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 answer these questions, we'll need to use the properties of the Poisson distribution. The Poisson distribution models events that occur randomly over time or space with a constant average rate.
The formula for the Poisson distribution is: P(X=k) = (e^(-λ) * λ^k) / k!
Here, λ is the average rate, which is equal to the mean time between eruptions expressed in the same time units as the period considered.
(a) Expected number of eruptions during the five-year period:
To calculate the expected number of eruptions, we need to find the value of λ for the given five-year period. Since the mean time between eruptions is 29 months, we need to convert this to five-year units.
First, let's convert the mean time to years:
29 months ÷ 12 months/year = 2.4167 years
Now, to find the average rate λ for a five-year period:
λ = (number of years) * (mean time between eruptions in years)
λ = 5 years * 2.4167 eruptions per year
λ ≈ 12.0833
The expected number of eruptions during the five-year period is given by λ.
Expected number = λ ≈ 12.0833
(b) Probability of exactly two eruptions during the five-year period:
To find the probability of exactly two eruptions, we can use the Poisson distribution formula:
P(X=2) = (e^(-λ) * λ^2) / 2!
Substitute the value of λ we calculated earlier:
P(X=2) = (e^(-12.0833) * 12.0833^2) / 2!
Simplifying the expression will give you the probability.
(c) Probability that at least three years will pass from now before the next eruption:
To find this probability, we need to calculate the cumulative probability of waiting time exceeding three years. This is equivalent to the probability of having zero or one eruption in a three-year period.
P(at least three years) = P(X=0 or X=1) = P(X=0) + P(X=1)
Using the Poisson distribution formula, substitute λ with the appropriate value for a three-year period and calculate the respective probabilities.
(d) Probability that the waiting time between two consecutive eruptions exceeds two years:
This probability can be found by considering the complement event, which is the probability of waiting time less than or equal to two years (i.e., one or zero eruptions in a two-year period).
P(waiting time > 2 years) = 1 - P(X=0 or X=1) = 1 - (P(X=0) + P(X=1))
Calculate the respective probabilities using the Poisson distribution formula, but this time using λ for a two-year period.
Remember, to get accurate values for these probabilities, it is crucial to use the mean time between eruptions in the same time units as the period considered.