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The Poisson Distribution

The Poisson Distribution. We can use the Poisson distribution to estimate the probability of arrivals at a car wash in one hour or the number of leaks in 100 miles of pipeline. Bell Labs uses it to model the arrival of phone calls. The Poisson Distribution.

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The Poisson Distribution

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  1. The Poisson Distribution We can use the Poisson distribution to estimate the probability of arrivals at a car wash in one hour or the number of leaks in 100 miles of pipeline. Bell Labs uses it to model the arrival of phone calls.

  2. The Poisson Distribution The Poisson distribution is defined by: • Where f(x) is the probability of x occurrences in an interval • is the expected value or mean value of occurrences within an interval e is the natural logarithm. e = 2.71828

  3. Properties of the Poisson Distribution • The probability of occurrences is the same for any two intervals of equal length. • The occurrence or nonoccurrence of an event in one interval is independent of an occurrence on nonoccurrence of an event in any other interval

  4. Example: Mercy Hospital MERCY • Poisson Probability Function Patients arrive at the emergency room of Mercy Hospital at the average rate of 6 per hour on weekend evenings. What is the probability of 4 arrivals in 30 minutes on a weekend evening?

  5. MERCY Example: Mercy Hospital • Poisson Probability Function  = 6/hour = 3/half-hour, x = 4

  6. MERCY Using Excel to ComputePoisson Probabilities • Formula Worksheet … and so on … and so on

  7. MERCY Using Excel to ComputePoisson Probabilities • Value Worksheet … and so on … and so on

  8. MERCY Poisson Probabilities 0.25 0.20 0.15 Probability 0.10 0.05 0.00 1 2 3 4 5 6 7 8 9 10 0 Number of Arrivals in 30 Minutes Example: Mercy Hospital • Poisson Distribution of Arrivals actually, the sequence continues: 11, 12, …

  9. Problem 31, p. 229 Consider a Poisson probability distribution with an average number of occurrences of two per period. • Write the appropriate Poisson distribution • What is the average number of occurrences in three time periods? • Write the appropriate Poisson function to determine the probability of x occurrences in three time periods. • Compute the probability of two occurrences in one time period. • Compute the probability of six occurrences in three time periods. • Compute the probability of five occurrences in two time periods.

  10. Problem 31, p. 229 (a) (b) (c) (d)

  11. Problem 31, p. 229 (d) (e)

  12. Problem 31, p. 229

  13. The Hypergeometric Distribution This is similar to the binominal distribution except: (1) the trials are NOT independent; and (2) the probability of success (ρ) changes from trial to trial.

  14. Hypergeometric Distribution Let r denote in the population size N labeled a success. N – r is the number of elements in the population labeled failure. The hypergeometric distribution is used to compute the probability that in a random selection of n elements, selected without replacement, we obtain x elements labeled success and N – x elements labeled failure.

  15. Notice that the x successes must be pulled from the r number of successes in the population and the n - x failures must be drawn from a population of N – r failures

  16. Hypergeometric Distribution Where n = the number of trials. N = number of elements in the population r = number of elements in the population labeled a success

  17. Hypergeometric Distribution Number of ways a sample of size n -x failures can be selected from a population of size N -r Number of ways a sample of size x successes can be selected from a population of size r Number of ways a sample of size n can be selected from a population of size N

  18. ZAP ZAP ZAP ZAP Example: Neveready • Hypergeometric Probability Distribution Bob Neveready has removed two dead batteries from a flashlight and inadvertently mingled them with the two good batteries he intended as replacements. The four batteries look identical. Bob now randomly selects two of the four batteries. What is the probability he selects the two good batteries?

  19. Example: Neveready • Hypergeometric Probability Distribution where: x = 2 = number of good batteries selected n = 2 = number of batteries selected N = 4 = number of batteries in total r = 2 = number of good batteries in total

  20. Using Excel to ComputeHypergeometric Probabilities • Formula Worksheet

  21. Using Excel to ComputeHypergeometric Probabilities • Value Worksheet

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