Chapter 11: Probability Distribution Functions and Inverses

binomial_cdf

Evaluates the binomial distribution function.

Synopsis

#include <imsls.h>

float imsls_f_binomial_cdf (int k, int n, float p)

The type double function is imsls_d_binomial_cdf.

Required Arguments

int k   (Input)
Argument for which the binomial distribution function is to be evaluated.

int n   (Input)
Number of Bernoulli trials.

float p   (Input)
Probability of success on each trial.

Return Value

The probability that k or fewer successes occur in n independent Bernoulli trials, each of which has a probability p of success.

Description

The imsls_f_binomial_cdf function evaluates the distribution function of a binomial random variable with parameters n and p. It does this by summing probabilities of the random variable taking on the specific values in its range. These probabilities are computed by the recursive relationship:

To avoid the possibility of underflow, the probabilities are computed forward from 0 if k is not greater than n ´ p; otherwise, they are computed backward from n. The smallest positive machine number, ɛ, is used as the starting value for summing the probabilities, which are rescaled by (1  p)nɛ if forward computation is performed and by pnɛ if backward computation is used.

For the special case of p = 0, imsls_f_binomial_cdf is set to 1; for the case p = 1, imsls_f_binomial_cdf is set to 1 if k = n and is set to 0 otherwise.

Example

Suppose X is a binomial random variable with n = 5 and p = 0.95. In this example, the function finds the probability that X is less than or equal to 3.

#include <imsls.h>

 

int main()

{

    int         k = 3;

    int         n = 5;

    float       p = 0.95;

    float       pr;

 

    pr = imsls_f_binomial_cdf(k,n,p);

    printf("Pr(x <= 3) = %6.4f\n", pr);

}

Output

Pr(x <= 3) = 0.0226

Informational Errors

IMSLS_LESS_THAN_ZERO                         Since “k” = # is less than zero, the distribution function is set to zero.

IMSLS_GREATER_THAN_N                         The input argument, k, is greater than the number of Bernoulli trials, n.


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