C Library Functions  - nbcdf (3)

NAME
nbcdf(3f) - [M_datapac:CUMULATIVE_DISTRIBUTION] compute the negative binomial cumulative distribution function

CONTENTS

Synopsis
Description
Note
Input Arguments
Output Arguments
Examples
Author
Maintainer
License
References

SYNOPSIS
SUBROUTINE NBCDF(X,P,N,Cdf) 

       REAL(kind=wp),intent(in)  :: X
       REAL(kind=wp),intent(in)  :: P
       INTEGER                   :: N
       REAL(kind=wp),intent(out) :: Cdf

DESCRIPTION

NBCDF(3f) computes the cumulative distribution function value at the REAL value X for the negative binomial distribution with REAL ’Bernoulli probability’ parameter = P, and integer

The negative binomial distribution used herein has mean = N*(1-P)/P and standard deviation = sqrt(N*(1-P)/(P*P))). this distribution is defined for all non-negative integer X-- X = 0, 1, 2, ... . This distribution has the probability function 

       f(X) = c(N+X-1,N) * P**N * (1-P)**X
Where c(N+X-1,N) is the combinatorial function equaling the number of combinations of N+X-1 items taken N at a time.

The negative binomial distribution is the distribution of the number of failures before obtaining N successes in an indefinite sequence of Bernoulli (0,1) trials where the probability of success in a precision trial = P.

NOTE

Even though the input to this cumulative distribution function subroutine for this discrete distribution should (under normal circumstances) be a discrete integer value, the input variable X is REAL in mode.

X has been specified as REAL so as to conform with the datapac convention that all input ****data**** (as opposed to sample size, for example) variables to all datapac subroutines are REAL. This convention is based on the belief that

1. A mixture of modes (floating point versus integer) is inconsistent and an unnecessary complication in a data analysis; and
2. Floating point machine arithmetic (as opposed to integer arithmetic) is the more natural mode for doing data analysis.

INPUT ARGUMENTS

X The value at which the cumulative distribution function is to be evaluated. X should be non-negative and integral-valued.
P The value of the ’Bernoulli probability’ parameter for the negative binomial distribution. P should be between 0.0 (exclusively) and 1.0 (exclusively).
N The integer value of the ’number of successes in Bernoulli
trials’ parameter.
  N should be a positive integer.

OUTPUT ARGUMENTS

CDF The cumulative distribution function value for the negative binomial distribution

EXAMPLES
Sample program: 

   program demo_nbcdf
   !@(#) line plotter graph of cumulative distribution function
   use M_datapac, only : nbcdf, plott, label
   implicit none
   real,allocatable  :: x(:), y(:)
   real              :: p
   integer           :: i
   integer           :: n
      call label(’nbcdf’)
      x=[(real(i),i=0,100,1)]
      if(allocated(y))deallocate(y)
      allocate(y(size(x)))
      p=0.50
      n=size(x)
      do i=1,size(x)
         call NBCDF(X(i),P,N,y(i))
      enddo
      call plott(x,y,size(x))
   end program demo_nbcdf
Results: 

    The following is a plot of Y(I) (vertically) versus X(I) (horizontally)
                      I-----------I-----------I-----------I-----------I
     0.1000000E+03 -                                             X X  X
     0.9583334E+02 I                                  X  X X  X
     0.9166666E+02 I                         X X X  X
     0.8750000E+02 I                 XX X X
     0.8333334E+02 I           XXX X
     0.7916667E+02 I       XXXX
     0.7500000E+02 -    XXX
     0.7083334E+02 I   XX
     0.6666667E+02 I  XX
     0.6250000E+02 I  X
     0.5833334E+02 I  X
     0.5416667E+02 I  X
     0.5000000E+02 -  X
     0.4583334E+02 I  X
     0.4166667E+02 I  X
     0.3750000E+02 I  X
     0.3333334E+02 I  X
     0.2916667E+02 I  X
     0.2500000E+02 -  X
     0.2083334E+02 I  X
     0.1666667E+02 I  X
     0.1250000E+02 I  X
     0.8333336E+01 I  X
     0.4166672E+01 I  X
     0.0000000E+00 -  X
                      I-----------I-----------I-----------I-----------I
              -0.1776E-14  0.1250E+00  0.2500E+00  0.3750E+00  0.5000E+00

AUTHOR
The original DATAPAC library was written by James Filliben of the Statistical Engineering Division, National Institute of Standards and Technology.

MAINTAINER
John Urban, 2022.05.31

LICENSE
CC0-1.0

REFERENCES
o National Bureau of Standards Applied Mathematics Series 55, 1964, page 945, Formulae 26.5.24 and 26.5.28, and page 929.
o Johnson and Kotz, Discrete Distributions, 1969, pages 122-142, especially page 127.
o Hastings and Peacock, Statistical Distributions--A Handbook for Students and Practitioners, 1975, pages 92-95.
o Feller, an Introduction to Probability Theory and Its Applications, Volume 1, Edition 2, 1957, pages 155-157, 210.
o Kendall and Stuart, the Advanced Theory of Statistics, Volume 1, Edition 2, 1963, pages 130-131.
o Williamson and Bretherton, Tables of the Negative Binomial Probability Distribution, 1963.
o Owen, Handbook of Statistical Tables, 1962, page 304.

Nemo Release 3.1 nbcdf (3) July 22, 2023
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