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QZ 3-A - bus to arrive P T 5> 0.5 = P X 0.5 ≤ 4 = P...

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STAT 400 Fall 2011 Version A Name ANSWERS . Quiz 3 (10 points) Be sure to show all your work; your partial credit might depend on it. No credit will be given without supporting work. 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 15 minutes. X t = number of buses in t hours. Poisson ( λ t ) T k = arrival time of the k th bus. Gamma, α = k . one bus per 15 minutes λ = 4. a) (4) Find the probability that Alex would have to wait longer than 30 minutes for the fifth
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Unformatted text preview: bus to arrive. P ( T 5 > 0.5 ) = P ( X 0.5 ≤ 4 ) = P ( Poisson ( 2 ) ≤ 4 ) = 0.947 . OR P ( T 5 > 0.5 ) = ( ) ∫ ∞ Γ--5 . 4 1 5 5 5 4 dt t t e = ∫ ∞-5 . 4 4 5 4 4 ! dt t t e = … b) (6) Find the probability that the fifth bus arrives during the second hour. P ( 1 < T 5 < 2 ) = P ( T 5 > 1 ) – P ( T 5 > 2 ) = P ( X 1 ≤ 4 ) – P ( X 2 ≤ 4 ) = P ( Poisson ( 4 ) ≤ 4 ) – P ( Poisson ( 8 ) ≤ 4 ) = 0.629 – 0.100 = 0.529 . OR P ( 1 < T 5 < 2 ) = ( ) ∫--Γ 2 1 4 1 5 5 5 4 dt t t e = ∫-2 1 4 4 5 4 4 ! dt t t e = …...
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