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Math Help - Poisson Process Question - Can't figure out.

  1. #1
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    Poisson Process Question - Can't figure out.

    I don't know how to approach part of this question:

    A boy throws a piece of bread to each bird that swims by.
    There are 2 types of birds: ducks and swans.
    Ducks and swans arrive in independent Poisson processes of rates a and b per minute, respectively.
    Let N(t) be the number of pieces of bread thrown by the boy from the time he arrives at the river bank, at time 0, until t minutes later. Let Td be the time until the first duck arrives.

    i) Name the distribution of N(t) and state it's mean

    I got this as N(t) ~ Poi (at + bt), where the mean is (at + bt)

    ii) Name the distribution of Td and state it's mean

    I got Td ~ exp (a), where the mean is 1/a

    iii) Find the probability that the first bird to arrive is a duck

    This is where I get stuck. I figure that this is P(Td < Ts), but then how would I do this?

    And then there's a couple more parts after it which I can't figure out:

    Consider the first 20 pieces of bread thrown by the boy. Let Nd be the number of these 20 pieces that are eaten by a duck.

    iv) Find the distribution of Nd and explain why it is.

    v) Find the probability that after 10 minutes, the boy has thrown a piece of bread to at least one duck and at least one swan.
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  2. #2
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    iii) Conditioning on T_s and apply total probability
    iv) After you're done with iii), think about this: is the type of the second bird that arrives dependent of the first bird that arrives?
    v) Ducks and swans are independent of each other, aren't they?
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  3. #3
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    For (iii) I got a/(a+b).

    iv) It's independent, but does that mean Nd ~ Poi (at)? Or Nd ~ Poi (20 - bt)?
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  4. #4
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    It's related to iii). The type of the bird of each arrival is independent of each other. What is the probability distribution that the nth bird to arrive is a duck?
    Sum 20 of these probability distributions and you get ... ?
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