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Discrete Distributions

Section 06. Discrete Distributions. Uniform. x is one specific outcome Parameters: N – number of points. Poisson. x is the number of events in a period of time Parameters: λ – all integers Lambda can be manipulated for periods of time

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Discrete Distributions

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  1. Section 06 Discrete Distributions

  2. Uniform • x is one specific outcome • Parameters: • N – number of points

  3. Poisson • x is the number of events in a period of time • Parameters: • λ – all integers • Lambda can be manipulated for periods of time • Ex: If λ represents the number of customers in an hour, then 2λ is the number in two hours, and λ/2 is the number in half an hour.

  4. Binomial • x is the number of total successes • Parameters: • n – number of trials • p – probability of success on one trial • denoted The Bernoulli distribution is a special case where n=1!

  5. Geometric • x is the number of failed trials until first success • Parameters: • p – probability of success on single trial

  6. Negative binomial • x is number of failures until r-th success occurs • Parameters: • p – probability of success on one trial • r – number of successes The geometric distribution is a special case where r =1!

  7. Multinomial • xi is the number of trials resulting in outcome i • Parameters: • n – number of trials • p1, p2, …, pk – probability of outcome i on one trial

  8. Hypergeometric • x is number of Type I objects in selected subset • Parameters: • M – number of objects • K – number of objects of Type I • n – number of objects selected

  9. Likelihood of distributions • DEFINITELY know • Uniform • Binomial • Poisson • Geometric • TRY TO know • Negative Binomial • Hypergeometric • Maybe not • Multinomial

  10. Sample Exam #96 A tour operator has a bus that can accommodate 20 tourists. The operator knows that tourists may not show up, so he sells 21 tickets. The probability that an individual tourist will not show up is .02, independent of all other tourists. Each ticket costs 50, and is non-refundable if a tourist fails to show up. If a tourist shows up and a seat is not available, the tour operator has to pay 100 (ticket cost + 50 penalty) to the tourist. What is the expected revenue of the tour operator?

  11. Sample Exam #30 An actuary has discovered that policyholders are three times as likely to file two claims as to file four claims. If the number of claims filed has a Poisson distribution, what is the variance of the number of claims filed?

  12. Sample Exam #67 A baseball team has scheduled its opening game for April 1. If it rains on April 1, the game is postponed and will be played on the next day that it does not rain. The team purchases insurance against rain. The policy will pay 1000 for each day, up to 2 days, that the opening game is postponed. The insurance company determines that the number of consecutive days of rain beginning on April 1 is a Poisson random variable with mean .6 What is the standard deviation of the amount the insurance company will have to pay?

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