1.Consider a linear birth–death process where the individual birth rate is =1, t
ID: 3200535 • Letter: 1
Question
1.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 . Please show all work
(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?
Explanation / Answer
a)
Initial population=p
Birth rate =1 and death rate =3
P+1-3=p-2
Immigration is happening with rate .
Hence, the population increase by (-2) if >2 and decrease by (2-) if <2 and remains stagnant if =2.
b)
For the population size to increase by 1 (i.e. 11) from 10 now, we need to have a migration of 3 because we have (-2). Therefore, we can put the formula as below:-
E^- . ^3/3!
c)
Because we have calculated (-2), population can become extinct when p=0,1,2. Hence, it would be calculated as below:-
E^- . ^0/0!+ E^- . ^1/1!+ E^- . ^2/2!
Substituting =1 we get 5/2e.
Related Questions
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.