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Math Help - Poisson

  1. #1
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    Poisson

    Cars arrive at a toll booth at a mean rate of 15 cars every half hour according to a poisson process. Find the probability that the toll collector will have to wait longer than 36.78 minutes before collecting the eleventh toll.
    Please Help!!!
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  2. #2
    Member
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    Jul 2008
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    I will use the formulation of Poisson from wikipedia.

    \lambda is the expected number of occurrences over the particular interval of interest. So in our case the expected rate of occurrence is 15/30=0.5 cars per minute. So the expected number of cars in 36.78 minutes is: \lambda=15/30*36.78=18.4

    If N is the number of cars that arrive in 36.78 minutes, then the question can be rephrased to: what is the probability that N<11

    The PDF is:
    f(k; \lambda)=\frac{\lambda^k e^{-\lambda}}{k!}

    So

    \mathrm{P}(N<11) = \sum_{k=0}^{10}\frac{\lambda^k e^{-\lambda}}{k!}=\ldots
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