Suppose that we discover an island with a volcano on it. Based on the type of vo

Question

Suppose that we discover an island with a volcano on it. Based on the type of volcano, we know that its volcanic eruptions occur according to a poisson process at a rate of 5 volcanic eruptions every 100 years. (a) What is the probability of having exactly 3 volcanic eruptions in the next 50 years? (b) What is the variance of the amount of time (in years) that we must wait until we see 9 eruptions? (c) Suppose that there are 8 volcanic in a period of 5 years. What is the variance of the amount of (in years) until the next volcanic eruption? (d) Suppose that we do not have any historical records about the volcano (since we just discovered it). How long ago do we expect the last eruption to have occurred? (In other words, what is the expected amount time between the last eruption and the present day?)

Explanation / Answer

Given, volcano rate

A.a) mean=5 per 100 years

mean=2.5 per 50 years

x=3

P(X=3)=e^-2.5*2.5^3/3!

solving we get

P(X=3)=0.2137

A.b) x=9

in Poisson distribution mean and variance are equal

for 100 years, we have 5 volcanoes

for 9 volcanoes, we need 100*9/5=180 years

hence the variance of amount of time is 180 years

A.c) 8 volcanoes erupted in 5 years

in previous problem we calculated, for 9 volcanoes to occur it takes 180 years

hence the variance in amount of time is 180-5=175 years

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