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Chapter 5: Discrete Probability Distributions

Chapter 5: Discrete Probability Distributions. Section 5.4: Poisson Distribution. A Poisson distribution is used when the sample size, n, is large and the probability, p, is small and when independent variables occur over time.

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Chapter 5: Discrete Probability Distributions

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  1. Chapter 5: Discrete Probability Distributions Section 5.4: Poisson Distribution

  2. A Poisson distribution is used when the sample size, n, is large and the probability, p, is small and when independent variables occur over time. It can also be used when a density of items is distributedover a given area or volume.

  3. Poisson Probability Formula The probability of X occurrences in an interval of time,volume, area, etc., for a variable where:λ (“lambda”) = mean number of occurrences per unit _e-λ∙ λX X! P(X;λ) = *e is a constant like π and is NOT something we will plug a number in Round to FOUR decimal placesv

  4. Different ways to find λ: λ = 1) The mean (average) number of occurrences λ = np (if the probability or percent is given) 2) The number of occurrences in the past total number of outcomes in the past λ = 3)

  5. EXAMPLE 1: A door-to-door salesman averages 2 sales per day.Find the probability of getting 5 sales in a day if thisapproximates a Poisson distribution. 5 2 X = λ = _e-λ∙ λX X! _e-2 ∙ 25 5! P(X;λ) = P(5; 2) = = = 0.0361

  6. EXAMPLE 2: There is a probability of 0.2% of a defective part in ashipment of computer components. If a shipment of500 components arrives, find the probability of 0 being defective. This approximates a Poisson distrib. 0 np = 500 (0.002) = 1 X = λ = _e-λ∙ λX X! _e-1 ∙ 10 0! P(X;λ) = P(0; 1) = = = 0.368

  7. EXAMPLE 3: 300 apples were checked for worms and 60 worms were found. What is the probability of an apple withone worm in it if this approximates a Poisson distribution? _# of occurrences_total # of outcomes _60300 1 X = λ = = = 0.2 _e-λ∙ λX X! _e-0.2 ∙ 0.21 1! P(X;λ) = P(1; 0.2) = = = 0.1637

  8. EXAMPLE 4: If there are 200 typographical errors randomly distributed in a 500-page manuscript, find the probability that a given page contains exactly threeerrors. Assume a Poisson distribution. _# of occurrences_total # of outcomes 200500 3 X = λ = = = 0.4 _e-λ∙ λX X! _e-0.4 ∙ 0.43 3! P(X;λ) = P(3; 0.4) = = = 0.0072

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