Poisson interarrival intervals, dual processes
- From: bloviatrix@xxxxxxxxxxx
- Date: 31 May 2006 06:27:48 -0700
Paging Rich and John2... :)
Previously, I'd asked if measuring interarrival times told me anything
about
the mean rate of a Poisson process. Turns out the answer is no.
Ok, now that we have THAT question answered, let me tell you why I want
the interarrival intervals anyway.
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 measure the interval between A events, and I want to know the
probability that when I see two A events closer than Tmin, there's
really an invisible B event hiding in there.
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? Thanks again...
.
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