beta_cdf

Evaluates the beta probability distribution function.

Synopsis

#include <imsl.h>

float imsl_f_beta_cdf (float x, float pin, float qin)

The type double function is imsl_d_beta_cdf.

Required Arguments

float x (Input)
Argument for which the beta probability distribution function is to be evaluated.

float pin (Input)
First beta distribution parameter. Argument pin must be positive.

float qin (Input)
Second beta distribution parameter. Argument qin must be positive.

Return Value

The probability that a beta random variable takes on a value less than or equal to x.

Description

Function imsl_f_beta_cdf evaluates the distribution function of a beta random variable with parameters pin and qin. This function is sometimes called the incomplete beta ratio and with p = pin and q = qin, is denoted by Ix (p, q). It is given by

 

where Γ(⋅) is the gamma function. The value of the distribution function by Ix (p, q) is the probability that the random variable takes a value less than or equal to x.

The integral in the expression above is called the incomplete beta function and is denoted by βx (p, q). The constant in the expression is the reciprocal of the beta function (the incomplete function evaluated at one) and is denoted by β(p, q).

Function beta_cdf uses the method of Bosten and Battiste (1974).

Example

Suppose X is a beta random variable with parameters 12 and 12. (X has a symmetric distribution.) This example finds the probability that X is less than 0.6 and the probability that X is between 0.5 and 0.6. (Since X is a symmetric beta random variable, the probability that it is less than 0.5 is 0.5.)

 

#include <imsl.h>

#include <stdio.h>

 

int main()

{

float p, pin, qin, x;

 

pin = 12.0;

qin = 12.0;

x = 0.6;

p = imsl_f_beta_cdf(x, pin, qin);

printf(" The probability that X is less than 0.6 is %6.4f\n",

p);

 

x = 0.5;

p -= imsl_f_beta_cdf(x, pin, qin);

printf(" The probability that X is between 0.5 and 0.6 is %6.4f\n",

p);

}

Output

 

The probability that X is less than 0.6 is 0.8364

The probability that X is between 0.5 and 0.6 is 0.3364