The distance between major cracks in a highway follows an exponential distributi

Question

The distance between major cracks in a highway follows an exponential distribution with a mean of four miles.

(a) What is the standard deviation of the distance between two major cracks?

(b) What is the probability that the first major crack occurs between 11 and 14 miles of the start of inspection?

(c) Given that there are no cracks in the first four miles inspected, what is the probability that there is no major crack in the next four miles stretch?

(d) What is the probability that there are no major cracks in two separate four-mile stretches?

Explanation / Answer

a)standard deviation of the distance between two major cracks =mean =4 miles

b)

probability that the first major crack occurs between 11 and 14 miles of the start of inspection=P(11<X<14)

=e-11/4-e14/4 =0.0337

c)from memoryless property: P(X>8|X>4) =P(X>4) =e-4/4 =0.3679

d)

due to indepedence probability that there are no major cracks in two separate four-mile stretches

=P(X1>4)*P(X2>4) =e-4/4 *e-4/4 =0.3679*0.3679 =0.1353

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