Poisson Process Question

zh4ngk · #1 $Old$ November 2nd, 2009, 10:53 AM

"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 $X$ be 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