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 need to use the principles of a Poisson process and the exponential distribution. The Poisson process is a mathematical model used to describe events that occur randomly over time or space.
(a) To find the expected number of eruptions during the five-year period, we need to use the average rate of eruptions, which is given as 29 months (or approximately 2.42 years).
The expected number of eruptions during a given time period can be calculated using the formula:
Expected number of events = Average rate * Time period
In this case, the time period is five years, so the calculation would be:
Expected number of eruptions = 2.42 * 5 = 12.1
Therefore, the expected number of eruptions during the five-year period is approximately 12.1.
(b) The probability of exactly two eruptions during a five-year period can be calculated using the Poisson distribution formula:
P(X = k) = (e^(-λ) * λ^k) / k!
Where:
- λ (lambda) is the average rate per unit time (in this case, 2.42 eruptions per year)
- k is the number of eruptions we are interested in (in this case, 2)
Using this formula, we can calculate the probability:
P(X = 2) = (e^(-2.42) * 2.42^2) / 2!
(c) The probability that at least three years will pass from now before the next eruption can be found using the exponential distribution. The exponential distribution is used to model the time between two consecutive events in a Poisson process.
The cumulative distribution function (CDF) for the exponential distribution is given by:
CDF(t) = 1 - e^(-λt)
Where:
- t is the time period we are interested in (in this case, three years)
- λ (lambda) is the average rate per unit time (in this case, 2.42 eruptions per year)
Using this formula, we can calculate the probability:
P(at least 3 years) = 1 - CDF(3)
(d) The probability that the waiting time between two consecutive eruptions exceeds two years can be calculated using the exponential distribution.
The survival function (SF) or complementary cumulative distribution function (CCDF) for the exponential distribution is given by:
SF(t) = e^(-λt)
Where:
- t is the time period we are interested in (in this case, two years)
- λ (lambda) is the average rate per unit time (in this case, 2.42 eruptions per year)
Using this formula, we can calculate the probability:
P(time between eruptions > 2 years) = SF(2) = e^(-2.42*2)