ioe265f11-Lec08 - Lec 08 - Poisson Distribution Topics...

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Lec 08 - Poisson Distribution IOE 265 F11 1 1 Poisson Probability Distribution 2 Topics I. Poisson Failure Processes II. Poisson Distribution III. Poisson Time Process IV. Poisson Industry Applications
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Lec 08 - Poisson Distribution IOE 265 F11 2 3 Multiple failures possible for one unit Separate failures are independent Probability of failure increases with exposure (e.g. time, area, volume) Examples: Part with many features # flaws in rolls of textiles (# snags in carpet) # paint chips on an automotive body # calls to a telephone exchange # bad sectors in a disk space I. Poisson Failure Processes 4 Poisson Example Suppose flaws or defects occur at random along the length of a thin copper wire. X = Random variable representing the number of flaws across L millimeters of wire. Goal ~ predict the probability of a flaw over some sub-interval x x x x Approach: Group overall L mm into subintervals of size n such that the prob. of a single defect in the subinterval is constant ( p = constant). E(X) = p*n = rate per unit of length, area, time, …)
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Lec 08 - Poisson Distribution IOE 265 F11 3 5 Developing Poisson Distribution Poisson distribution derives from the binominal pmf under the following conditions. As n infinity and p 0 (large # of subintervals with a small probability of occurrence), np approaches a value > 0. Then: b(x; n, p) p(x: ) General rules for using poisson over binomial: For any binomial experiment with large n and small p, we may approximate using poisson. In general, if n (# of subintervals) >= 100 p <= 0.01 and np <= 20. (and np ~ > 0) 6 II. Poisson Distribution = rate per unit time or unit area x = rv (x = 0,1,2, … ) where x is the number of defects * Law of small numbers: events with low frequency in a large population follow a Poisson Distribution 0 !
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This note was uploaded on 01/05/2012 for the course IOE 265 taught by Professor Jin during the Fall '07 term at University of Michigan.

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ioe265f11-Lec08 - Lec 08 - Poisson Distribution Topics...

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