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 ï¬rst computing the expected number of explosions conditioned on the duration of an active period, then taking the average with respect to the duration.
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 ï¬rst computing the expected number of explosions conditioned on the duration of an active period, then taking the average with respect to the duration.
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