Question

The number of major faults on a randomly chosen 1 km stretch of highway has a Poisson distribution with mean 1.8.

I know that the distance between two successive major faults follows X ~ exponential(mean = 1/1.8)

What is the probability you must travel more than 3 km before encountering the next four major faults? Give your answer to 3 decimal places.

By expressing the problem as a sum of independent Exponential random variables and applying the Central Limit Theorem, find the approximate probability that you must travel more than 25 km before encountering the next 33 major faults