This function evaluates the Poisson cumulative distribution function.

Function Return Value

POIDF — Function value, the probability that a Poisson random variable takes a value less than or equal to K.   (Output)

Required Arguments

K — Argument for which the Poisson cumulative distribution function is to be evaluated.   (Input)

THETA — Mean of the Poisson distribution.   (Input)
THETA must be positive.

FORTRAN 90 Interface

Generic:                              POIDF (K, THETA)

Specific:                             The specific interface names are S_POIDF and D_POIDF.

FORTRAN 77 Interface

Single:                                POIDF (K, THETA)

Double:                              The double precision name is DPOIDF.

Description

The function POIDF evaluates the cumulative distribution function of a Poisson random variable with parameter THETA. THETA, which is the mean of the Poisson random variable, must be positive. The probability function (with θ = THETA) is

f(x) = e−θθx/x!,       for x = 0, 1, 2,…

The individual terms are calculated from the tails of the distribution to the mode of the distribution and summed. POIDF uses the recursive relationship

f(x + 1) = f(x)θ/(x + 1),          for x = 0, 1, 2, …k − 1,

with f(0) = e−θ.

Comments

Informational error

Type Code

1         1                  The input argument, K, is less than zero.

Example

Suppose X is a Poisson random variable with θ = 10. In this example, we evaluate the distribution function at 7.

 

      USE UMACH_INT

      USE POIDF_INT

      IMPLICIT   NONE

      INTEGER    K, NOUT

      REAL       DF, THETA

!

      CALL UMACH (2, NOUT)

      K     = 7

      THETA = 10.0

      DF    = POIDF(K,THETA)

      WRITE (NOUT,99999) DF

99999 FORMAT (' The probability that X is less than or equal to ', &

            '7 is ', F6.4)

      END

Output

 

The probability that X is less than or equal to 7 is 0.2202

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