1. 381 Discrete Probability Distributions (The Poisson and Exponential Distributions) QSCI 381 – Lecture 15 (Larson and Farber, Sect 4.3)

2. 381 The Poisson Distribution-I The Poisson distribution is the sampling distribution. It is used to determine the probability that a specific number of occurrences takes place within a given unit of time or space. A random variable X is Poisson distributed if: 1. The experiment consists of counting the number of times, x, an event occurs in a given interval (the interval can relate to time, area or volume). 2. The probability of the event occurring is the same for each interval. 3. The number of occurrences in one interval is independent of the number of occurrences in other intervals.

3. 381 The Poisson Distribution-II Notation: Note that x can be any non-negative integer.

4. 381 An Example of the Poisson Distribution-I The mean catch of dolphins per month in the Eastern Tropical Pacific is 3. What is the probability that the catch will be 4 in a given month? What is the probability that the catch will be 2 or less in a given month.

5. 381 An Example of the Poisson Distribution-II The mean catch of dolphins per month in the Eastern Tropical Pacific is 3. What is the probability that the catch will be 4 in a given month?

6. 381 An Example of the Poisson Distribution-III The mean catch of dolphins per month in the Eastern Tropical Pacific is 3. What is the probability that the catch will be 4 in a given month? What is the probability that the catch will be 2 or fewer in a given month? P[X  2]=P[X=0]+P[X=1]+P[X=2]=

7. 381 Modeling Typhoons-I Assuming that typhoons are Poisson distributed, find the mean number of typhoons per year and the probability that there will be three or more in a single year.

8. 381 Modeling Typhoons-II Mean number of typhoons: Probability of x typhoons a year: The probability of the number of typhoons being at least 3 is 0.015. Later we will examine methods to determine whether it is reasonable to assume that these data are Poisson distributed.

9. 381 The Poisson Distribution (Caveats) The Poisson distribution would seem to be a good distribution to assume for catch data in fisheries. However, fish school. What does this imply regarding the assumptions of the Poisson distribution? Be clear when using the Poisson distribution what the interval is.

10. 381 The Poisson Distribution Graphically  =2  =5  =8

11. 381 Summary of Discrete Distributions DistributionNumber of trialsRandom Variable Parameters BinomialPre-specifiedNumber of successes n, p Negative Binomial Repeated until a given number of successes Number of failures r, p GeometricRepeated until one success occurs Number of failures p PoissonN/ANumber of occurrences 

12. 381 Discrete Probability Distributions and EXCEL-I BINOMDIST(x,n,p,cum) Evaluates the probability of obtaining x successes from n trials when the probability of a success is p. Set “cum” to True to calculate the cumulative probability that the number of successes is x or fewer. NEGBINOMDIST(x,s,p) Evaluates the probability of obtaining the s th success after x failures when the probability of a success is p. Set s=1 for the Geometric distribution.

13. 381 Discrete Probability Distributions and EXCEL-II POISSON(x, ,cumu) Evaluates the probability of obtaining x occurrences in an interval when the mean number is . Set “cum” to True to calculate the cumulative probability that the number of occurrences is x or less.