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Discrete Probability Distributions (The Poisson and Exponential Distributions)

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

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Discrete Probability Distributions (The Poisson and Exponential Distributions)

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

  2. The Poisson Distribution-I • The Poisson distribution is the samplingdistribution.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: • 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). • The probability of the event occurring is the same for each interval. • The number of occurrences in one interval is independent of the number of occurrences in other intervals.

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

  4. 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. 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. 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. 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. 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. 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. The Poisson Distribution Graphically =2 =5 =8

  11. Summary of Discrete Distributions

  12. 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. 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.

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