Data collected on volcanic eruptions give a mean time between eruptions of 29 months. Assume these eruptions occur as a Poisson process in time, find the:

(a) Expected number of eruptions during the five-year period beginning from the time you have read this question.
(b) Probability that there will be exactly two eruptions during this five-year period.
(c) Probability that at least three years will pass from now before the next erupt
(d) Probability that the waiting time between two consecutive earthquakes exceeds two year

To find the answers to these questions, we can use the properties of a Poisson process and the formula for the Poisson distribution.

(a) To find the expected number of eruptions during a five-year period, we need to calculate the mean number of eruptions within this time frame. Since the mean time between eruptions is given as 29 months, we need to convert this to years by dividing by 12:

Mean time between eruptions = 29 months = 29/12 years

Now, we can calculate the expected number of eruptions:

Expected number of eruptions = (number of years) * (mean number of eruptions per year)

Expected number of eruptions = 5 * (1 / mean time between eruptions)

Expected number of eruptions = 5 / (29/12)

Expected number of eruptions ≈ 2.069

So, the expected number of eruptions during the five-year period is approximately 2.069.

(b) To find the probability of exactly two eruptions during the five-year period, we can use the Poisson distribution formula:

P(X = k) = (e^(-λ) * λ^k) / k!

where X is the random variable representing the number of eruptions, λ is the average number of eruptions, and k is the desired number of eruptions.

In this case, λ is the expected number of eruptions calculated in part (a), which is approximately 2.069, and k is 2.

P(X = 2) = (e^(-2.069) * 2.069^2) / 2!

P(X = 2) ≈ 0.270

So, the probability that there will be exactly two eruptions during this five-year period is approximately 0.270.

(c) To find the probability that at least three years will pass from now before the next eruption, we need to calculate the probability of no eruptions occurring within three years.

P(no eruptions in three years) = P(X = 0)

Using the Poisson distribution formula with λ = 2.069 and k = 0, we get:

P(X = 0) = e^(-2.069) * 2.069^0 / 0!

P(X = 0) ≈ 0.125

So, the probability that at least three years will pass from now before the next eruption is approximately 0.125.

(d) To find the probability that the waiting time between two consecutive eruptions exceeds two years, we can calculate the probability of no eruptions occurring within a two-year period.

P(no eruptions in two years) = P(X = 0)

Using the Poisson distribution formula with λ = 2.069 and k = 0, we get:

P(X = 0) = e^(-2.069) * 2.069^0 / 0!

P(X = 0) ≈ 0.125

So, the probability that the waiting time between two consecutive eruptions exceeds two years is approximately 0.125.