1. THE POISSON RANDOM VARIABLE

2. POISSON DISTRIBUTION ASSUMPTIONS Can be used to model situations where: –No two events occur simultaneously –The probabilities for the number of events that will occur in intervals of the same length are the same –The time until the next event occurs is independent of when the last event occurred The Poisson distribution can be completely characterized by knowing only the average number of events that occur in the interval = μ

3. POISSON DISTRIBUTION Mean, Variance, Standard Deviation If μ is the average number of events that will occur in the time interval, then –Mean = μ –Variance = μ –STD DEV = SQRT(μ)

4. POISSON PROBABILITY DISTRIBUTION Prob (x events in the time interval) = f(x)

5. Adjusting μ Suppose the average number arrivals in an hour is 15, but we wish to know the probability of the number of arrivals in 20 minutes For 60 minutes, μ = 15. For 20 minutes, μ = (20/60)15 = 5For 20 minutes, μ = (20/60)15 = 5

6. EXAMPLE Customers arrive to a bank on the average of once every 4 minutes What is the mean and standard deviation of the number of arrivals in –1 hour?  = 60/4 = 15 so Std. Dev. = SQRT(15) = 3.873 –20 minutes?  = 20/4 = 5 so Std. Dev. = SQRT(5) = 2.236

7. Calculating a Point Probability What is the probability that there will be exactly 9 arrivals in one hour to the bank?

8. POISSON PROBABILITIES USING EXCEL Mean number of events in the interval = μ Point Probability P(EXACTLY x events in the interval) = POISSON(x, μ,FALSE) Cumulative Probability P(x or less events in the interval) =POISSON(x, μ,TRUE)

9. Point and Cumulative Probabilities for the Poisson Distribution with μ = 5 P(X = 2) = =POISSON(2,5,FALSE) P(X <= 6) = =POISSON(6,5,TRUE)

10. hour In a given hour, the average number of arrivals, μ = 15 20 minuteWhat is the probability that in a 20 minute period there will be: –6 or less arrivals –between 5 and 8 arrivals –8 or more arrivals –Exactly 4 arrivals Note for 20 minutes, μ = 5Note for 20 minutes, μ = 5 Example

11. Typical Poisson Probabilities “Less Than or Equal”Prob. =POISSON(6,5,True) “Between” Prob. =POISSON(8,5,TRUE)-POISSON(4,5,TRUE) “Greater Than or Equal to” Prob =1–POISSON(7,5,TRUE) “Equal To” Prob. “Equal To” Prob.=POISSON(4,5,FALSE)

12. REVIEW Poisson Random Variable –Definition –Mean, Variance –Point Probabilities From Formula By Excel –Cumulative Probabilities By Excel