The IMSL_POISSONCDF function evaluates the Poisson distribution function.

The IMSL_POISSONCDF function evaluates the distribution function of a Poisson random variable with parameter theta. The mean of the Poisson random variable, theta, must be positive.

The probability function (with θ = theta) is as follows:

f(x) = (eθx)/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. The IMSL_POISSONCDF function uses the recursive relationship:

f(x + 1) = f(x)(θ/(x + 1)), for x = 0, 1, 2, ..., k - 1

with:

f(0) = e

## Example

Suppose X is a Poisson random variable with θ = 10. This example evaluates the probability that X ≤ 7.

`p = IMSL_POISSONCDF(7, 10)`
`PM, 'Pr(x <= 7) = ', p, FORMAT = '(a13,f7.4)'`
` `
`Pr(x <= 7) = 0.2202`

## Syntax

Result = IMSL_POISSONCDF(K, Theta [, /DOUBLE])

## Return Value

The probability that a Poisson random variable takes a value less than or equal to k.

## Arguments

### K

Parameter for which the Poisson distribution function is to be evaluated.

### Theta

Mean of the Poisson distribution. Parameter theta must be positive.

## Keywords

### DOUBLE (optional)

If present and nonzero, double precision is used.

## Errors

### Informational Errors

STAT_LESS_THAN_ZERO: Input parameter, k, is less than zero.

## Version History

 6.4 Introduced