g01ee computes the upper and lower tail probabilities and the probability density function of the beta distribution with parameters

a

and

b

Syntax

C#
public static void g01ee( double x, double a, double b, double tol, out double p, out double q, out double pdf, out int ifail )

Visual Basic
Public Shared Sub g01ee ( _ x As Double, _ a As Double, _ b As Double, _ tol As Double, _ <OutAttribute> ByRef p As Double, _ <OutAttribute> ByRef q As Double, _ <OutAttribute> ByRef pdf As Double, _ <OutAttribute> ByRef ifail As Integer _ )

Visual Basic

Public Shared Sub g01ee ( _
	x As Double, _
	a As Double, _
	b As Double, _
	tol As Double, _
	<OutAttribute> ByRef p As Double, _
	<OutAttribute> ByRef q As Double, _
	<OutAttribute> ByRef pdf As Double, _
	<OutAttribute> ByRef ifail As Integer _
)

Visual C++
public: static void g01ee( double x, double a, double b, double tol, [OutAttribute] double% p, [OutAttribute] double% q, [OutAttribute] double% pdf, [OutAttribute] int% ifail )

F#
static member g01ee : x : float * a : float * b : float * tol : float * p : float byref * q : float byref * pdf : float byref * ifail : int byref -> unit

Parameters

x: Type: System..::..Double
On entry: $β$ , the value of the beta variate.

Constraint: $0.0 \leq x \leq 1.0$ .

a: Type: System..::..Double
On entry: $a$ , the first parameter of the required beta distribution.

Constraint: $0.0 < a \leq 10^{6}$ .

b: Type: System..::..Double
On entry: $b$ , the second parameter of the required beta distribution.

Constraint: $0.0 < b \leq 10^{6}$ .

tol: Type: System..::..Double
On entry: this parameter is no longer referenced, but is included for backwards compatability.

p: Type: System..::..Double%
On exit: the lower tail probability, $P (B \leq β : a, b)$ .

q: Type: System..::..Double%
On exit: the upper tail probability, $P (B \geq β : a, b)$ .

pdf: Type: System..::..Double%
On exit: the probability density function, $f (B : a, b)$ .

ifail: Type: System..::..Int32%
On exit: $ifail = 0$ unless the method detects an error or a warning has been flagged (see [Error Indicators and Warnings]).

Description

The probability density function of the beta distribution with parameters

a

and

b

is:

f (B : a, b) = \frac{Γ (a + b)}{Γ (a) Γ (b)} B^{a - 1} {(1 - B)}^{b - 1}, 0 \leq B \leq 1; a, b > 0 .

The lower tail probability,

P (B \leq β : a, b)

is defined by

P (B \leq β : a, b) = \frac{Γ (a + b)}{Γ (a) Γ (b)} \int_{0}^{β} B^{a - 1} {(1 - B)}^{b - 1} d B = I_{β} (a, b), 0 \leq β \leq 1; a, b > 0 .

The function

I_{x} (a, b)

, also known as the incomplete beta function is calculated using (S14CCF not in this release).

References

Hastings N A J and Peacock J B (1975) Statistical Distributions Butterworth

Error Indicators and Warnings

Note: g01ee may return useful information for one or more of the following detected errors or warnings.

Errors or warnings detected by the method:

$ifail = 1$

On entry,	$x < 0.0$ ,
or	$x > 1.0$ .

$ifail = 2$

On entry,	$a \leq 0.0$ ,
or	$a > 10^{6}$ ,
or	$b \leq 0.0$ ,
or	$b > 10^{6}$ .

$ifail = 4$: x is too far out into the tails for the probability to be evaluated exactly. The results returned are $0$ and $1$ as appropriate. These should be a good approximation to the required solution.

$ifail = -9000$: An error occured, see message report.

Accuracy

The accuracy is limited by the error in the incomplete beta function. See [Accuracy] in s14cc for further details.

Parallelism and Performance

None.

Further Comments

None.

Example

This example reads values from a number of beta distributions and computes the associated upper and lower tail probabilities and the corresponding value of the probability density function.

Example program (C#): g01eee.cs

Example program data: g01eee.d

Example program results: g01eee.r