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 solve these problems, we'll be using the concept of Poisson processes. In a Poisson process, events (in this case, volcanic eruptions) occur randomly and independently over time.
(a) The mean time between eruptions is given as 29 months. We need to find the expected number of eruptions during a 5-year period.
To find the expected number of eruptions, we can use the formula for the mean of a Poisson distribution which is given by λ, where λ is the average rate of events per unit time. In this case, λ is given by the reciprocal of the mean time between eruptions.
λ = 1 / 29 months
Now, we need to find the expected number of eruptions in a 5-year period. Since there are 12 months in a year, the 5-year period contains 5 * 12 = 60 months.
Expected number of eruptions = λ * 60 months
Plug in the value of λ to calculate the expected number of eruptions.
(b) To find the probability of exactly two eruptions during the 5-year period, we can use the Poisson probability formula. The probability of observing k events in a given interval is given by:
P(k; λ) = (e^(-λ) * λ^k) / k!
Here, k represents the number of events (2 in our case), and λ is again the average rate of events per unit time.
Substitute the values of k and λ into the formula to calculate the probability.
(c) To find the probability that at least three years pass before the next eruption, we need to consider the time between eruptions. We know that the average time between eruptions is 29 months.
To find the probability of at least three years passing (36 months), we want to find the probability that the waiting time between two eruptions is greater than or equal to 36 months.
The waiting time between two consecutive eruptions follows an exponential distribution with rate parameter λ. The probability that the waiting time exceeds a certain value t is given by:
P(T > t) = e^(-λ * t)
Here, t represents the time duration in months. Substituting the given values, we can find the probability.
(d) The probability that the waiting time between two consecutive eruptions exceeds two years (24 months) is similar to part (c). We can use the same exponential distribution formula:
P(T > t) = e^(-λ * t)
Substitute the value of t (24 months) and calculate the probability.
Remember to convert the results to appropriate units (e.g., percentages) if required.