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Geometric Distribution

Geometric Distribution. A probability distribution to determine the probability that success will occur on the nth trial of a binomial experiement. Geometric Distribution. Repeated binomial trials Continue until first success Find probability that first success comes on nth trial

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Geometric Distribution

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  1. Geometric Distribution A probability distribution to determine the probability that success will occur on the nth trial of a binomial experiement

  2. Geometric Distribution • Repeated binomial trials • Continue until first success • Find probability that first success comes on nth trial • Probability of success on each trial = p

  3. Geometric Probability

  4. A sharpshooter normally hits the target 70% of the time. • Find the probability that her first hit is on the second shot. • Find the mean and the standard deviation of this geometric distribution.

  5. A sharpshooter normally hits the target 70% of the time. • Find the probability that her first hit is on the second shot. • P(2)=p(1-p) n-1 = .7(.3)2-1 = 0.21 • Find the mean •  = 1/p = 1/.7 1.43 • Find the standard deviation

  6. Poisson Distribution A probability distribution where the number of trials gets larger and larger while the probability of success gets smaller and smaller

  7. Poisson Distribution • Two outcomes : success and failure • Outcomes must be independent • Compute probability of r occurrences in a given time, space, volume or other interval •  (Greek letter lambda) represents mean number of successes over time, space, area

  8. Poisson Distribution

  9. The mean number of people arriving per hour at a shopping center is 18. • Find the probability that the number of customers arriving in an hour is 20. r = 20  = 18 Find P(20) e = 2.7183

  10. The mean number of people arriving per hour at a shopping center is 18.

  11. Poisson Probability Distribution Table Table 4 in Appendix II provides the probability of a specified value of r for selected values of .

  12. Using the Poisson Table •  = 18, find P(20):

  13. Poisson Approximation to the Binomial Distribution The Poisson distribution can be used as a probability distribution for “rare” events.

  14. “Rare” Event The number of trials (n) is large and the probability of success (p) is small.

  15. If n  100 and np < 10, then • The distribution of r (the number of successes) has a binomial distribution which is approximated by a Poisson distribution . • The mean  = np.

  16. Use the Poisson distribution to approximate the binomial distribution: • n = 240 • p = 0.02 • Find the probability of at most 3 successes.

  17. Using the Poisson to approximate the binomial distribution for n = 240 and p = 0.02 Note that n  100 and np = 4.8 < 10, so the Poisson distribution can be used to approximate the binomial distribution. Find the probability of at most 3 successes: Since  = np = 4.8, we use Table 4 to find P( r  3) =.0082 + .0395 + . 0948 + . 1517 = .2942

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