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

a

and

b

Syntax

[p, q, pdf, ifail] = g01ee(x, a, b)

[p, q, pdf, ifail] = nag_stat_prob_beta(x, a, b)

Note: the interface to this routine has changed since earlier releases of the toolbox:

At Mark 23:

tol was removed from the interface

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 nag_specfun_beta_incomplete (s14cc).

References

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

Parameters

Compulsory Input Parameters

1: $x$ – double scalar: $β$ , the value of the beta variate.

Constraint: $0.0 \leq x \leq 1.0$ .
2: $a$ – double scalar: $a$ , the first parameter of the required beta distribution.

Constraint: $0.0 < a \leq 10^{6}$ .
3: $b$ – double scalar: $b$ , the second parameter of the required beta distribution.

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

Optional Input Parameters

None.

Output Parameters

1: $p$ – double scalar: The lower tail probability, $P (B \leq β : a, b)$ .
2: $q$ – double scalar: The upper tail probability, $P (B \geq β : a, b)$ .
3: $pdf$ – double scalar: The probability density function, $f (B : a, b)$ .
4: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

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

Errors or warnings detected by the function:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

$ifail = 1$

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	$b > 10^{6}$ .

W $ifail = 4$: 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.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

Accuracy

The accuracy is limited by the error in the incomplete beta function. See Accuracy in nag_specfun_beta_incomplete (s14cc) for further details.

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.

Open in the MATLAB editor: g01ee_example

function g01ee_example


fprintf('g01ee example results\n\n');

% Lower and upper tail probabilities for Beta distribution
x = [0.25; 0.75; 0.50];
a = [1.00; 1.50; 2.00];
b = [2.00; 1.50; 1.00];

fprintf('    x       a       b       p       q       pdf\n');
for j = 1:numel(x)

  [p, q, pdf, ifail] = g01ee( ...
                              x(j), a(j), b(j));

 fprintf('%8.4f', x(j), a(j), b(j), p, q, pdf);
 fprintf('\n')
end

g01ee example results

    x       a       b       p       q       pdf
  0.2500  1.0000  2.0000  0.4375  0.5625  1.5000
  0.7500  1.5000  1.5000  0.8045  0.1955  1.1027
  0.5000  2.0000  1.0000  0.2500  0.7500  1.0000

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox