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In these examples we made the assumption that the expected number
of background events is well known. If this is not the case,
we can quantify our uncertainty about it by a pdf ,
whose modeling depends on our best knowledge about .
Taking account of this uncertainty in a probabilistic approach
is rather simple, at least conceptually (calculations can
be quite complicate, but this is a different question).
In fact, applying probability theory we get:
We recognize in this formula that the pdf that takes
into account all possible values of
is a weighted average of all dependent pdf's,
with a weight equal to .