neg_binomial_pdf

Evaluates the probability density (mass) function (PDF) of the negative binomial distribution.

Synopsis

#include <imsls.h>

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

The type double function is imsls_d_neg_binomial_pdf.

Required Arguments

int k (Input)
Argument for which the PDF 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 negative binomial random variable takes on a value 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.

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