neg_binomial_cdf

Evaluates the cumulative distribution function (CDF) of the negative binomial distribution.

Synopsis

#include <imsls.h>

float imsls_f_neg_binomial_cdf (int k, float r, float p)

The type double function is imsls_d_neg_binomial_cdf.

Required Arguments

int k (Input)
Argument for which the CDF is to be evaluated. k must be non-negative.

float r (Input)
The negative binomial parameter. If r is equal to an integer, it represents the number of target successes. r must be greater than 0.

float p (Input)
The negative binomial parameter. If r is equal to an integer, p represents the probability of success in each independent Bernoulli trial. p must be a valid probability.

Return Value

The probability that a discrete negative binomial random variable takes on a value less than or equal to k. A value of NaN is returned if an error occurs.

Description

A discrete random variable taking values

has the negative binomial distribution with parameters

when

where Γ(a) is the complete gamma function,

If r is an integer, an equivalent expression is

and in this case X can be interpreted as the number of failures before obtaining r successes in binomial trials with probability of success, p.

For the negative binomial distribution with parameters (r,p) , the cumulative distribution function (CDF) is

Where Ip(a,b) is the regularized incomplete beta function.

Where

are the incomplete and complete beta functions, respectively.

Example

This example illustrates calling the negative binomial pdf, cdf, and inverse cdf with p = 0.5 and r = 1.0.

#include <imsls.h>
#include <stdio.h>

int main()
{
  int k;
  float p = 0.5, r = 1.0;

  float pdf, cdf;
  int invcdf;

  printf("\nr: %f\n", r);
  printf("p: %f\n", p);
  printf("          k        pdf          cdf    quantile \n");

  for (k = 0; k < 12; k++) {
      pdf = imsls_f_neg_binomial_pdf(k, r, p);
      cdf = imsls_f_neg_binomial_cdf(k, r, p);
      invcdf = imsls_f_neg_binomial_inverse_cdf(cdf, r, p);
      printf(" %10d %12.8f %12.8f %5d \n", k, pdf, cdf, invcdf);
  }
}

Output

r: 1.000000
p: 0.500000
          k        pdf          cdf    quantile
          0   0.50000000   0.50000000     0
          1   0.25000000   0.75000000     1
          2   0.12500000   0.87500000     2
          3   0.06250000   0.93750000     3
          4   0.03125000   0.96875000     4
          5   0.01562500   0.98437500     5
          6   0.00781250   0.99218750     6
          7   0.00390625   0.99609375     7
          8   0.00195313   0.99804688     8
          9   0.00097656   0.99902344     9
         10   0.00048828   0.99951172    10
         11   0.00024414   0.99975586    11

Informational Errors