15.2 You will again need a general minimizing program for this exercise. In Exercise 8.1, you worked out a method for generating Monte Carlo events with a Breit– Wigner (B-W) distribution in the presence of a background. Here we will use a somewhat simpler background. Suppose the background is proportional to a + bE with a = b = 1. The B–W resonance has energy and width (E0, ). The number of events in the resonance is 0.22 times the total number of events over the energy range 0−2. We further set: E0 = 1,  = 0.2. Generate 600 events. Save the energies on a file for use in calculating maximum likelihood, and for making a histogram for the next problem. Ignore any B–W points with energy past 2. (Regenerate the point if the energy is past 2.) Now suppose that you are trying to fit the resultant distribution of events to a B–W distribution and a background term of the above form with

b, E0,  to be determined from your data. Take a to be the actual value 1, and introduce a new parameter f ract, the fraction of events in the B–W. The likelihood for each event will be f = f ract × fBW + (1 − f ract) × fback, where both density functions are properly normalized. Using the maximum likelihood method, fit the events you have generated and see how closely the fitted parameters correspond to the actual ones. You will have to take account here that for the parameters given, only about 93% of the B–W is within the interval 0–2 GeV. As noted in Sect. 15.7, you need to recalibrate the errors since 1 standard deviation corresponds to wmax − w = 0.5 rather than the value of 1.0 for a χ2 fit. Directions for doing this for MINUIT are given in the example MINUIT fit in Appendix A.