Re: counts in an interval

[Ludwik Kowalski <KowalskiL@MAIL.MONTCLAIR.EDU>]

John Denker wrote:

> Observation:  If we "round off" 2.77 to 3, then the
> middle rt value is 5 times bigger than the small one, and
> 5 times smaller than the big one.  This leads to a hypothesis,
> namely repeated mistakes in converting between "counts per
> bin" and "counts per second".  However, my crystal ball
> is a bit murky at the moment, so other possible explanations
> can't be ruled out.

Not murky at all. I was not able to check everything, but
one thing is clear, the mean counting rate was 11.8 and not
3.08. I wish I could honestly say that "it was a typo". In any
case let us compare experimental data and predictions now.

According to Poisson formula  [g(m,t) = (rt)^m exp(-rt) / m!]
with r*t=11.8*0.2=2.36 one finds:

> m=0 -->  253 outcomes (instead of experimental 123)
> m=1 -->  596 outcomes (instead of experimental 467)
> m=2 -->  704 outcomes (instead of experimental 674)
> m=3 -->  554 outcomes (instead of experimental 604)
> m=4 -->  327 outcomes (instead of experimental 503)
> m=5 -->  154 outcome  (instead of experimental 172)

This is reasonable but far from being satisfactory. Let me
check this on another saved experimental file (Timing 3).
It had 7004 events. The first 20 events are shown below;
these are times of arrivals in seconds. The last digits are
likely to be affected by rounding or truncations. As you
can see, there were 5 counts in the first 0.3 s interval,
3 in the next, 4 in the next, etc.

The first waiting time was only 3 ms, the next was 70 ms,
the next was 25 ms, etc. The distribution of waiting times
is exponential, the distribution of counts per dt=0.3 s has
the poissonian shape. The mean counting rate was 12.59
cnts/sec. Do the experimental data agree EXACTLY with
the Poisson formula? I will be happy to send this text file
to anybody who asks for it. Use any software you want
and share observations.

The only person who responded to my question about
this activity being useful was John D. Do you agree with
his statement that this is mathematics, not physics? I
disagree; counting experiments are the ways to explore
the random nature of radioactive transformations. How
else can it be explored?

Would you use the described activity with students if
the program to collect data on the arriving times (and to
display the distributions while data are collected) were
built-in into the Vernier software? As I wrote before,
I am trying to persuade Vernier to add such routine to
their package. So please answer this question on
Phys-L; Vernier people are physics teachers and some
of them are on our list. Elaborate on your opinion and
suggest what else would you  do with such software,
if it were made available.

The Timing3 file (see the sample below) was checked for
the constancy of counting rates. I did this to be sure that
nobody changed geometry (moved something) when I
was in the next room. The mean cnts/sec at the end of
the run were practically the same as at the beginning.
                    Ludwik kowalski
0.048
0.051
0.121
0.146
0.283
0.320
0.494
0.506
0.670
0.711
0.733
0.797
0.916
1.009
1.035
1.039
1.180
1.276
1.440
1.483