Re: statistical error of correlated events?

On May 26, 8:51 pm, Richard Ulrich <Rich.Ulr...@xxxxxxxxxxx> wrote:
On 25 May 2007 12:40:08 -0700, Markus <iandj...@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, wpi...@xxxxxxxxxxxx://www.pitt.edu/~wpilib/index.html

Of course, you are right!
I'm sorry that I made typos in both cases, for 2 and for n passengers.
What I really wanted to say is:
If every car had exactly "n" passengers then the answer is:
sigma = n*sqrt(Nn) or sigma^2 = n^2 *Nn

But my final formula was right:
sigma^2 = Sum_n (n^2 *Nn)

.... and this seems to be in agreement if I extrapolate your formula
for the n=2 case to the general case of arbitrary n.
Would you agree?
Also: Would you consider this formula to be trivial or obvious (in
other words,
one does not need any special reference for it), or do you know any
literature
in which this case is discussed?

Thanks for your reply!
Markus

.

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