poissonCdf

Evaluates the Poisson distribution function.

Synopsis

poissonCdf (k, theta)

Required Arguments

int k (Input)

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

float theta (Input)

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

Return Value

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

Description

The function poissonCdf 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

\[f(x) = e^{-q} θ^x/x!, for x = 0, 1, 2, …\]

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

\[f(x + 1) = f(x)q/(x + 1), for x = 0, 1, 2, …, k - 1\]

with \(f(0)=e^{-q}\).

Figure 9.24 — Plot of \(F_p(k,\theta)\)

Example

Suppose X is a Poisson random variable with \(\theta=10\). This example evaluates the probability that \(X\leq 7\).

from __future__ import print_function
from numpy import *
from pyimsl.math.poissonCdf import poissonCdf

k = 7
theta = 10.0

p = poissonCdf(k, theta)
print("Pr(x <= 7) = %6.4f" % (p))

Output

Pr(x <= 7) = 0.2202

Informational Errors

IMSL_LESS_THAN_ZERO

The input argument, k, is less than zero.