅓ median = ------— α + β - ⅔

See also

https://en.wikipedia.org/wiki/Beta_distribution#Median

Returns

float

mode()

MathPHP\Probability\Distribution\Continuous\Beta::mode

(

)

Mode of the distribution.

α - 1 mode = ------— α, β > 1 α + β - 2

mode = 0 α = 1, β > 1 mode = 1 α > 1, β = 1

Returns

float

pdf()

MathPHP\Probability\Distribution\Continuous\Beta::pdf

(

float

$x

)

Probability density function.

xᵃ⁻¹(1 - x)ᵝ⁻¹ pdf = -----------— B(α,β)

Parameters

float

$x

x ∈ (0,1)

Returns

float

Implements MathPHP\Probability\Distribution\Continuous\ContinuousDistribution.

variance()

MathPHP\Probability\Distribution\Continuous\Beta::variance

(

)

Variance of the distribution.

αβ var[X] = ----------------— ⟮α ＋ β⟯²⟮α ＋ β ＋ 1⟯

Returns

float

Member Data Documentation

PARAMETER_LIMITS

const MathPHP\Probability\Distribution\Continuous\Beta::PARAMETER_LIMITS

Initial value:

= [
        'α' => '(0,∞)'

SUPPORT_LIMITS

const MathPHP\Probability\Distribution\Continuous\Beta::SUPPORT_LIMITS

Initial value:

= [
        'x' => '[0,1]'

---

The documentation for this class was generated from the following file:

• src/Probability/Distribution/Continuous/Beta.php