C Library Functions - poicdf (3)

NAME

poicdf(3f) - [M_datapac:CUMULATIVE_DISTRIBUTION] compute the Poisson cumulative distribution function

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

SYNOPSIS

SUBROUTINE POICDF(X,Alamba,Cdf)

       REAL(kind=wp),intent(in)  :: X
       REAL(kind=wp),intent(in)  :: Alamba
       REAL(kind=wp),intent(out) :: Cdf

DESCRIPTION

POICDF(3f) computes the cumulative distribution function value at the precision value X for the Poisson distribution with precision tail length parameter = alamba.
The Poisson distribution used herein has mean = ALAMBA and standard deviation = sqrt(ALAMBA).
This distribution is defined for all discrete non-negative integer X-- X = 0, 1, 2, ... .
This distribution has the probability function
       f(X) = exp(-ALAMBA) * ALAMBA**X / X!
The Poisson distribution is the distribution of the number of events in the interval (0,ALAMBA) when the waiting time between events is exponentially distributed with mean = 1 and standard deviation = 1.

INPUT ARGUMENTS

X The value at which the cumulative distribution function is to be evaluated. x should be non-negative and integral-valued.

ALAMBA The value of the tail length parameter. alamba should be positive. The tail length parameter alamba is not restricted to only integer values. ALAMBA can be set to any positive real value --integer or non-integer.

OUTPUT ARGUMENTS

CDF The cumulative distribution function value. For the Poisson distribution

NOTE

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

EXAMPLES

Sample program:

   program demo_poicdf
   !@(#) line plotter graph of cumulative distribution function
   use M_datapac, only : poicdf, plott, label
   implicit none
   real,allocatable :: x(:), y(:)
   real             :: alamba
   integer          :: i
      call label(’poicdf’)
      x=[(real(i),i=0,100,1)]
      if(allocated(y))deallocate(y)
      allocate(y(size(x)))
      alamba=29.5
      do i=1,size(x)
         call poicdf(X(i),Alamba,y(i))
      enddo
      call plott(x,y,size(x))
   end program demo_poicdf

Results:

The following is a plot of Y(I) (vertically) versus X(I) (horizontally) I-----------I-----------I-----------I-----------I 0.1000000E+03 - X 0.9583334E+02 I X 0.9166666E+02 I X 0.8750000E+02 I X 0.8333334E+02 I X 0.7916667E+02 I X 0.7500000E+02 - X 0.7083334E+02 I X 0.6666667E+02 I X 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 XX 0.3750000E+02 I XXXX 0.3333334E+02 I X X X X 0.2916667E+02 I X X X X 0.2500000E+02 - X X X X X 0.2083334E+02 I XXX X 0.1666667E+02 I XX 0.1250000E+02 I X 0.8333336E+01 I X 0.4166672E+01 I X 0.0000000E+00 - X I-----------I-----------I-----------I-----------I 0.1543E-12 0.2500E+00 0.5000E+00 0.7500E+00 0.1000E+01

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 Johnson and Kotz, Discrete Distributions, 1969, pages 87-121, especially page 114, Formula 93.

o Hastings and Peacock, Statistical Distributions--A Handbook for Students and Practitioners, 1975, page 112.

o National Bureau of Standards Applied Mathematics Series 55, 1964, page 941, Formulae 26.4.4 and 26.4.5, and page 929.

o Feller, An Introduction to Probability Theory and Its Applications, Volume 1, Edition 2, 1957, pages 146-154.

o Cox and Miller, The Theory of Stochastic Processes, 1965, page 7.

o General Electric Company, Tables of the Individual and Cumulative Terms of Poisson Distribution, 1962.

o Owen, Handbook of Statistical Tables, 1962, pages 259-261.

Nemo Release 3.1

poicdf (3)

February 23, 2025

Generated by manServer 1.08 from 9ca64be3-20c7-42fe-be4a-2adb2022e420 using man macros.

X	The value at which the cumulative distribution function is to be evaluated. x should be non-negative and integral-valued.
ALAMBA	The value of the tail length parameter. alamba should be positive. The tail length parameter alamba is not restricted to only integer values. ALAMBA can be set to any positive real value --integer or non-integer.

o	Johnson and Kotz, Discrete Distributions, 1969, pages 87-121, especially page 114, Formula 93.
o	Hastings and Peacock, Statistical Distributions--A Handbook for Students and Practitioners, 1975, page 112.
o	National Bureau of Standards Applied Mathematics Series 55, 1964, page 941, Formulae 26.4.4 and 26.4.5, and page 929.
o	Feller, An Introduction to Probability Theory and Its Applications, Volume 1, Edition 2, 1957, pages 146-154.
o	Cox and Miller, The Theory of Stochastic Processes, 1965, page 7.
o	General Electric Company, Tables of the Individual and Cumulative Terms of Poisson Distribution, 1962.
o	Owen, Handbook of Statistical Tables, 1962, pages 259-261.