3. (Value: 20 marks, due in class Monday Dec 5, 2011)
A new particle has been discovered in a lab by colliding two beams of leptons and measuring how many particles are produced as a function of the energy E of the beams. The table below gives the results.

where M = 91.01 GeV, Γ = 2.6 GeV, and σ
pol = 0.00009846 events. Create a random number
generator that produces random numbers according this distribution. Plot these random numbers in a histogram and superpose the data points on your distribution. You may chose one free floating normalization parameter to get the overall scale of the Monte Carlo events
to agree with the data (state what normalization you use). However, the other parameters in the distribution are fixed to the values of M, Γ, σpol given above. Discuss the results of your
comparison between data and Monte Carlo events. (d) It may be possible to improve the comparison by "smearing" your Monte Carlo results by a Gaussian distribution, i.e. once you have generated your Monte Carlo event, then add to it a second component which is another random number which follows a Gaussian distribution centered at zero but with a small width. This effectively widens your distribution. Smear your Monte Carlo data with a Gaussian of width 1 GeV and once again compare to the data. Discuss the results of your comparison between data and Monte Carlo events. Try smearing with Gaussians of different widths and explain what you see. What width best describes the data? (e) Given your work in parts (a)-(d), propose a better smearing function than a regular Gaussian.
Explain your choice and show that this function works well at describing the data.