STAT 400 Lecture AL1 Spring 2015 Dalpiaz Examples for 2 6 Poisson Distribution X the number of occurrences of a particular event in an interval of time or space P X x x e x E X Var X Table III pp 580 582 1 x 0 1 2 3 P X x gives 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 1 week 1 4 b P X 0 0 0 2466 What is the probability that there will be exactly 3 accidents next week 1 week 1 4 c 1 4 0 e 1 4 P X 3 1 4 3 e 1 4 0 1128 3 What is the probability that there will be at most 2 accidents next week 1 week 1 4 P X 2 P X 0 P X 1 P X 2 1 4 0 e 1 4 0 1 4 1 e 1 4 1 4 2 e 1 4 1 0 2466 0 3452 0 2417 0 8335 2 d What is the probability that there will be at least 2 accidents during the next two weeks 2 weeks 2 8 2 8 0 e 2 8 2 8 1 e 2 8 P X 2 1 P X 0 P X 1 1 0 1 1 0 0608 0 1703 0 7689 e What is the probability that there will be exactly 5 accidents during the next four weeks 4 weeks 5 6 f 0 1697 P X 2 0 2 2 e 0 2 2 0 0164 What is the probability that the next accident will not occur for three days 3 days 0 6 h 5 What is the probability that there will be exactly 2 accidents tomorrow 1 day 0 2 g P X 5 5 6 5 e 5 6 P X 0 0 6 0 e 0 6 0 0 5488 What is the probability that there will be exactly three accident free weeks during the next eight weeks Success an accident free week 1 week 1 4 p P Success P X 0 1 4 0 e 1 4 0 0 2466 P exactly 3 accident free weeks in 8 weeks 8 C 3 0 2466 3 0 7534 5 0 20384 Binomial distribution i What is the probability that there will be exactly five accident free days during the next week Success an accident free day 1 day 0 2 p P Success P X 0 0 2 0 e 0 2 0 81873 0 P exactly 5 accident free days in 7 days 7 C 5 0 81873 5 0 18127 2 0 25385 Binomial distribution 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 2 n p 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 50 P X 2 0 01 2 0 99 48 0 075618 2 b Use Poisson approximation to find the probability that the sample contains exactly 2 defective parts n p 0 5 P X 2 0 5 2 e 0 5 0 075816 2 c Find the probability that the sample contains at most 1 defective part P X 1 P X 0 P X 1 50 50 0 01 0 0 99 50 0 01 1 0 99 49 0 910565 0 1 d Use Poisson approximation to find the probability that the sample contains at most 1defective part P X 1 P X 0 P X 1 0 5 0 e 0 5 0 5 1 e 0 5 0 909796 0 1
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