Re: Geiger, not binomial ?

["Glenn A. Carlson" <gcarlson@MAIL.WIN.ORG>]

My first inclination is to respond that even if we don't know anything
about radioactive decay, the 10^23 nuclei in your sample do "know."
But I think I understand your question and will give it more thought.
Interesting question.

Glenn

Glenn A. Carlson, P.E.
gcarlson@mail.win.org

> Subject: Re: Geiger, not binomial ?
> Date: Thu, 23 Mar 2000 18:50:09 -0500
> From: Ludwik Kowalski <KowalskiL@MAIL.MONTCLAIR.EDU>
>
> "Glenn A. Carlson" wrote:
>
> > There is every reason to expect your data to fit a binomial
> > distribution since it is the correct distribution.  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.  ....
>
> 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.
>
> The experimental distribution gives me the mean number of
> counts; it is = 2.20 per interval. 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.
> Something must be wrong with this logic (see below) but
> I do not see it.
>
> If this logic is correct then I must use n=2.2/0.24=9.1. So
> strating with n=10 is reasonable. The resulting distribution
> is narrower than the experimental, as shown below. But
> with n=170 (I had to use Sterling's formula for this) I got
> a rather good fit. This is also shown below. The n=10^23
> would give a slightly better fit but n=170 is good enough
> for me, and for my computer program. Thanks to those
> who responded.
>
> cnts  binom  experim n=10  exper n=170
>
>  0     1932    3321         3249
>  1     6102    7278         7277
>  2     8671    7990         8099
>  3     7302    5994         5974
>  4     4036    3260         3285
>  5     1529    1439         1436
>  6      402     557          520
>  7       73     163          161
>  8        9      50           43