The number of large cracks in a length of pavement along a certain street has a Poisson distribution with a mean of 1 crack per 100 m.
a. What is the probability that there will be exactly 8 cracks in a 500 m length of pavement?
b. What is the probability that there will be no cracks in a 100 m length of pavement?
c. Let T be the distance in meters between two successive cracks. What is the probability density function of T?
d. What is the probability that the distance between two successive cracks will be more than 50 m?
(a) Let be X the number of cracks in a 500 meter length of pavement.
Then the mean of Xis 5, so X~ Poisson(5).
(b) Let Y be the number of cracks in a 100 meter length of pavement.
Then the mean of Y is 1, so Y~ Poisson(1).
(c) Since T is the distance between two consecutive events in a Poisson process, T has an exponential distribution.
Since the rate at which events occur is 0.01 per meter, T~ Exp (0.01).
The probability density function is 
(d) 
.
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