Poisson Distribution Assume discrete events occur randomly throughout a continuous interval according to: 1.the probability of more than one occurrence.

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Presentation transcript:

Poisson Distribution Assume discrete events occur randomly throughout a continuous interval according to: 1.the probability of more than one occurrence in a subinterval is zero; 2.the probability of one occurrence in a subinterval is the same for all subintervals and proportional to the length of the subinterval; 3.the occurrence of an event in one subinterval has no effect on the occurrence or non-occurrence in another non- overlapping subinterval.

Poisson Random Variable λ = Mean (expected) number of occurrences in an interval X = Observed number of occurrences in that interval is said to have Poisson distribution with parameter λ.

Poisson PMF p (x) = x = 0, 1, 2, ….

Factorial m! = m x (m – 1) x …. x 2 x 1 0! = 1 1! = 1 2! = 2 x 1 = 2 3! = 3 x 2 x 1 = 6 4! = 4 x 3 x 2 x 1 = 24

Expectation E(X) = λ

Variance Var (X) = λ

Example 1 The number of cracks requiring repair in a section of motor way follows a Poisson distribution with a mean of 2 cracks/mile. What is the probability that there are: A)no cracks that require repair in 5 miles of motor way ( )? B)at least one crack requires repair inone-half mile of motor way (0.632)? C)exactly three cracks that requirerepair in 2 miles of motor way (0.195)?

Example 2 One-hundred-year records of earthquakes in a region show that there were12 earthquakes of intensity 6 or more. What is the probability that A)such earthquakes will occur in the next 3 years (0.302)? B)no such earthquake will occur in the next 10 years (0.301)?

Example 3 Suppose that in one year the number of industrial accidents follows a Poisson distribution with mean 3. What is the probability that in a given year there will be at least 1 accident (0.95)?

Example 4 The number of surface flaws in plastic panels used in the interior of automobiles has a Poisson distribution with a mean of 0.05/square foot. Assume an automobile interior contains 10 square feet of plastic panel. A)What is the probability that there are no surface flaws in an auto's interior (0.607)? B)If 10 cars are sold to a rental company, what is the probability that none of the 10 cars has any surface flaws (0.007)?