Consider monitoring the number of critical cracks in the I-77 Interstate Highway

Question

Consider monitoring the number of critical cracks in the I-77 Interstate Highway starting at Columbia, SC (Mile Marker 0). Assume that the number of critical cracks could be modeled according to a homogeneous 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 ?rst 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 ?nd no more than 3 critical cracks?

(c) Starting at Mile Marker 0 in Columbia, SC, what is the probability that the ?rst critical crack will be found on or before the ?fth mile?

(d) On the average, how many miles, starting from Mile Marker 0, do you have to examine before you will ?nd the ?rst critical crack? [Note: At this point this is a challenge problem.]

Explanation / Answer

for poisson distribution  ? = .5 per mile

a) probability that in the ?rst mile of I-77 there will be zero critical cracks =P(X=0)=e-0.5*0.50/0! =0.6065

b)expected number of cracks in 10 miles =10*0.5 =5

probability that in a span of I-77 covering 10 miles, you will ?nd no more than 3 critical 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 number of cracks in 5 miles =5*0.5 =2.5

probability that the ?rst critical crack will be found on or before the ?fth mile =1-P(no crack till 5th mile)

=1-e-2.5*2.50/0! =1-0.821 =0.9179

d)

expected distance to find first crack=1/? =1/0.5 =2 miles

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