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Unformatted text preview: The hypergeometric probability distribution Suppose a population of N units has a of the N
possessing a certain characteristic, N—a not
possessing the characteristic (a 2 outcome setup).
A sample of n units is to be selected from the
population without replacement. (There are a “S
outcomes” and N—a “F outcomes in the population). The random variable of interest is X, the number of
units in the sample possessing the characteristic.
X is called a hypergeometric random variable and
its f(X) is denoted h(x;n,a,N), a three parameter
probability distribution. h0gn¢gN)=—El::£—gxzﬁhh2wunmﬁan)(ummﬂyn) (if) Notel: In acceptance sampling(lot sampling,
inspection sampling), N is the lot size, a is the
number of defectives in the lot, n is the sample
size, and f(x) is the probability of exactly X
defectives in the sample. 114a Note2: From Vandermonde’s theorem, 2 :
vox n—x n ihﬁhngyN):l.
x20 '  :b :/N
Note3: hHl lﬁxﬂhabD (Xnﬂ) a > ; ROT nSlQHO.
N—ém There is an important connection between the
binomial model and sampling. Suppose we are
planning to sample n items, and observe whether the
selected item has a certain characteristic or not.
Before selecting an item think of the population
from which the sample is being selected as being
thoroughly mixed with respect to the
characteristic. Interest is in the random variable, X, the number of items in the sample with the
characteristic. Under each of the following
situations X is a binomial RV: 1) The population is finite_§£§1each sampled
item is replaced back into the population after its
possession of the characteristic or not is noted.
The population is thoroughly mixed after each item
is returned to the population. Here p is the
proportion in the population with the
characteristic. 2) The population is infinitewandeach sampled
item is replaced back into the population after its
possession of the characteristic or not is noted.
The population is thoroughly mixed after each item
is returned to the population. Here p is the
probability an item possesses the characteristic. 3) Same as 2) except the item is not replaced. In these three situations the binomial model
applies for X. There is a fourth case, of course, and it's a
practical case: The population is finite and items
are not replaced. The special distribution that
models this situation is called the hypergeometric
distribution, which leads to our second special
discrete random variable. (4.3 in the book). ...
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This note was uploaded on 10/26/2011 for the course INDE 2333 taught by Professor Staff during the Spring '08 term at University of Houston.
 Spring '08
 Staff

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