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Math Help - Poisson Process Question

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    Poisson Process Question

    "particles follow poisson proocess with average rate of \lambda per unit of time. Each particle has probability p of being detected independantly of other particles. Let Xbe the number of particles emitted in time interval T and Y the number of those particles which are detected"


    So the first part i think i have done and that is to show the following;


    Probability(X=m andY=s)= \frac{\mu^{m}}{m!}e^{-\mu}\left(\stackrel{m}{s}\right)p^{s}q^{m-s}


    where mu = lambda x t and q=1-p.


    Its the following i'm not sure how to do;


    I have to deduce that


    \P(y=s)=\frac{p^{s}q^{-s}e^{-\mu}}{s!}\sum^{\infty}_{m=s}\frac{\left(q\mu\right )^{m}}{\left(m-s\right)!}


    and on summing the series show that Y has a Poisson distribution with parameter p\mu And say why this would be expected.


    All help is very much appreciated. thanks
    Last edited by zh4ngk; November 2nd 2009 at 10:04 AM.
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