Suppose that globally, there are an average of 60 earthquakes per day and an average of 0.12
volcanic eruptions per day. Suppose that the occurrences of earthquakes are independent
of other earthquakes and the occurrences of volcanic eruptions are independent of other
volcanic eruptions. Further suppose that the occurrences of earthquakes and volcanic eruptions are independent of each other.
(a) Let E and V be the counts of earthquakes and volcano eruptions,
respectively, on a randomly chosen day. Justify why E follows a Poisson distribution
with \lambda = 60 and V follows a Poisson distribution with \lambda = 0.12.
(b) Let S be the count of earthquakes and volcanic eruptions on a randomly
chosen day. That is, let S = E + V . Use the moment generating function provided below to show that S follows a Poisson distribution with \lambda = 60.12.
Note: A Poisson random variable X with parameter \lambda has moment generating function:
MX(t) = e^\lambda (et−1)
(c) Find the probability that on a randomly chosen day, there are a combined
60 earthquakes and volcanic eruptions (i.e., S = 60). You may derive the answer analytically or by using R.
(d) From today, let A be the time until the 100th earthquake, and B be the
time until the first volcanic eruption. Justify why A follows a gamma distribution
with \alpha = 100 and \lambda = 60 and B follows an exponential distribution with \lambda = 0.12.
(e) Using R, estimate the probability that the first volcanic eruption comes
before the 100th earthquake. To do this:
• Randomly generate 100,000 values from A with the rgamma function.
• Randomly generate 100,000 values from B with the rexp function.
• Calculate a vector of differences B−A, and find the proportion of these differences
which are less than 0.
Provide all R commands used and evidence of your empirical
computation (screenshot of RStudio or copy & paste your R code).= e\lambda (et−1)