CS2 ch1 Q1.4(ii) when compound poisson process is also a poisson process

Discussion in 'CS2' started by iamminime, Apr 26, 2023.

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  1. iamminime

    iamminime Keen member

    ch1 Q1.4(ii) it states that
    when St = X1 + X2 + ... + XNt is a compound Poisson process, if Xj can only take value 0 or 1, then St is also Poisson process.

    I can understand when Xj = 1 for all j, St is a Poisson Process, because all Xj then sum up to Nt, which is a Poisson Pcoess.
    But why when some of Xj = 0, St is also Poisson Process? Thanks a lot!
     
    iamminime, Apr 26, 2023
    #1
  2. Andrew Martin

    Andrew Martin ActEd Tutor Staff Member

    Hi Iamminime

    Good question! This is similar to the idea of thinning a Poisson process. Say we have that Xj takes the value 0 with probability 50% and 1 with probability 50%. The process St = X1 + X2 + ... + XNt where Nt ~ Pois(lambda * t) is the same as a Poisson process with parameter lambda / 2, ie it is the same as the following process:

    Rt = Y1 + Y2 + ... + YMt
    where Yj = 1 with certainty and Mt ~ Pois(lambda / 2 * t)

    Hope that helps!

    Andy
     
    Andrew Martin, May 15, 2023
    #2

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