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Re: Geiger, not binomial ?



Hi all-
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. In other words,
nothing but the total no. of counts depends on N.
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.
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:

"Glenn A. Carlson" wrote:

There is every reason to expect your data to fit a binomial
distribution since it is the correct distribution.

Right.

However, as Mr.
Cleyet correctly points out, the Poisson distribution is more useful
here because of the huge number of nuclei in your sample (on the
order of 10^23) and the virtually zero probability that any one
nucleus will decay during the counting interval. ....

Exactly right, again.

Then at 06:50 PM 3/23/00 -0500, Ludwik Kowalski wrote:
>
Suppose we know nothing about radioactive decay at all. The
counter counts something. Perhaps these are cars crossing a
line on a busy highway in consecutive seconds. Who cares
how many cars are there in the entire country. I can ignore
all cars beyond a certain limit (the limit depends on how long
I am counting) and the distribution is exactly the same.

That's right, if you pass to the "certain limit" in a way that captures the
correct physics.

The experimental distribution gives me the mean number of
counts; it is = 2.20 per interval.

OK. That's the quantity that people have been calling pn (the probability
of arrival of a given car, times the number of cars in the country). The
limit you spoke of is
n -> large
while
pn = constant.

which is the limit in which the binomial distribution becomes
indistinguishable from the Poisson distribution. As you said, n doesn't
directly matter, as long as it is large enough. It is pn that directly
matters.

It also gives me the relative
frequency of 0.24 for recording only one count per interval.
For my practical purpose the relative frequency is the
probabiliity p which appear in the binomial distribution.

Nope. p is very small, because pn is medium-sized and n is huge.