Statistics 312 13 Poisson Distribution 1 A Poisson random variable is a count of the number of successes nonconformities defects and the like per sampled continuum or area of opportunity a continuous unit or interval of time volume or such area in which more than one occurrence of an event may be noted A Poisson process is said to exist if we can observe discrete events in an area of opportunity in such a manner that if we subdivide the area of opportunity into very small equal sized subareas of opportunity 1 The probability of observing exactly one success in any subarea of opportunity is stable 2 The probability of observing two or more successes in any subarea of opportunity is zero 3 The occurrence of a success in any one subarea of opportunity is statistically independent of that in any other The Poisson probability distribution is characterized by only one parameter the average number of successes per area of opportunity P X x e x X 0 1 2 x Example Suppose the number of flaws in a 100 foot roll of paper is a Poisson random variable with 10 Then the probability that there are eight flaws in a 100 foot roll is P X 8 10 e x x 8 10 2 71828 10 100 000 000 e 10 1126 40 320 8 The probability of seven flaws in a 50 foot roll is P X 7 5 e x x 7 5 2 71828 5 78 125 e 5 1044 5 040 7 Statistics 312 13 Poisson Distribution The Mean and the Standard Deviation x x Example 100 foot roll x 10 10 3 16 Read pp 167 171 Prob 4 36 4 37 4 40 2
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