Re: [Phys-L] Coincidence Statistics
- From: John Denker <jsd@av8n.com>
- Date: Sat, 18 Nov 2023 13:52:50 -0700
On 11/16/23 4:39 PM, Paul Nord via Phys-l wrote:
Subtle statistics question is bothering me. Assume that I have two
stochastic processes that take place once every 10 seconds. If I watch for
100 seconds, I expect to see the "thing" happen 10 times (a Poisson
distribution around 10, actually).
My "timing resolution" for watching these is 1 second. So, I'll put them
into 100 bins depending on the moment in the 100 second observation window
that the event happened.
What is the rate at which I will expect to observe coincident events?
I'm looking at a lab writeup that suggests the coincidence rate Rc will be
a function of the two rates (R1 and R2) and the timing resolution (t).
Rc = R1*R2*t
In my case:
Rc = 0.1 * 0.1 * 1
or 1 event every 100 seconds
My sense is that the birthday paradox should apply here and the time
between pulses will have an exponential probability density.
Or that the binning described above is actually 1/2 second timing
resolution. Two pulses of 0.5 second length would overlap if they were
just under 0.5 seconds apart.
Maybe I'm missing something, but this doesn't look too hard.
I assume the definition of coincidence is one or more counts in the
channel 1 *and* one or more counts in channel 2, in a particular small
bin.
The chance of one or more is 1 minus the chance of zero, which is easy
to calculate from the standard Poisson formula. So
Pc = (1 - exp R1)(1 - exp R2) [1]
When both rates are small, that reduces to the product R1 R2.
More generally you need the full equation [1].
Obviously when both rates are huge the probability of coincidence
flatlines at 100%.