hassignment Instructions: Deadline: Feb 15th, 2022, assignments handed in after the deadline will not be graded. Assignments should be handed in via email. Submission should include a (short) report (lab report format) including all the appropriate code (in separate files for each part of the code). The code should be documented and runnable (in either MATLAB or python). Code that does not run will not be graded or given partial credit. The assignment is individual. 1. The secretary problem: You would like to hire someone to work for you. N applicants have applied for the job. The deal is that you are allowed to interview them sequentially, and then decide whether to hire the applicant or not. If you hire them, that's it, if you do not, you will never see them again (cannot go back). In order to maximize your chances of hiring the best person, it's clear you should interview some r of the applicants, and then select the first applicant you see who is better than all the previous ones. Given that you choose this method (optimal stopping), simulate the process for various values of N and r and determine the optimal r. Given this r, what is the probability of hiring the best person? How does this change when you would like to hire one of the two best people? 2. MC Integration (a) Use Monte-Carlo hit-miss integration to determine the value of the integral e'dr. fraction" . Estimate the uncertainty on the value as a function of the number of generated events. (b) We will now use a different method to calculate the same integral. It is clear that s()dx = (6a) (8) Use this to calculate the integral. What is the uncertainty in this case? 1 3. You are given the PDF f(z) = 1 + a sin(6) + bcos(), OS S 27 The file SinCos.CSV contains numbers sampled from this distribution which you will use in this problem. (a) Write the log of the PDF. (b) Assuming that a and b are small perform a Taylor expansion of the logarithm of the PDF (expand to second order). (c) Write the minimization condition for the log likelihood function for a and b. (d) Write the equations in matrix form. (e) Now use the data in the file and the equations you derived to calculate a and b. This method is known as linearization of the log likelihood function. 4. Given the distribution The file Dat1.csy contains 100K events drawn from an exponential distribution. Use Maximum likelihood methods to determine the decay constant and the uncertainty. Histogram the data before before you perform the fit in the following ways and compare the results: 10 bins 50 bins. 100 bins. 20 equal count bins 50 equal count bins. 5. The file Dat2.csv contains two sets of data (side by side). drawn from two exponential distributions. Use the Kolmogorov-Smirnov and x? tests two check whether they are drawn from the same distribution, what are the different p-Values? 6. The file Dat3.csv contains the results of a measurement in the form x,y,dy). The function that the results are expected to follow is of the form f() = ln(1 + I)e-3(2-)? Perform a x? fit to find the parameters of the fit. Report the full results
