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Re: Geiger (a challenge)



At 11:48 PM 7/8/00 -0400, Ludwik Kowalski wrote:
According to Bob Sciamanda:

fm(t) = (t^m)*r^(m+1)*exp(-rt)/m!

where fm(t) is the probability of recording m counts in time t
and r is the rate of decay (dN/dt).

No, that's not what Bob wrote. Bob's formula is the answer to a different
question.

One can see from dimensional analysis that fm(t) is not a probability; it
is a probability density (probability per unit time). BTW it is probably
better to write fm(t) as f(m,t).

In particular, f(0,t) dt is the probability that one will see an
inter-event spacing of exactly t. More generally, f(m,t) dt is the
probability that the timing between two pulses with exactly m other pulses
between them will be exactly t.

==========

The probability of recording m counts in time t given an average counting
rate of r is given by
g(m,t) = (rt)^m exp(-rt) / m!

This has the property that the RHS depends on rt, not on r or t
separately. This is as it should be! Any other dependence is wrong.

BTW this has an amusing derivative:
(d/dt) g(m,t) = f(m-1,t) - f(m,t)

where the RHS can be interpreted as the probability of creating a group of
m (by adding one to a smaller group) minus the probability of destroying a
group of m (by adding one too many). So we see that Bob's analysis is
quite relevant to the situation under discussion.