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¶

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 as follows:

$f(x|\theta) = e^{-\theta} \theta^x / x!, \phantom{...} \text{for } x = 0,1,2,\ldots$

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

$f(x+1|\theta) = f(x|\theta)(\theta/(x+1)) \phantom{...} \text{for } x = 0,1,2,\ldots,k-1$

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

Figure 11.1 — Plot of Fp (k, θ)

Example¶

Suppose X is a Poisson random variable with $\theta=10$ . In this example, we evaluate the probability that X is less than or equal to 7.

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

k = 7
theta = 10.0
pr = poissonCdf(k, theta)
print("Pr(x <= 7) = %6.4f" % pr)

Output¶

Pr(x <= 7) = 0.2202

Informational Errors¶

IMSLS_LESS_THAN_ZERO Since “k” = # is less than zero, the distribution function is set to zero.