Notes_for_19th_feb - The hypergeometric probability...

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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)=—E-l::£—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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  • Spring '08
  • Staff
  • Probability distribution, Binomial distribution, Discrete probability distribution, Hypergeometric Distribution

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