STAT 400 Lecture AL1 Examples for 2.6 Spring 2015 Dalpiaz Poisson Distribution: X = the number of occurrences of a particular event in an interval of time or space. P( X = x ) = !λλ xx e −⋅, x = 0, 1, 2, 3, … . E( X ) = λ, Var( X ) = λ. Table III ( pp. 580 – 582 ) gives P( X ≤ x ) EXCEL: = POISSON( x , λ , 0 ) gives P( X = x ) = POISSON( x , λ , 1 ) gives P( X ≤ x ) 1. Traffic accidents at a particular intersection follow Poisson distribution with an average rate of 1.4 per week. a) What is the probability that the next week is accident-free? b) What is the probability that there will be exactly 3 accidents next week?c) What is the probability that there will be at most 2 accidents next week? d) What is the probability that there will be at least 2 accidents during the next two weeks? e) What is the probability that there will be exactly 5 accidents during the next four weeks? f) What is the probability that there will be exactly 2 accidents tomorrow?g) What is the probability that the next accident will not occur for three days? h) What is the probability that there will be exactly three accident-free weeks during the next eight weeks? i) What is the probability that there will be exactly five accident-free days during the next week?When n is large ( n ≥ 20 ) and p is small ( p ≤ 0.05 ) and n ⋅ p ≤ 5, Binomial probabilities can be approximated by Poisson probabilities. For this, set λ = n ⋅ p. 2. Suppose the defective rate at a particular factory is 1%. Suppose 50 parts were selected from the daily output of parts. Let X denote the number of defective parts in the sample. a) Find the probability that the sample contains exactly 2 defective parts. b) Use Poisson approximation to find the probability that the sample contains exactly 2 defective parts. c) Find the probability that the sample contains at most 1 defective part. d) Use Poisson approximation to find the probability that the sample contains at most 1defective
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