CERN-THESIS-2012-153 26/07/2012 - CERN Document Server

means:
Q = L( X,s + b)
L( , (8.5)
X,b)
the ratio of probability density for the signal+background and the background-only hypotheses. The likeli-
hood ratio for a combination of two independent channels, like the 2ID+TL and 3ID, is simply the product of
the likelihood ratios of the individual channels, so the combination is straightforward. The likelihood ratio
Q for an experiment with Nchannels independent channels and measurements of a discriminating variable
(number of observed events in this case) is:
Q = e −stot
Nchannels
i=1
1 + si
ni , (8.6)
bi
where ni is the number of observed candidates in each channel, and si and bi are the integrated signal
and background rates per channel, stot is the total signal events for all channels. Equation 8.6 follows
from Poisson statistics. This can be further simplified to a form of “counting weighted events” by a simple
derivation. Since P(Q ≤ Qobs) = P(ln(Q) ≤ ln(Qobs)), given that Q ≥ 0, Equation 8.6 can ve written as:
Nchannels
ln(Q) = −stot +
where n is the total number of events observed in all channels and the weight for each channel i is:
i=1
niwi
(8.7)
wi = ln 1 + si
. (8.8)
bi
Since the constant stot appears in both side of the expression ln(Q) ≤ ln(Qobs), the final test-statistic
consists basically of comparing the observed number of weighted events with the distribution expected for
the signal+background and the background-only hypotheses. This is:
Xd =
Nchannels
i=1
ni ln 1 + si
. (8.9)
bi
When data events are more similar to the signal events, the Xd variable takes higher values. The Xd test-
statistic is then compared to 10 5 pseudo-experiments of the hypotheses of signal+background, Xs+b and
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