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Re: Geiger, not binomial ?

High their!

I've analyzed your data. It's almost too good to be true. I get a
reduced Chi square of ~ 0.6 w/ dF of 9 The probability of doing worse
than this is ~80%. So I conclude your multi-channel scalar is accurate
and you aren't having any spurious counts or dead time problems. The
only problem I had with your data is that you should have had ~ 2.5
times with ten counts, 0.5 times with 11 counts, etc. instead of zero.

I used the Poisson distrib., as that is the std. one to use (the assumed
distribution with which to compare) . You have discovered why in
counting experiments the binomial distrib. is not used! To get a good
fit n must be very large and p very small, typically 1E+20 and 1E-20
!!! This is because the number of trials is the number of radioactive
nuclei and the prob. of success is very low (i.e. the decay).

bc

If you want a reference, write.

Ludwik Kowalski wrote:

I have a problem with the distribution of counts from a
Geiger counter.It must be a binomial distribution but I
can not fit it with a binomial distribution.

A weak source is placed in front of a Geiger counter
from Vernier. That counter is connected to a computer
via ULI (also from Vernier) and the event counter
software (also Vernier) is used to count the number
of counts in consecutive time intervals of 0.5 seconds.

Typical counts are like 2, 0, 1, 4, 2, 1, 6, 1 etc. After
counting 30,038 times (for nearly 5 hr.) the distribution
of counts is as follows:

0 counts per 0.5 s occurred 3321 times
1 count per 0.5 s occurred 7278 times
2 occurred 7990 times
3 occurred 5994 times
4 occurred 3260 times
5 occurred 1439
6 occurred 557
7 occured 163
8 occurred 50
9 occurred 13

The same experiment was performed 7 times and each
time the counts are practically identical, except for
the last one which fluctuates a lot (as expected).
So everything is perfectly stable. But my attempts to
fit the distribution with a binomial model were not
successful. I do not know why.

For example, the first attempt was made after noticing
that the relative frequency of counting 1 is 7278/30038
=0.244 which corresponds to n=9 (because the mean is
2.20 counts. This predicts the distribution:

0 2425
1 7043
2 9092
3 6847
4 3315
5 1068
6 230
7 31
8 2

I expected to match the first 6 experimental counts
better than about 5% (and first 5 better than 2%) but
this turned out to be impossible. I tried many different
n for the binomial distribution. For example n=24 and
p=0.092 (to get the mean=24*0.092=2.2) gives

0 3125
1 7240
2 8439
3 6120
4 3178
5 1257
6 394
7 100
8 21
9 4

The general shape is always reasonable but details
are never satisfactory. Any comments? I expected
a nearly perfect fit because Geiger counters satisfy
all criteria under which the binomial distribution
should be a very good model.

Ludwik Kowalski