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Math Help - Poisson, Exponential problem

  1. #1
    Super Member craig's Avatar
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    Poisson, Exponential problem

    Little stuck on a Poisson problem.

    Sightings of a particular species of rare bird occur at a rate of 6 per year. Assume that sightings can be modelled by a Poisson process (N(t), t>0) where time t is measured in years and time 0 is the present time.

    Let T_1 be the time from time 0 until the 1st sighting. What is the probability that T_1 is at most:

    i) 1.
    ii) 2.
    I remember something about the exponential distribution being the time between Poisson processes, but not sure how to apply it here.

    Would T_1 be distributed Ex(12)?

    Thanks in advance
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  2. #2
    Moo
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    Hello,

    Why 12 ? I think the Ti follow an exponential(lambda), and here, lambda=6

    P(T1<1)=1-P(T1>1)=1-P(N(1)=0), because T1 is the time of first sighting. So if it's >1, it means that between 0 and 1, there have been no sighting.

    And N(1) follows a Poisson distribution with parameter lambda.
    So P(T1<1)=1-exp(-lambda)

    Similarly, P(T1<2)=1-P(N(2)=0), and N(2) follows a Poisson distribution with parameter 2*lambda
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  3. #3
    Super Member craig's Avatar
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    Cheers for that, was a typo in the original post.
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