Re: statistical error of correlated events?

On 25 May 2007 12:40:08 -0700, Markus <iandjohn@xxxxxxxxx> wrote:

I have a statistics problem for which (I think) I know the answer.
But I would like to know if this is correct, and if you can point me
to some reference in the literature.

The problem is how to determine the statistical uncertainty for a
counting experiment in case that (some of) the counts are produced
correlated.

Example:
I am standing at a street and I'm counting the passengers sitting in
cars.
I would like to determine the expectation value of the number of
passenegers per hour and it's statistical uncertainty.
A car with 1 passenger will contribute 1 count, whereas a car with 4
passengers will contribute 4 counts. After one hour I counted Ntot
passengers (which is the expectation value) - but what is the
corresponding statistical uncertainty?

Let me define:
Ntot ... number of total counts (=total passengers)
sigma ... the statistical uncertainty of Ntot
Nn ... the number of cars with "n" passengers
where: Ntot = Sum_n (n * Nn)

The "naive" (but wrong) estimate of the statistical uncertainty is:
sigma = sqrt(Ntot) or sigma^2 = Ntot = Sum_n (n *Nn)

Towards the correct answer:
If every car had only one single passenger then the answer is
trivial:
sigma = sqrt(Ntot) or sigma^2 = Ntot
If every car had exactly two passengers then the answer is again
trivial:
sigma = 2*sqrt(Ntot) or sigma^2 = 2^2 *Ntot

Maybe I'm not paying the right attention, but isn't
the variance computation properly based on the number
of cars, Nn, so that
sigma = 2*sqrt(Nn) in this case?
rather than 2*sqrt(Ntot) . And so on.

[snip, rest]

--
Rich Ulrich, wpilib@xxxxxxxx
http://www.pitt.edu/~wpilib/index.html
.

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