NAG Library Function Document

nag_prob_beta_dist (g01eec)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

▸▿ 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1 Purpose

nag_prob_beta_dist (g01eec) computes the upper and lower tail probabilities and the probability density function of the beta distribution with parameters

a

and

b

2 Specification

#include <nag.h>

#include <nagg01.h>

void	nag_prob_beta_dist (double x, double a, double b, double tol, double p, double q, double pdf, NagError fail)

3 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_incomplete_beta (s14ccc).

4 References

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

5 Arguments

1: $x$ – doubleInput: On entry: $β$ , the value of the beta variate.

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

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

Constraint: $0.0 < b \leq 10^{6}$ .
4: $tol$ – doubleInput: On entry: this argument is no longer referenced, but is included for backwards compatability.
5: $p$ – double *Output: On exit: the lower tail probability, $P (B \leq β : a, b)$ .
6: $q$ – double *Output: On exit: the upper tail probability, $P (B \geq β : a, b)$ .
7: $pdf$ – double *Output: On exit: the probability density function, $f (B : a, b)$ .
8: $fail$ – NagError *Input/Output: The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.2.1.2 in the Essential Introduction for further information.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.

An unexpected error has been triggered by this function. Please contact NAG.
See Section 3.6.6 in the Essential Introduction for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 3.6.5 in the Essential Introduction for further information.
NE_REAL_ARG_GT: On entry, $x = 〈value〉$ .
Constraint: $x \leq 1.0$ .
NE_REAL_ARG_LE: On entry, $a = 〈value〉$ and $b = 〈value〉$ .
Constraint: $a > 0.0$ .

On entry, $a = 〈value〉$ and $b = 〈value〉$ .
Constraint: $b > 0.0$ .
NE_REAL_ARG_LT: On entry, $x = 〈value〉$ .
Constraint: $x \geq 0.0$ .

7 Accuracy

The accuracy is limited by the error in the incomplete beta function. See Section 7 in nag_incomplete_beta (s14ccc) for further details.

8 Parallelism and Performance

Not applicable.

9 Further Comments

None.

10 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.

NAG Library Function Documentnag_prob_beta_dist (g01eec)