Consider a linear birth-death process where the individual birth rate is =1, the individual death rate is = 3, and there is constant immigration into the population according to a Poisson process with rate .

(a) State the rate diagram and the generator.

(b) Suppose that there are 10 individuals in the population.

What is the probability that the population size increases to 11 before it decreases to 9?

(c) Suppose that = 1 and that the population just became extinct. What is the expected time until it becomes extinct again?

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