Page 117 - Applied statistics and probability for engineers

Section 3-8/Hypergeometric Distribution 95

What is the probability that at least one part in the sample is from the local supplier?
⎛100 ⎞ ⎛200 ⎞
⎜ ⎟ ⎜ ⎟
P X ≥ ) = − ( 0 1 ⎝ 0 ⎠ ⎝ 4 ⎠ = 0 804.
(
P X = ) = −
1
1
⎛300 ⎞
⎜ ⎟
⎝ 4 ⎠
Practical Interpretation: Sampling without replacement is frequently used for inspection and the hypergeometric

distribution simpliies the calculations.
The mean and variance of a hypergeometric random variable can be determined from the
trials that compose the experiment. However, the trials are not independent, so the calculations

are more dificult than for a binomial distribution. The results are stated as follows.

Mean and Variance
If X is a hypergeometric random variable with parameters N,K, and n, then
⎛ N − ⎞ n
2
μ = ( ) = np and σ = ( ) = (1 − ) ⎜ ⎟ (3-14)
p
E X
np
V X
⎝ N − ⎠ 1
where p = K N.
Here p is the proportion of successes in the set of N objects.

Example 3-28 Mean and Variance In Example 3-27, the sample size is four. The random variable X is the
number of parts in the sample from the local supplier. Then, p = 100 /300 = 1/3. Therefore,

⎛ 100 ⎞
E X ( ) = 4 ⎜ ⎟ ⎠ = 1 33.
and ⎝ 300

− ⎞
⎛ ⎞ ⎛ ⎞ ⎛ 300 4
1
2
V X ( ) = 4 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ = 0 88.
3
3
⎝ ⎠ ⎝ ⎠ ⎝ 299 ⎠
For a hypergeometric random variable, E X ( ) is similar to the mean of a binomial random vari-
able. Also, V X ( ) differs from the result for a binomial random variable only by the following term.

Finite Population
Correction Factor The term in the variance of a hypergeometric random variable
N − n
(3-15)
N −1

is called the inite population correction factor.

Sampling with replacement is equivalent to sampling from an ininite set because the proportion
of success remains constant for every trial in the experiment. As mentioned previously, if sampling
were done with replacement, X would be a binomial random variable and its variance would be
(
np 1 − p). Consequently, the inite population correction represents the correction to the binomial

variance that results because the sampling is without replacement from the inite set of size N.

If n is small relative to N, the correction is small and the hypergeometric distribution is similar
to the binomial distribution. In this case, a binomial distribution can effectively approximate the
hypergeometric distribution. A case is illustrated in Fig. 3-13.