HW7_08_soln

HW7_08_soln - IE 111 Fall 2008 Homework Assignment#7...

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IE 111 Fall 2008 Homework Assignment #7 Solutions Question 1. The total amount of snow during a winter in the Lehigh valley is a random variable with mean 40 inches and variance 100 inches. My son is wishing for 100 or more inches of snow this winter. What can you tell him about the probability of his wish? μ = 40 σ = 10 If we choose k=6 then Chebyshev’s inequality becomes: P( μ -k σ < X < μ +k σ ) 1 - (1/k 2 ) P(40 -6*10 < X < 40+6*10) 1 - (1/6 2 ) P(-20 < X < 100) 0.97222 So his chances are less than 0.028 Question 2. On Tuesdays, patients arrive to the hospital at a rate of 2.5 per hour according to a Poisson process. I work an 8 hour shift at the hospital. a) Find the probability that exactly 20 patients arrive during my shift. α = 2.5 pat/hr T = 8 hrs. λ = α T = 20 X~Poisson( λ = 20) 0.088835 ! 20 20 ) 20 ( 20 20 = = = - e X P (From excel POISSON(20,20,FALSE) (Note that the EXCEL function POISSON is very useful for evaluating Poisson prob- abilities) b) Find the probability that 20 or fewer patients arrive during my shift. P(X≤20) = 0.559093 (From excel POISSON(20,20,TRUE) c) Find the probability that 10 or more patients arrive during my shift. P(X≥10) = 1 – P(X≤9) = 1 – POISSON(9,20,TRUE) = 0.995005

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d) On Mondays the patient arrival rate is 1.5 patients per hour (rather than 2.5 as on Tuesdays). Let X be the total number of patients that arrive during my shift on Monday and Tuesday. Find the expected value of X. λ Mon = 1.5*8 = 12 λ Tue = 2.5*8 = 20 E(X Mon + X Tue ) = E(X Mon ) + E(X Tue ) = λ Mon + λ Tue = 12+20 = 32 Question 3. Street lights in a certain city burn out according to a Poisson process with rate 6 per day. What is the probability that 50 or more burn out in a week? X~Poisson( λ = α T = 6*7 = 42) P(X≥50) = 1- P(X≤49) = 1-POISSON(49,42,TRUE) = 0.125024 Question 4. Forest fires in the state of Wyoming occur according to a Poisson process with rate 2 per day. a) A certain forest fire takes 8 hours to put out. What is the probability that no other fires start during that time? X~Poisson( λ = α T = 2 perday* 1/3 day = 2/3) P(X=0) = POISSON(0,0.6666,FALSE) = e -0.66666 = 0.513417 Question 5.
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This note was uploaded on 11/30/2008 for the course IE 111 taught by Professor Storer during the Spring '07 term at Lehigh University .

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HW7_08_soln - IE 111 Fall 2008 Homework Assignment#7...

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