Applied Statistics and Probability for Engineers

3-8 HYPERGEOMETRIC DISTRIBUTION 87

and

For a hypergeometric random variable, is similar to the mean a binomial random

variable. Also, differs from the result for a binomial random variable only by the term

shown below.

V 1 X 2

E 1 X 2

V 1 X 2  (^411)  (^3212)  3231300  (^42)  2994 0.88
Sampling with replacement is equivalent to sampling from an infinite set because the propor-
tion of success remains constant for every trial in the experiment. As mentioned previously, if
sampling were done with replacement, Xwould be a binomial random variable and its vari-
ance would be np(1 p). Consequently, the finite population correction represents the cor-
rection to the binomial variance that results because the sampling is without replacement from
the finite set of size N.
If nis small relative to N, the correction is small and the hypergeometric distribution is sim-
ilar to the binomial. In this case, a binomial distribution can effectively approximate the distribu-
tion of the number of units of a specified type in the sample. A case is illustrated in Fig. 3-13.
EXAMPLE 3-29 A listing of customer accounts at a large corporation contains 1000 customers. Of these, 700
have purchased at least one of the corporation’s products in the last three months. To evaluate
a new product design, 50 customers are sampled at random from the corporate listing. What is
0.0
0
Hypergeometric N = 50, n = 5, K = 25
Hypergeometric probability
Binomial probability
0
0.025
0.031
1
0.149
0.156
2
0.326
0.321
3
0.326
0.312
4
0.149
0.156
5
0.025
0.031
Binomial n = 5, p = 0.5
(x)
0.1
0.2
0.3
1 2 3 4 5
x
Figure 3-13
Comparison of hyper-
geometric and binomial
distributions.
The term in the variance of a hypergeometric random variable
is called the finite population correction factor.
Nn
N 1
Finite
Population
Correction
Factor
PQ220 6234F.Ch 03 13/04/2002 03:19 PM Page 87