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Re: Geiger, not binomial ?



The Poisson probability has to do with the occurrence of events
at a detector. It has nothing do do with decaying nuclei except in some
very special circumstances. Here is a derivation:
Let rdt be the probability of an event in a detector during an
infinitesimal time interval dt. Then (1-rdt) is the probability of no
events during such a time interval.
Turn the detector on at time zero. After a time t the probability
of n events can be written P_n(t). At time t+dt the probability of n+1
events is
probability of n+1 at t times probability of no events in time dt
plus probability of n at t times " " of 1 event in time dt.

In symbols,
P_<n+1>(t+dt) =P_<n+1>(t)(1 + rdt)+P_n(t)rdt
do the Taylor expansion of P_<n+1>(t+dt> to get:

dP_<n+1>(t)/dt = r[P_n(t)-P_<n+1>(t)]
The solution that satisfies
Sum from 0 to infinity of P_n(t)=1 is

the Poisson probability P_n=[(rt)^n/n!]exp(-rt).

John's comment, below, has to do with the special case where
the detector events can be related to some assemblage of decaying atoms.
Regards,
Jack


Adam was by constitution and proclivity a scientist; I was the same, and
we loved to call ourselves by that great name...Our first memorable
scientific discovery was the law that water and like fluids run downhill,
not up.
Mark Twain, <Extract from Eve's Autobiography>

On Thu, 23 Mar 2000, John Denker wrote:

At 09:47 PM 3/23/00 -0600, Jack Uretsky wrote:
Something is wrong here. Let N_n be the number of occurrences
of n counts in the interval T, and let r be the counting rate. Then,
for a Poisson distribution the total number of counts N is:
N=Sum of n= 0 to infinity of N_n,
where N_n = N{(rT)^n/n!}exp(-rT)
and both the mean and variance of the distribution = rT.

OK.

In other words,
nothing but the total no. of counts depends on N.

I'm not sure what that means. "Nothing" seems like a too-sweeping term.

In this case, r is the probability per unit time that one nucleus
will decay - that in fact is one way to derive the Poisson distribution.

If r is the counting rate (which I take to mean average counting rate) then
don't you mean r/N (not plain r) is the probability per unit time that one
nucleus will decay?