STAT 400 1 Discussion 7 Spring 2015 Suppose that number of accidents at the Monstropolis power plant follows the Poisson process with the average rate of 0 40 accidents per day Assume all days are independent of each other d Find the probability that the first accident would occur during the fourth day e Find the probability that the second accident would occur during the fifth day f Find the probability that the third accident would occur during the second week 2 As Alex is leaving for college his parents give him a car but warn him that they would take the car away if Alex gets 6 speeding tickets Suppose that Alex receives speeding tickets according to Poisson process with the average rate of one ticket per six months a Find the probability that it would take Alex longer than two years to get his sixth speeding ticket b Find the probability that it would take Alex less than four years to get his sixth speeding ticket c Find the probability that Alex would get his sixth speeding ticket during the fourth year d Find the probability that Alex would get his sixth speeding ticket during the third year 3 6 Let the joint probability density function for X Y be f x y 12 x y 3 5 0 y 1 y x 2 zero otherwise Do NOT use a computer You may only use and on a calculator Show all work Example 2 1 1 12 6 2 3 x 2 24 3 6 5 3 5 x y dx dy 5 x y x y dy 5 y 5 y dy 0 y 0 0 1 1 6 1 6 y 1 y4 y6 1 y 0 5 5 5 5 3 f x y is a valid joint p d f a Sketch the support of X Y b Find the marginal probability density function of X f X x c Find the marginal probability density function of Y f Y y 4 Find the probability P X 2 Y 5 Find the probability P X Y 2 6 Find the probability P X Y 1 That is sketch 0 y 1 y x 2 1 Suppose that number of accidents at the Monstropolis power plant follows the Poisson process with the average rate of 0 40 accidents per day Assume all days are independent of each other d Find the probability that the first accident would occur during the fourth day T 1 has Exponential distribution with 0 40 or 1 0 4 2 5 4 P 3 T1 4 0 4 e 0 4 t dt e 1 2 e 1 6 0 0993 3 OR P 3 T1 4 P T1 3 P T1 4 P X3 0 P X4 0 P Poisson 1 2 0 P Poisson 1 6 0 0 301 0 202 0 099 OR no accidents during P the first three days AND at least one accident during the fourth day P X 3 0 P X 1 1 0 301 1 0 670 0 0993 OR Day 1 Day 2 Day 3 Day 4 no accident no accident no accident accident s 0 670 0 670 0 670 0 330 0 0993 e Find the probability that the second accident would occur during the fifth day T 2 has Gamma distribution with 2 and 0 40 or 1 0 4 2 5 P 4 T2 5 P T2 4 P T2 5 P X4 1 P X5 1 P Poisson 1 6 1 P Poisson 2 0 1 0 525 0 406 0 119 OR 5 P 4 T2 5 4 0 4 2 2 1 0 4 t t e dt 2 5 0 16 t e 0 4 t dt 0 118925 4 OR one accident during P the first four days AND at least one accident during the fifth day no accidents during P the first four days AND at least two accidents during the fifth day P X4 1 P X1 1 P X4 0 P X1 2 0 525 0 202 1 0 670 0 202 1 0 938 0 119 f Find the probability that the third accident would occur during the second week T 3 has Gamma distribution with 3 and 0 40 or 1 0 4 2 5 P 7 T 3 14 P T 3 7 P T 3 14 P X 7 2 P X 14 2 P Poisson 2 8 2 P Poisson 5 6 2 0 469 0 082 0 387 OR 14 P 7 T 3 14 7 14 0 4 3 3 1 0 4 t t e dt 3 7 0 4 3 2 t 2 e 0 4 t dt 0 387065 OR T 3 has Gamma distribution with 3 and 2 8 1 week P 1 T3 2 P T3 1 P T3 2 P X1 2 P X2 2 P Poisson 2 8 2 P Poisson 5 6 2 0 469 0 082 0 387 OR 2 P 1 T3 2 1 2 8 3 3 1 2 8 t t e dt 3 2 1 2 8 3 2 t 2 e 2 8 t dt 0 387065 OR two accidents during P the first week AND during the second week at least one accident one accident during P the first week AND at least two accidents during the second week no accidents during P the first week AND at least three accidents during the second week 2 As Alex is leaving for college his parents give him a car but warn him that they would take the car away if Alex gets 6 speeding tickets Suppose that Alex receives speeding tickets according to Poisson process with the average rate of one ticket per six months X t number of speeding tickets in t years Poisson t T k time of the k th speeding ticket Gamma k one ticket per six months 2 If T has a Gamma 1 distribution where is an integer then P T t P X t and P T t P X t 1 where X t has a Poisson t distribution a Find the probability that it would take Alex longer than two years to get his sixth speeding ticket P T 6 2 P X 2 5 P Poisson 4 5 0 785 OR P T6 2 2 b 26 t 6 1 e 2 t dt 6 2 2 6 5 2 t t e dt 5 Find the probability that it would take Alex less than four years to get his sixth speeding ticket P T6 4 P X4 6 P X4 6 1 P X4 5 1 P Poisson 8 5 1 0 191 0 809 OR 4 P T6 4 0 c 26 t 6 1 e 2 t dt 6 4 0 26 5 t 5 e 2 t dt Find the probability that Alex would get his sixth speeding ticket during the fourth year P 3 T6 4 P T6 3 P T6 4 P X3 5 P X4 5 P Poisson 6 5 …
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