# Can I get additional info from interarrival times?

1. May 21, 2006

Oh wise ones --

I have a Poisson process (think clicks in a Geiger counter) for which I have a fixed but short time window, sufficient to get a smallish (10-50) number of events.

I do NOT have the average arrival rate; that's what I want to estimate.

Is my estimate any better if I measure the distribution of interarrival times than if I just take the average (number of events detected divided by the width of the time window)? Can someone with more math horsepower than I have tell me if so, and (for extra credit) by how much?

I know I have quantization issues with the simple average because the arrival times are independent of my window start/stop times. I know the interarrival times are independent of each other, but they are of course not independent of the total in the window. Can any of you tell me if there's extra info hiding in there if I can get the interval distribution?

Thanks from a poor dumb engineer who only flunked two courses in college -- both math. :)

2. May 22, 2006

A little more info: I can easily get the timing between events to fairly good precision (1 part in 1000 or so). I understand that the variance of the number of counts in my time window is equal to the number of counts, hence standard deviation is the square root of that. So that's the uncertainty in my estimate of the mean arrival rate the simple way.

I also know that the probability distribution of the arrival intervals is a declining exponential for a Poisson process, correct? So if I measure the intevals, it *seems* like I should have more information than a simple count. Is there a way to fit the interval distribution to derive the exponent, with better standard deviation than root-N?

3. May 30, 2006

Ok, I've gotten info in another forum that answers my first question (no, the simple average is the best you can do, no extra information in arrival-time distribution other than verifying that the process really is a simple Poisson). Now let me tell you the other purpose for looking at interarrival times, and ask a second question.

Let's suppose I have two distinguishable classes of events, A and B, with different relative probabilities which are known. They are both independent Poisson processes. I can always detect A events. I can detect an event of class B in between two events of class A only if the interval between the A events is long enough. Call that minimum interval Tmin. I am specifically interested in 3-event sequences.

I can calculate the probability of getting exactly 3 events of either class, all separated by no more than Tmin, readily enough. If I know the combined total event rate R = R(A) + R(B), then from the first event, it's P(event within Tmin) times P(event within Tmin) times P(no event for Tmin), or (1 - e^(-RTmin)) times (1-e^(-RTmin)) times e^(-RTmin).

Now I can arrange all 8 possible orderings of 3 events of either type:

AAA, AAB, ABA, ABB, BAA,BAB,BBA,BBB

and, since they're independent events, I can assign each sequence a probability based on the known relative probabilities of type A or type B events.

So the bad case is ABA. I can only see an event of type B in between two A's if the interval between the A's is greater than Tmin, otherwise the A's mask it. So.... what I *think* is true is that because A and B events are independent, the interval between A events is also independent. I can take the overall probability of my ABA sequence derived from their combined rate R, and multiply it by the independent probability that the two A's are separated by Tmin, which is just e^-(R(A)Tmin), where R(A) is the A event rate. Then 1 - e^-(R(A)Tmin) is the probability that I miss the B event, times the probability of the ABA sequence, gives me the overall loss rate for B events. In all the other triple-event sequences, the B's are detectable.

Can anyone confirm or deny this logic?