"particles follow poisson proocess with average rate of per unit of time. Each particle has probability of being detected independantly of other particles. Let be the number of particles emitted in time interval 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;

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

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

I have to deduce that

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

All help is very much appreciated. thanks