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 7?

d. What is the probability that the distance between two successive cracks will be more than 50 m?

Solution 2SE

Step1 of 5:

Let us consider a random variable X it presents the number of large cracks in a length of pavement along a certain street and it has poisson distribution with mean of 1 crack per 100 m.

Let X follows poisson distribution with parameters “ ”.

Then the probability mass function of poisson distribution is given by:

, x = 0,1,2,...,n.

Where,

= parameter

n = sample size

x = random variable

e = mathematical constant.

Here our goal is:

a).We need to find the probability that there will be exactly 8 cracks in a 500 m length of pavement.

b.We need to find the probability that there will be no cracks in a 100 m length of pavement.

c).We need to find the probability density function of 7, When T be the distance in meters between two successive cracks.

d).We need to find the probability that the distance between two successive cracks will be more than 50 m.

Step2 of 5:

a).

Here we have

X = 8 and

=

= 5

Now,

The probability that there will be exactly 8 cracks in a 500 m length of pavement is given by

Consider,

P(X = 8) =

=

=

= 0.0653

Hence, P(X = 8) = 0.0653.

Therefore, The probability that there will be exactly 8 cracks in a 500 m length of pavement is 0.0653.

Step3 of 5:

b).

Here we have

X = 8 and

=

= 1

Now,

The probability that there will be no cracks in a 100 m length of pavement is given by

Consider,

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