Question

The occurrence of rust attacks along a gas pipeline can be modeled as a Poisson process with intensity λ per kilometer. This means that the number of rust attacks per kilometer along the pipeline can be assumed to be Poisson distributed with expectation value λ.
In the first instance, suppose that it is known that λ = 5.
a)

What is the probability of just two rust attacks in half a kilometer?
What is the probability of more than two rust attacks in half a kilometer?
A section of 7.5 kilometers has been closed down to improve rust attacks. What is the probability that they will find more than 40 rust attacks in this section?
The distance between two subsequent rust attacks is exponentially distributed by parameter λ (the distance between subsequent events in a Poisson process is exponentially distributed), and we still assume that λ = 5 per kilometer.

b) What is the expected distance between two subsequent rust attacks?
What is the probability that the distance between two subsequent rust attacks is more than 200 meters?
What is the probability that the distance between two subsequent rust attacks is less than 300 meters?
What is the probability that the distance between two subsequent rust attacks is between 200 and 300 meters?
In reality, the value of λ is unknown, but it can be estimated from the result of the improvement of the section of 7.5 kilometers. In this section, 34 rust attacks were found.
c) Set up an estimator for λ and explain why it is reasonable to assume that this estimator is approximately normally distributed.
Find the expectation value and variance for the estimator, and explain how we can also find from this an (approximate) 90% confidence interval for λ.
Use the confidence interval to determine the test at 10%:
H0: λ = 5 versus H1: λ̸ = 5
Briefly comment on what to think about to determine whether the results of the test and the confidence interval are valid for the rest of the pipeline.

Answer 1