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Math Help - Birth Process/Poisson Process

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    Birth Process/Poisson Process

    An Island is initially uninhabitated. Creatures arrive as a poisson process of rate a per month. Once there each creature produces offspring as a poisson process of rate b per month. Assume that the Poisson process for arrivals and births are all independent. Assume also that there are no deaths and no creatures leave the island. let X(t) be the number of creatures on the island at time t months and P_{n}(t)=P(X(t)=n).
    What is the probability that the island is inhabitated by time 3 months?
    Do they mean there is at least one creature after 3 months? I have a formula for P_{n}(t) but I am not sure what is n=? when t=3.
    thanks for any help.
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    Quote Originally Posted by charikaar View Post
    What is the probability that the island is inhabitated by time 3 months?
    Do they mean there is at least one creature after 3 months? I have a formula for P_{n}(t) but I am not sure what is n=? when t=3.
    thanks for any help.
    Yes. They are asking for \mathbb{P}(X_3\geq 1). You seem confused, P_n(t) is the probability that at time t, you have n inhabitants. n does not depend on t, but the P_n(t) varies with t and n. In this case it may be easier to compute the probability that the island is empty at t=3, and then take away from 1. In your notation, \mathbb{P}(X_3\geq 1)=1-\mathbb{P}(X_3=0)=1-P_0(3)
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    that really hepled so many thanks.

    f) what is the expectation of the time at which the population size first reaches 10?

    can you give me a hint to answer this one please?

    thanks
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    Quote Originally Posted by charikaar View Post
    that really hepled so many thanks.

    f) what is the expectation of the time at which the population size first reaches 10?

    can you give me a hint to answer this one please?

    thanks
    Let T_{n,m}=\inf\{t \geq 0, X_t=m \mbox{ given that } X_0=n\}.

    Note that the waiting times are independently exponential, that is, the amount of time you spend at n is exponential and is independent from the amount of time you spent in any previous state.

    This allows you to calculate  \mathbb{E}[T_{0,1}].

    What can you say about T_{n,n+1}?

    Finally, T_{0,n}=\sum_{i=0}^{n-1}T_{i,i+1}.
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    Thanks for your help. I am still having trouble understanding the question.
    f) what is the expectation of the time at which the population size first reaches 10?
    Do they want me to find the time, t when there are n=10 inhabitants. thanks
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    Quote Originally Posted by charikaar View Post
    Thanks for your help. I am still having trouble understanding the question.
    f) what is the expectation of the time at which the population size first reaches 10?
    Do they want me to find the time, t when there are n=10 inhabitants. thanks
    Yes. But it is the first time that there are 10 inhabitants. The Poisson process stays put at any level for an exponential time, so just because there are 10 inhabitants does not mean that it is the first time.

    In mathematical terms they want  \mathbb{E}[\inf\{t>0:X_t=10\}].

    The previous post might be useful if you draw a Poisson process and see which each thing refers to.
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