Question

Fully describe how to use the unified approach for a Poisson distribution describing signal and background events. Illustrate this by constructing a 90% confidence level interval for the number of observed events given a signal yield µ of 2 events and an assumed background of 1 events. You may wish to consider total event yields between zero and ten.

Answer 1

• Rare processes are described by Poisson statistics. In the limit of small numbers, where the Gaussian approximation is invalid, it is useful to consider one and two sided intervals of a Poisson distribution to obtain intervals and upper limits. The following discussion mirrors the previous section. Two significant differences between the Poisson and Gaussian distributions are that r, unlike x, is a discrete parameter and the Poisson distribution

is not symmetric about its mean value whereas a Gaussian distribution is. This fact is relevant when constructing a two-sided interval and in particular when determining the ±1" uncertainty
interval on a measured observable in the limit of small statistics. Such a two-sided interval can be constructed by integrating the Poisson distribution for a given r such that the limits ​ 1 and ​ 2 are equally probable in order to obtain the desired CL. In doing so we naturally determine an asymmetric interval about the mean value ​ . If we aremeasuring some quantitywherewe wish to express 90% coverage about a mean, then an asymmetric PDF such as the Poisson distribution naturally leads to an asymmetric uncertainty. As with the Gaussian case, a one-sided interval is a matter of integrating f(x, ​ ) for a given observed number of events x, to obtain a limit with the desired coverage.
Figure 6.3 shows the one and two sided confidence intervals obtained for ​ in a counting experiment as a function of the number of observed signal events r. The upper limit is quoted in terms of both 90% and, as these are commonly found as the levels of coverage used in many scientific publications. The
corresponding two-sided interval plot also includes the90% CL contours in order to be able to enable a
comparison with the Gaussian ". One sided integral tables of the PoissonPDF can be found inappendix D.The case study described in section 6.8.2 gives an example of using a Poisson distribution to set a confidence interval. While these intervals are represented by a smooth distribution, one should note that the possible
outcomes of an experiment are in terms of discrete numbers of events. The situation encountered where one has a non-zero background component modifies the previous discussion
on computing limits. For such a scenario, where one observes Nsig signal events and Nbg background events, both of which are distributed according to a Poisson distribution with means ​ sig and ​ bg, respectively. One can show that the sum of these two components is also a Poisson distribution with a mean of ​ sig + ​ bg. Given sufficient knowledge of ​ bg, one can proceed to compute limits on $sig. This situation is discussed in Cowan (1998), which also highlights issues surrounding measurements involving large backgrounds with small numbers of observed signal events.