hypergeometricCdf¶

Evaluates the hypergeometric distribution function.

Synopsis¶

hypergeometricCdf (k, n, m, l)

Required Arguments¶

int k (Input): Argument for which the hypergeometric distribution function is to be evaluated.
int n (Input): Sample size n must be greater than or equal to k.
int m (Input): Number of defectives in the lot.
int l (Input): Lot size l must be greater than or equal to n and m.

Return Value¶

The probability that k or fewer defectives occur in a sample of size n drawn from a lot of size l that contains m defectives.

Description¶

The function hypergeometricCdf evaluates the distribution function of a hypergeometric random variable with parameters n, l, and m. The hypergeometric random variable x can be thought of as the number of items of a given type in a random sample of size n that is drawn without replacement from a population of size l containing m items of this type. The probability function is

\[\Pr(x=j) = \frac{\binom{m}{j} \binom{l-m}{n-j}}{\binom{l}{n}} \text{ for } j=i,i+1,\ldots, \min(n,m)\]

where \(i=\max(0,n-l+m)\).

If k is greater than or equal to i and less than or equal to \(\min(n,m)\), hypergeometricCdf sums the terms in this expression for j going from i up to k. Otherwise, 0 or 1 is returned, as appropriate.

To avoid rounding in the accumulation, hypergeometricCdf performs the summation differently, depending on whether k is greater than the mode of the distribution, which is the greatest integer in \((m+1) (n+1)/(l+2)\).

Example¶

Suppose X is a hypergeometric random variable with \(n=100\), \(l=1000\), and \(m=70\). This example evaluates the distribution function at 7.

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

k = 7
l = 1000
m = 70
n = 100

p = hypergeometricCdf(k, n, m, l)
print("Pr(x <= 7) = %6.4f" % (p))

Output¶

Pr(x <= 7) = 0.5995

Informational Errors¶

`IMSL_LESS_THAN_ZERO`	The input argument, k, is less than zero.
`IMSL_K_GREATER_THAN_N`	The input argument, k, is greater than the sample size.

Fatal Errors¶

IMSL_LOT_SIZE_TOO_SMALL Lot size must be greater than or equal to n and m.