I've learned and/or relearned a lot since this thread began, and I
appreciate everyone's insight. One thing I have noticed is that some
messages mention the binomial distribution, some mention the Poisson
distribution, some mention the Gaussian distribution, and lately we have
mention of a t-distribution.
At one time (i.e. graduate school) I had this all sorted out. This current
discussion has made it clear that I had forgotten all these details. That
realization prompted me to blow the dust off my copy of "Data Reduction and
Error Analysis for the Physical Sciences" by Philip R. Bevington. This 1969
book has been mentioned on this list before (I think). It remains one of my
most trusted sources, even though it is not exactly easy reading. I
suppose it is good, once in a while, for professors to study a book with the
same attention and care we insist our students should apply when studying
their textbooks. Having just done so with Bevington, I would like to add a
couple points to the discussion which I am not sure have been made, or
perhaps not made clearly enough that I recognized it.
(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.
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?
(1a) Because the binomial distribution cannot be evaluated exactly for
situations involving large n because the binomial distribution requires the
evaluation of n!. We have to use some approximation method for finding n!
when n is large. One possibility is Sterling's approximation. Someone
already mentioned the use of Sterling's approximation in this situation.
Another possibility is Poisson's approximation in which the
binomial-distribution terms containing n! are approximated by a different
method (i.e. not using Sterling).
(1b) Sometimes, approximation methods lead to some insight that might be
more difficult to notice if the approximation were not performed.
(2) What insights might be included in (1b)?
(2a) Poisson gives us easier ways (compared to binomial) to find or
understand the mean and variance of the distribution. To quote Bevington,
"The advantage of using the Poisson distribution for analyzing any such
experiment comes from the fact that the distribution is specified completely
by the mean (mu), and the standard deviation (sigma) is uniquely determined
by that mean" [page 43].
(2b) It helps us understand the shape of the distribution. The Poisson
distribution typically applies to a situation in which the possible values
of data are strictly bound on one side, but not on the other, leading to a
distribution curve that is asymmetric. For example, in a counting
experiment we don't have negative counts during a time interval; but on the
high side there is no definite limit to how many counts could be detected in
a time interval given a "hot" enough source and a detector with fast
resolving time. Hence the distribution has a "long tail" on the high side,
but cuts off abruptly at zero counts on the low side. Bevington says this
type of asymmetry is characteristic of the Poisson distribution... that is,
characteristic of the binomial distribution in which n is very large and p
is very small.
I found I had forgotten most of this. The only things I had remembered
were... Poisson statistics are generally used for radiation counting
experiments, and the standard deviation is the square-root of the mean.
Michael D. Edmiston, Ph.D. Phone/voice-mail: 419-358-3270
Professor of Chemistry & Physics FAX: 419-358-3323
Chairman, Science Department E-Mail edmiston@bluffton.edu
Bluffton College
280 West College Avenue
Bluffton, OH 45817