
Poisson Process Question
"particles follow poisson proocess with average rate of $\displaystyle \lambda$ per unit of time. Each particle has probability $\displaystyle p$ of being detected independantly of other particles. Let $\displaystyle X$be the number of particles emitted in time interval $\displaystyle T$ 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;
$\displaystyle Probability(X=m andY=s)= \frac{\mu^{m}}{m!}e^{\mu}\left(\stackrel{m}{s}\right)p^{s}q^{ms}$
where mu = lambda x t and q=1p.
Its the following i'm not sure how to do;
I have to deduce that
$\displaystyle \P(y=s)=\frac{p^{s}q^{s}e^{\mu}}{s!}\sum^{\infty}_{m=s}\frac{\left(q\mu\right )^{m}}{\left(ms\right)!}$
and on summing the series show that Y has a Poisson distribution with parameter $\displaystyle p\mu$ And say why this would be expected.
All help is very much appreciated. thanks