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## Engineering Statistics ECIV 2305

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**Engineering Statistics ECIV 2305**Chapter 3 DISCRETE PROBABILITY DISTRIBUTIONS • 3.1 The Binomial Distribution • 3.2 The Geometric Distribution • 3.3 The Hypergeometric Distribution • 3.4 The Poisson Distribution 3.3 The Hypergeometric Distribution**Engineering Statistics ECIV 2305**Section 3.3 The Hypergeometric Distribution 3.3 The Hypergeometric Distribution**Normal kind item**Special kind item • Consider the following example: N = total number of items =15 , r = Number of special kind items = 4 • If one item is chosen at random, the probability that it is of the special kind is p = r/N = 4/15 • If “n” items (OR “n” trials) are chosen at random with replacement, then the number of defective items chosen is a binomial RV , i.e. X~B(n, r/N) Examples of special kind items: - some black balls within white balls - some defective items within satisfactory items 3.3 The Hypergeometric Distribution**N = total number of items =15 ,**r = Number of special kind items = 4 • What if “n” items were selected without replacement? • The binomial distribution will not work. Instead, the number of defective items chosen will have a distribution called the hypergeometric distribution. 3.3 The Hypergeometric Distribution**The pmf of the hypergeometric distribution is:**For example: If n = 7, r = 4 , and N = 15 ►► 0 ≤ x ≤ 4 If n = 7, r = 4 , and N = 10 ►► 1 ≤ x ≤ 4 With an expectation of : and a variance of : 3.3 The Hypergeometric Distribution**Example**• Recall the example of the 500 chips (in sections 1.5 &1.7); nine of which are defective. Three chips are drawn at random without replacement. What is the probability that exactly one defective chip is drawn? N = 500 , r = 9 (this is the number of special kind items) n = 3 (three trials were performed) 3.3 The Hypergeometric Distribution**When the total number of items “N” (population size) is**much larger than the sample size n, then sampling without replacement is very similar to sampling with replacement. This is because the reduction of the total population “N” by the items selected does not appreciably change the selection probabilities of additional items. ►►So, it becomes more convenient to use the binomial distribution; X~B(n, r/N) You can also use the hypergeometric, but it will be just more work. 3.3 The Hypergeometric Distribution**Example Milk Container Contents (page 182)**• A box contains 16 milk containers. Six of them are underweight. • Five containers are chosen at random for inspection. • What is the distribution of the number of underweight milk containers in the sample chosen by the inspector? 3.3 The Hypergeometric Distribution**… continute Milk Container Contents example…**3.3 The Hypergeometric Distribution**Example Fish Tagging and Recapture (page 182)**• A small lake contains 50 fish. Ten of them are tagged. • A fisherman catches eight fish. • What is the distribution of the number of tagged fish caught by the fisherman? 3.3 The Hypergeometric Distribution