1. 1 Discrete Probability Distributions Hypergeometric & Poisson Distributions Dr. Jerrell T. Stracener, SAE Fellow Leadership in Engineering EMIS 7370/5370 STAT 5340 : PROBABILITY AND STATISTICS FOR SCIENTISTS AND ENGINEERS Systems Engineering Program Department of Engineering Management, Information and Systems

2. 2 Hypergeometric Distribution

3. 3 Conditions: Population size = N K members are “successes” N - K members are “failures” Sample size = n Obtained without replacement = the number of successes in n trials Hypergeometric Distribution

4. 4 where x = 0, 1,..., n if n  k k if n > k Probability Mass Function

5. 5 Mean or Expected Value of Standard Deviation of Mean & Standard Deviation

6. 6 Example: Five individuals from an animal population thought to be near extinction in a certain region have been caught, tagged and released to mix into the population. After they have had an opportunity to mix in, a random sample of 10 of these animals is selected. Let = the number of tagged animals in the second sample. If there are actually 25 animals of this type in the region, Find: a) b) Hypergeometric Distribution

7. 7 Example Solution: The parameter values are n = 10 K = 5 (5 tagged animals in the population) N = 25 X = 0, 1, 2, 3, 4, 5 Hypergeometric Distribution

8. 8 Example Solution: a. b. Hypergeometric Distribution

9. 9 Similar to the Binomial Dist in Excel: –Click the Insert button on the menu bar (at the top of the Excel page) Go to the function option Choose Statistical from the Function Category window (a list of all available statistical functions will appear in the Function Name window) Choose the HYPGEOMDIST function Type in parameters: –Sample_s => x –Number_sample => n –Population_s => k –Number_pop => N Hypergeometric Distribution

10. 10 Poisson Distribution

11. 11 1. The number of outcomes occurring in one time interval or specified region is independent of the number that occurs in any other disjoint time interval or region of space. In this way we say that the Poisson process has no memory. 2. The probability that a single outcome will occur during a very short time interval or in a small region is proportional to the length of the time interval or the size of the region and does not depend on the number of outcomes occurring outside this time interval or region. Properties

12. 12 3. The probability that more than one outcome will occur in such a short time interval or fall in such a small region is negligible. Remark: The Poisson distribution is used to describe a number of processes, including the distribution of telephone calls going through a switchboard system, the demand (needs) of patients for service at a health institution, the arrivals of trucks and cars at a tollbooth, the number of accidents at an intersection, etc. Properties

13. 13 Definition - If is the number of outcomes occurring during a Poisson experiment, then has a Poisson distribution with probability mass function where  = t and is the average number of outcomes per unit time, t is the time interval and e = 2.71828... Poisson Distribution

14. 14 Mean or Expected Value of Variance and Standard Deviation of Poisson Distribution

15. 15 Example: When a company tests new tires by driving them over difficult terrain, they find that flat tires externally caused occur on the average of once every 2000 miles. What is the probability that in a given 500 mile test no more than one flat will occur? Poisson Distribution

16. 16 Example Solution: Here the variable t is distance, and the random variable of interest is = number of flats in 500 miles Since E(X) is proportional to the time interval involved in the definition of X, and since the average is given as one flat is 2000 miles, we have Poisson Distribution

17. 17 Example Solution: The values assigned to and t depend on the unit of distance adopted. If we take one mile as the unit, then t = 500, = 0.0005, and  = t = 1/4. If we take 1000 miles as the unit, then t = 1/2, = 1/2, and again t = 1/4, and so on. The important thing is that t = 1/4, no matter what unit is chosen. Poisson Distribution

18. 18 Example Solution Poisson Distribution Is this acceptable?