namespace RooStats::NumberCountingUtils

 Expected P-value for s=0 in a ratio of Poisson means.
 Here the background and its uncertainty are provided directly and
 assumed to be from the double Poisson counting setup described in the
 BinomialWithTau functions.
 Normally one would know tau directly, but here it is determiend from
 the background uncertainty.  This is not strictly correct, but a useful
 approximation.


 This is based on code and comments from Bob Cousins
  based on the following papers:

 Statistical Challenges for Searches for New Physics at the LHC
 Authors: Kyle Cranmer
 http://arxiv.org/abs/physics/0511028

  Measures of Significance in HEP and Astrophysics
  Authors: J. T. Linnemann
  http://arxiv.org/abs/physics/0312059

 In short, this is the exact frequentist solution to the problem of
 a main measurement x distributed as a Poisson around s+b and a sideband or
 auxiliary measurement y distributed as a Poisson around \taub.  Eg.
 L(x,y|s,b,\tau) = Pois(x|s+b)Pois(y|\tau b)

 Expected P-value for s=0 in a ratio of Poisson means.
 Based on two expectations, a main measurement that might have signal
 and an auxiliarly measurement for the background that is signal free.
 The expected background in the auxiliary measurement is a factor
 tau larger than in the main measurement.

 P-value for s=0 in a ratio of Poisson means.
 Here the background and its uncertainty are provided directly and
 assumed to be from the double Poisson counting setup.
 Normally one would know tau directly, but here it is determiend from
 the background uncertainty.  This is not strictly correct, but a useful
 approximation.

 P-value for s=0 in a ratio of Poisson means.
 Based on two observations, a main measurement that might have signal
 and an auxiliarly measurement for the background that is signal free.
 The expected background in the auxiliary measurement is a factor
 tau larger than in the main measurement.

 See BinomialExpP

 See BinomialWithTauExpP

 See BinomialObsZ

 See BinomialWithTauObsZ