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.
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.
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)