STAT 400 Lecture AL1 Spring 2015 Dalpiaz Answers for 3 2 Part 2 Gamma Distribution f x f x x 1 e x x 1 e x 0 x 1 0 x E X E X Var X 2 Var X 2 If T has a Gamma 1 distribution where is an integer then F T t P T t P X t P T t P X t 1 t where X t has a Poisson t distribution 1 Alex is told that he needs to take bus 5 to the train station He misunderstands the directions and decides to wait for the fifth bus Suppose that the buses arrive to the bus stop according to Poisson process with the average rate of one bus per 20 minutes X t number of buses in t hours Poisson t T k arrival time of the k th bus Gamma k one bus per 20 minutes a 3 Find the probability that Alex would have to wait longer than 1 hour for the fifth bus to arrive P T 5 1 P X 1 4 P Poisson 3 4 0 815 OR P T5 1 1 35 t 5 1 e 3 t dt 5 5 3 4 3t 4 t e dt 1 b Find the probability that the fifth bus arrives during the second hour P 1 T5 2 P T5 1 P T5 2 P X1 4 P X2 4 P Poisson 3 4 P Poisson 6 4 0 815 0 285 0 530 OR 2 P 1 T5 2 1 c 2 35 t 5 1 e 3 t dt 5 1 35 4 t 4 e 3 t dt Find the probability that the fifth bus arrives during the third hour P 2 T5 3 P T5 2 P T5 3 P X2 4 P X3 4 P Poisson 6 4 P Poisson 9 4 0 285 0 055 0 230 OR 3 P 2 T5 3 2 1 3 35 t 5 1 e 3 t dt 5 3 2 3 5 4 3t t e dt 4 Traffic accidents at a particular intersection follow Poisson distribution with an average rate of 1 4 per week 0 2 per day a What is the probability that the next accident will not occur for three days Exponential 0 2 P T 3 e 0 2 3 e 0 6 0 5488 OR Poisson 3 days t 0 2 3 0 6 P X 0 0 6 0 e 0 6 0 0 5488 b What is the probability that the next accident will occur during the third day That is the time until the next accident is more than two days but less than three days Exponential 0 2 P 2 T 3 P T 2 P T 3 e 0 2 2 e 0 2 3 e 0 4 e 0 6 0 1215 OR Poisson 1 day t 0 2 1 0 2 1st day No accident 0 81873 c 2nd day No accident 0 81873 P X 0 0 2 0 e 0 2 0 0 81873 3rd day Accident s 0 18127 0 1215 What is the probability that the second accident will occur before the end of the third day P the second accident will occur before the end of the third day P at least 2 accidents will occur in three days 1 0 6 0 e 0 6 0 6 1 e 0 6 0 1219 0 1 OR 3 1 3 1 0 2 2 1 0 2 x x e 0 2 x 2 e 0 2 x 0 1219 dx 0 2 2 1 x e 0 2 0 2 0 0 d What is the probability that the fourth accident will occur after the end of the seventh day Gamma 4 0 2 P T 4 7 days 7 0 2 4 3 0 2 x x e dx 3 OR Gamma 4 1 4 P T 4 1 week 1 1 4 4 3 1 4 x x e dx 3 OR P the fourth accident will occur after the end of the seventh day P at most 3 accidents will occur in seven days e 1 4 0 e 1 4 1 4 1 e 1 4 1 4 2 e 1 4 1 4 3 e 1 4 0 9463 0 1 2 3 What is the probability that the third accident will occur during the fourth day P the third accident will occur during the fourth day P the third accident will occur after the end of the third day P the third accident will occur after the end of the fourth day P at most two accidents will occur in three days P at most two accidents will occur in four days 0 6 0 e 0 6 0 6 1 e 0 6 0 6 2 e 0 6 0 1 2 0 8 0 e 0 8 0 8 1 e 0 8 0 8 2 e 0 8 0 0243 0 1 2 OR 4 P 3 T3 4 0 2 3 2 0 2 x dx 0 0243 2 x e 3 1 7 During a radio trivia contest the radio station receives phone calls according to Poisson process with the average rate of five calls per minute X t number of phone calls in t minutes Poisson t T k time of the k th phone call Gamma k five calls per minute a 5 Find the probability that we would have to wait less than two minutes for the ninth phone call P T 9 2 P X 2 9 1 P X 2 8 1 P Poisson 10 8 1 0 333 0 667 OR 2 P T9 2 0 b 59 t 9 1 e 5 t dt 9 2 0 5 9 8 5 t t e dt 8 Find the probability that the ninth phone call would arrive during the third minute P 2 T9 3 P T9 2 P T9 3 P X2 8 P X3 8 P Poisson 10 8 P Poisson 15 8 0 333 0 037 0 296 OR 3 P 2 T9 3 2 59 t 9 1 e 5 t dt 9 3 2 5 9 8 5t t e dt 8 2 a Mistakes that David makes in class occur according to Poisson process with the average rate of one mistake per 10 minutes Find the probability that the third mistake David makes occurs during the last 15 minutes of a 50 minute class Notations X t number of mistakes in t minutes T k time of the k th mistake 1 min 10 1 0 10 10 P 35 T 3 50 P T 3 35 P T 3 50 P X 35 2 P X 50 2 5 min 2 1 0 50 2 P 7 T 3 10 P T 3 7 P T 3 10 P X 7 2 P X 10 2 10 min 1 1 P 3 5 T 3 5 P T 3 3 5 P …
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