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Math Help - Poisson process question

  1. #1
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    Poisson process question

    Given a poisson process Z(t) with a given rate lamda, k and m nonnegative integers and t and c real and positive numbers, calculate the probability:
    P(Z(t-c)=m | Z(t)=k)

    thank you
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  2. #2
    MHF Contributor
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    Re: Poisson process question

    This is a poisson process so we have the following properties:

    [A]~Z_0 = 0
    [B]~Z_t \sim Po(\lambda t)
    [C]~Z_c - Z_b \sim Po(\lambda (c-b)) \text{ and is independent of } Z_b

    Start from the usual formula for conditional probabilities

    P(Z_{t-c} = m | Z_t =k) = \frac{P\left(Z_{t-c} = m \cap Z_{t} = k \right)}{P(Z_t)=k}

    The denominator is easy using property [B].

    The numerator can be re-written as follows:
    P\left(Z_{t-c} = m \cap Z_{t} = k \right) = P\left(Z_{t-c} = m \cap Z_{t}-Z_{t-c} = k -m \right)

    using property [C], the events Z_{t-c} =m and Z_t -Z_{t-c} =k-m are independent, and hence

    P\left(Z_{t-c} = m \capZ_{t}-Z_{t-c} = k -m \right) = P\left(Z_{t-c} = m \right) \times P \left(Z_{t}-Z_{t-c} = k -m \right)

    Both of those terms follow a poisson distribution. (unless k-m is negative, then the probability is zero).


    So the final answer is:
    P(Z_{t-c} = m | Z_t =k) = \frac{P\left(Z_{t-c} = m \right) \times P \left(Z_{t}-Z_{t-c} = k -m \right)}{P(Z_t = k)}

    Where:
    Z_t \sim Po(\lambda t)
    Z_t - Z_{t-c} \sim Po(\lambda c)
    Z_{t-c} \sim Po(\lambda (t-c))

    you can subsitute the formulae for the PMFs to finish. Dont forget that the probability is zero if m > k
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