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Combination of results: general case
The previous case is rather artificial and can be used, at most,
to combine several measurements of the same experiment repeated
times, each with the same running time. In general,
experiments differ in size, efficiency, and
running time. A result on is no
longer meaningful.
The quantity which is independent from these contingent
factors is the rate, related to by
where indicates the efficiency, the generic `size'
(either area or volume, depending on whatever is relevant for the
kind of detection) and the running time: all the
factors have been grouped into a generic `integrated luminosity'
which quantify the effective exposure of the experiment.
As seen in the previous case, the combined result can be achieved
using Bayes' theorem iteratively, but now one has to pay attention
to the fact that:
 the observable is Poisson distributed, and the each experiment
can infer a parameter;
 the result on must be translated^{9.2}into a result on .
Starting from a prior on (e.g. a monopole flux) and
going from experiment 1 to we have
 from
and
we get
;
then, from the data we perform the inference on
and then on :
 The process is iterated for the second experiment:
 and so on for all the experiments.
Lets us see in detail the case of null
observation in all experiments
(
) ,
starting from a uniform distribution.
 Experiment 1:








(9.7) 


at 95% probability 
(9.8) 
 Experiment 2:

 Experiment :


(9.9) 
The final result is insensitive to the data grouping.
As the intuition suggests, many experiments give the
same result of a single experiment with equivalent luminosity.
To get the upper limit, we calculate, as usual, the cumulative
distribution and require a certain probability for to be below
[i.e.
]:
obtaining the following rule for the combination of
upper limits on rates:

(9.10) 
We have considered here only the case in which no background is
expected, but it is not difficult to take background into account,
following what has been said in Section .
Next: Including systematic effects
Up: Poisson model: dependence on
Previous: Combination of results from
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Giulio D'Agostini
20030515