A certain volcano can be in one of two states: active (a) and dormant (d), and s

Question

A certain volcano can be in one of two states: active (a) and dormant (d), and switches between the two in an apparently sudden and random way. Suppose that it switches from d to a in exponential time with mean 100 days, and from a back to d in exponential time with mean 1 day. In addition, while active, it produces small explosions at the times of a Poisson process with rate 10 per day.

i) Find the proportion of time that the volcano spends in the active state.

ii) Find the expected number of explosions produced during a single active period. Justify your answer by first computing the expected number of explosions conditioned on the duration of an active period, then taking the average with respect to the duration.

Explanation / Answer

ans=

(i) The proportion of time that the volcano spends in the active state= 10/100=0.1 per day.

(ii) Expected number of explosions produced during a single active period = (100+1)/10=10.1

Then expected number is greater than the proportion of time that the volcano spends in the active state.

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