Question

1. Rock noise in an underground mine occurs approximately according to a Poisson process at an average rate of three per hour.

a. What is the probability that the waiting time to record rock noise once is at least half an hour?

b. What is the average waiting time to record rock noise once? Determine also the standard deviation.

c. The waiting time to record rock noise twice is reasonably modeled with a gamma distribution with α = 2 and scale β = 1/3. Determine the mean and standard deviation of the waiting time.

Answer 1

1)

a)

We know that waiting time follows exponential distribution. Let X denote the waiting time to record rock noise once (in hours).

Then

Required probability =

b)

Let X denote the waiting time to record rock noise once (in hours).

Then

So, and

c)

We know that the mean and standard deviation Gamma distribution are given by: