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Re: Geiger, not binomial ?



On Mon, 27 Mar 2000, John S. Denker wrote:

--------------------------------------snip-----------------------------------

Yes indeed.

(1) Concerning whether the Geiger counter distribution is binomial or
Poisson.

It is both. The Poisson distribution is simply an approximation to the
binomial distribution.

Indeed the PD is a limiting case of the BD.

Not so "simply".

If the Poisson distribution is just an approximation to the binomial
distribution, why bother with the Poisson distribution... why not just use
the binomial distribution?

The Poisson distribution, as I showed in an earlier posting, can
be derived independently of the binomial distribution. In fact, Poisson
and binomial may be viewed as describing two physically different
contexts:
1. Poisson is appropriate for data that are describable by
time series where events accumulate with the increase of a continuous
variable such as time. If the probability per unit time of an event is
sonstant then one obtains the usual Poisson distribution.
2. Binomial is appropriate for data that are describable by
selection with replacement from an urn with two possible outcomes (say
black and white markers). The outcome of a finite sequence of such
selections is given by the binomial distribution.
The "large number" limit that gives Poisson is provided by the
transition from discrete sequences (binomial) to continuous sequences
(Poisson), but one need not know this mathematical fact to invoke the
appropriate distribution. The Geiger counter is therefore, to a very
good approximation, a Poisson device because it accumulates events
continuously (except for dead time).





(1a) Because the binomial distribution cannot be evaluated exactly for
situations involving large n because the binomial distribution requires the
evaluation of n!.

Actually there's a better reason: for large N (the total number of atoms)
and small p (probability of decay per atom, over relevant timescales), then
only the product of pN matters, so eliminating p and N in favor of pN
actually captures a valuable concept.

(1b) Sometimes, approximation methods lead to some insight that might be
more difficult to notice if the approximation were not performed.

Right.

(2) What insights might be included in (1b)?

See my comment under (1a).

This misses the point. See above.