5. Consider monitoring the number of critical cracks in the 1-77 Interstate High

Question

5. Consider monitoring the number of critical cracks in the 1-77 Interstate Highway starting at Columbia, SC (Mile Marker 0) Assume that the number of critical cracks cod b modeled according toa hotnogeneous Poisson process (HHP) with rate .5 per mile [that is, on average, there are 0.5 critical cracks in a span of one mile of I-77. (a) What is the probability that in the first mile of I-77 there will be zero critical cracks? (b) What is the probability that in a span of I-77 covering 10 miles, you will find no more than 3 critical eracks? will be found on or before the fifth mile? on will find the first critical crack? Note: At this point this is a challenge problem.] (c) Starting at Mile Marker 0 in Columbia, SC, what is the probability that the first critical crack (d) On the average, how many miles, starting from Mile Marker 0, do you have to examine before

Explanation / Answer

a)P(0 critical cracks) =P(X=0) =e-0.5*0.50/0! =0.6065

b)for 10 miles expected cracks =10*0.5 =4

P(no more than 3 cracks) =P(X<=3) =P(X=0)+P(X=1)+P(X=2)+P(X=3)

=e-5*50/0!+e-5*51/1!+e-5*52/2!+e-5*53/3! =0.2650

c)expected value of cracks in 5 mile =5*0.5 =2.5

P(first crack found in 5 miles)=1-P(no crack in 5 miles) =1-e-2.5*2.50/0! =1-0.0821 =0.9179

d)expected number of miles before first crack=1/0.5 =2

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