Independence in Poisson Process

[filter54321]

I'm studying the Poisson Process (rate R) and I'm hung up on the issue of dependence. This seems like and easy question but I have no background in probability whatsoever.

By definition, the number of events in disjunction time intervals are independent. Okay. Fine. But say we have an overall "observation window" of time [0,t]. By definition, the expected number of events in t is

E[N(t)] = Rt

and the probability of exactly n events in time t is

P{N(t)=n} = e^-Rt*(Rt)^n/n!


Say we have a small "peek window" of [0,s] where s<t. Will the number of observations when you get to peek be independent of the total number of observations? My intuition is no, but a colleague with considerably more expertise is saying yes.

If we know there are exactly n observations at time s, it would seem that the likelihood of having exactly that same n at a later time would have to decrease (relative to the odds if you didn't get to peak). If you know for certain that you ALREADY have n, then you'd be less likely to END with n because the Poisson Process doesn't decrease.

Any thoughts? Links to resources?

Thanks

 

[CRGreathouse]

Divide the observation space O into two parts, A and B. The number of Poisson events in B is independent of the number of Poisson events in A (where A is the area you "peeked" at), but the number of Poisson events in O is dependent.

If you observe 10 events in A, then O cannot have 0 to 9 events, so clearly they're dependent. But the essence of the Poisson distribution is that A does not influence B.

My guess is that you colleague is saying that A ans B are independent while you are taking about A and O. In that case both of you could be right.