Re: [Phys-L] Coincidence Statistics

[John Denker <jsd@av8n.com>]

On 11/20/23 11:55 AM, Paul Nord wrote:
> I'm not quite following that.  Your equation doesn't include the time bin.

My bad. You used a particular notation with a particular meaning. I used the 
same notation with a different meaning ... leading to unnecessary confusion. 
Sorry.

> Are we assuming that the units are those of time bins?

Well, I was. AFAICT you were using flashes per unit time (fput) whereas I was 
using flashes per bin (fpb).

Converting from one approach to the other is messier than one would like. See 
below.

For a simple Poisson process the conversion is easy. Multiply the fput by the 
bin-size to get the fpb. Note that the fpb is the only thing that enters the 
formula for the probability; neither the bin-size nor the fput enters directly. 
So formulating things in terms of fpb is not crazy.

Now let's segue to coincidences. I assume we are talking about accidental, coincidental 
coincidences, devoid of any physical connection. This is analogous to "optical 
double" stars in astronomy.

In this case we can't just say coincidences per bin (cpb) is the cput times the bin-size. 
That's because the bin-size enters the problem twice: once as the gate on the counter in 
the obvious way, but also as the definition of coincidence. Changing the bin-size changes 
what qualifies as a coincidence. So the whole idea of "rate of coincidences" 
hurts my brain. YMMV but I am more comfortable with /probability/ of coincidence. I know 
how to multiply probabilities.

Define λ to be the average i.e. the expectation value:
        λ := E[fpb]
then the probability of a coincidence in any given bin is:
        P = (1 - exp(-λ₁)) * (1 - exp(-λ₂))   [1]

If you're careful you can divide both sides by the bin-size to get something that 
sorta looks like a rate. When both λs are small this reduces to the rate 
formula given in the reference.

Equation [1] makes sense to me, as the product of probabilities.

> For the probability of one or more coincident events would we need to put
> that time resolution number into the equation (t)?
>         [2]
> Pc = (1 - e^(-R1*t))(1 - e(-R2*t))
>
> That gives us the same result when time resolution is 1 unit.  But when you
> change it to 2 units, the probability of coincidence goes to 3.28%.  And
> not the 2% calculated by the method given in the Canberra lab writeup.

In equation [2] you need to divide both sides by t if you want to calculate a 
rate.

===============================

Tangential remark:  IEEE floating point defines a function expm1(x) which 
calculates exp(x)-1 to high accuracy even when x is small. POSIX exposes this 
to the programmer. Also gnumeric provides it. Equation 1 is the perfect 
opportunity to use it.

See also log1p(x).

=====

Personal note: I got hit by a car. Between the pain and the painkillers I can't 
think clearly. And it's hard to type with your hand in a cast. It should all 
heal OK eventually.