Poisson Distribution (Not in Reif’s book)

1. Poisson Distribution(Not in Reif’s book)

2. The Poisson Probability Distribution Simeon Denis Poisson • "Researches on the probability of criminal civil verdicts" 1837  • Looked at the form of the binomial distribution When the Number of Trials is Large.  • He derived the cumulative Poisson distribution as the Limiting case of the Binomial when the probabilityp of success tends to zero.

3. The Poisson Probability Distribution Simeon Denis Poisson • "Researches on the probability of criminal civil verdicts" 1837  • Looked at the form of the binomial distribution When the Number of Trials is Large.  • He derived the cumulative Poisson distribution as the Limiting case of the Binomial when the probabilityp of success tends to zero. Simeon Denis “Fish”!

4. The Poisson Distribution • Poisson Distribution: An approximation to the binomial distribution for the SPECIAL CASEwhen the average number (mean µ) of successes is very much smaller than the possible numbern. i.e.µ << n because p << 1. • This distribution is important for the study of such phenomena as radioactive decay. This distribution is NOTnecessarily symmetric! Data are usually bounded on one side & not the other. An advantage of this distribution is that σ2 = μ µ = 10.0 σ = 3.16 µ = 1.67 σ = 1.29

5. The Poisson Distribution Models Counts. • If events happen at a constant rate over time, the Poisson Distributiongives The Probability of X Number of Events Occurring in a time T. • This distribution tells us the Probability of All Possible Numbers of Counts, from 0 to Infinity. • If X= # of counts per second, then the Poisson probability that X = k(a particular count) is: l λ ≡ the average number of counts per second.

6. Mean and Variance for thePoisson Distribution • It’s easy to show that for this distribution, The Mean is: • Also, it’s easy to show that The Variance is: • l • L •  TheStandard Deviation is:  For a Poisson Distribution, the variance and mean are equal!

7. More on thePoisson Distribution Terminology: A “Poisson Process” • The Poisson parameter can be given as the mean number of events that occur in a defined time period OR, equivalently,  can be given as a rate, such as  = 2 events per month.  must often be multiplied by a time t in a physical process (called a “Poisson Process” ) μ = t σ = t

8. Example 1. If calls to your cell phone are a Poisson process with a constant rate  = 2 calls per hour, what is the probability that, if you forget to turn your phone off in a 1.5 hour class, your phone rings during that time? Answer: If X = # calls in 1.5 hours, we want P(X ≥ 1) = 1 – P(X = 0) p P(X ≥ 1) = 1 – .05 = 95% chance

9. Example Continued 2. How many phone calls do you expect to get during the class? <X> = t = 2(1.5) = 3 Editorial comment: The students & the instructor in the class will not be very happy with you!!

10. Conditions Required for the Poisson Distributionto hold: l 1. The rate is a constant, independent of time. 2. Two events never occur at exactly the same time. 3. Each event is independent. That is, the occurrence of one event does not make the next event more or less likely to happen. 10

11. Example • A production line produces 600 parts per hour with an average of 5 defective parts an hour. If you test every part that comes off the line in 15 minutes,what is the probability of finding no defective parts (and incorrectly concluding that your process is perfect)? λ = (5 defects/hour)*(0.25 hour) = λ = 1.25  p(x) = (xe-)/(x!) x = given number of defects P(x = 0) = (1.25)0e-1.25)/(0!) = e-1.25 = 0.287 = 28.7%

12. Comparison of the Binomial & Poisson Distributions with Mean μ = 1 N N Clearly, there is not much difference between them! For N Large & m Fixed: Binomial  Poisson

13. Poisson Distribution: As λ (Average # Counts) gets large, this also approaches a Gaussian λ = 15 λ = 5 l λ = 35 λ = 25