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. Argument n must be greater than or equal to k.
int m (Input): Number of defectives in the lot.
int l (Input): Lot size. Argument 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¶

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

\[F(k|n,m,l) = \sum_{i=0}^{k} \Pr(X=j)\]

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 or not k is greater than the mode of the distribution, which is the greatest integer less than or equal to \((m+1)(n+1)/(l+2)\)

Example¶

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

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

k = 7
l = 1000
m = 70
n = 100
pr = hypergeometricCdf(k, n, m, l)
print("Pr(x <= 7) = %6.4f" % pr)

Output¶

Pr(x <= 7) = 0.5995

Informational Errors¶

`IMSLS_LESS_THAN_ZERO`	Since “`k`” = # is less than zero, the distribution function is set to zero.
`IMSLS_K_GREATER_THAN_N`	The input argument, `k`, is greater than the sample size.

Fatal Errors¶

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