The distance between major cracks in a highway follows an...

The distance between major cracks in a highway follows an exponcntial distribution with a mean of 5 miles. (a) What is the probability that there are no major cracks in a 10 -mile stretch of the highway? (b) What is the probability that there are two major cracks in a 10 -mile strctch of the highway? (c) What is the standard deviation of the distance between major cracks? (d) What is the probability that the first major crack occurs between 12 and 15 miles of the start of inspection? (e) What is the probability that there are no major cracks in two separate 5 -mile stretches of the highway? (f) Given that there are no cracks in the first 5 miles inspected, what is the probability that there are no major cracks in the next 10 miles inspected?

The distance between major cracks in a highway follows an exponcntial distribution with a mean of 5 miles.
(a) What is the probability that there are no major cracks in a 10 -mile stretch of the highway?
(b) What is the probability that there are two major cracks in a 10 -mile strctch of the highway?
(c) What is the standard deviation of the distance between major cracks?
(d) What is the probability that the first major crack occurs between 12 and 15 miles of the start of inspection?
(e) What is the probability that there are no major cracks in two separate 5 -mile stretches of the highway?
(f) Given that there are no cracks in the first 5 miles inspected, what is the probability that there are no major cracks in the next 10 miles inspected?

Key Concepts

Exponential Distribution

The exponential distribution is a continuous probability distribution often used to model the time or distance between independent events that occur at a constant average rate. It is characterized by its mean parameter (or rate parameter, which is the reciprocal of the mean). A key property of the exponential distribution is that its variance (and standard deviation) is equal to the square of the mean, and it is frequently applied in reliability analysis and waiting time problems.

Poisson Process

A Poisson process is a stochastic process that models the occurrence of events in a fixed interval of time or space, where these events happen independently and at a constant average rate. The number of events in a given interval follows a Poisson distribution, and the time between consecutive events is exponentially distributed. This relationship makes the Poisson process particularly useful for problems dealing with counts of rare events over specified intervals.

Memoryless Property

The memoryless property is a unique feature of the exponential distribution, indicating that the probability of an event occurring in the future is independent of how much time has already elapsed. In other words, past events have no effect on future probabilities. This property simplifies many calculations in probability, especially in assessing the likelihood of events following a waiting period without occurrences.

Transcript

00:01 In this exercise, we are told that x is exponentially distributed with a mean of 5 miles.

00:07 That is, the distance between cracks in a highway is exponentially distributed with a mean of 5 miles.

00:15 For part a, we are asked for the probability that there are no cracks in a 10 -mile stretch of highway.

00:22 That is the same as the probability that we go at least 10 miles before encountering a crack, which is the probability that x is greater than 10.

00:31 Is equal to 1 minus the cdf of x, evaluated at 10.

00:44 The cdf of an exponential random variable is given by this formula, and lambda is equal to 1 over the mean.

01:09 So this comes out to e to the negative 0 .2 times 10, and that gives us a probability of 0 .1353.

01:26 For part b, we are asked for the probability that there are two major cracks in a 10 -mile stretch of highway.

01:34 If the distance between cracks is exponentially distributed, then the number of cracks in a given distance is a poisson process.

01:44 And so we can say what is the probability that the number of cracks is equal to 2.

01:51 This is where x is equal to 10 miles.

01:59 This is equal to e to the minus lambda x times lambda x to the exponent n over n factorial.

02:17 Gives us e to the minus 2 times 2 squared over 2 factorial.

02:27 And this comes out to 0 .2707.

02:32 That's the probability of 2 cracks in length of 10 miles.

02:41 Part c asked for the standard deviation of the distance between cracks.

02:47 For an exponential random variable, the standard deviation is equal to 1 over lambda, or it's equal to the mean, which we know is 5 miles...

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