Transforming to a normal distribution We are...

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EE364a: Convex Optimization I S. Boyd June 7–8 or 8–9, 2017 Final Exam Solutions This is a 24 hour take-home final. Please turn it in at Bytes Cafe in the Packard building, 24 hours after you pick it up. You may use any books, notes, or computer programs, but you may not discuss the exam with anyone until 5PM June 9, after everyone has taken the exam. The only exception is that you can ask us for clarification, via the course staff email address. We’ve tried pretty hard to make the exam unambiguous and clear, so we’re unlikely to say much. Please make a copy of your exam, or scan it, before handing it in. Please attach the cover page to the front of your exam. Assemble your solutions in order (problem 1, problem 2, problem 3, . . . ), starting a new page for each problem. Put everything associated with each problem ( e.g. , text, code, plots) together; do not attach code or plots at the end of the final. We will deduct points from long, needlessly complex solutions, even if they are correct. Our solutions are not long, so if you find that your solution to a problem goes on and on for many pages, you should try to figure out a simpler one. We expect neat, legible exams from everyone, including those enrolled Cr/N. When a problem involves computation you must give all of the following: a clear discussion and justification of exactly what you did, the source code that produces the result, and the final numerical results or plots. Files containing problem data can be found in the usual place, Please respect the honor code. Although we allow you to work on homework assignments in small groups, you cannot discuss the final with anyone, at least until everyone has taken it. All problems have equal weight. Some are (quite) straightforward. Others, not so much. Be sure you are using the most recent version of CVX, CVXPY, or Convex.jl. Check your email often during the exam, just in case we need to send out an important announcement. Some problems involve applications. But you do not need to know anything about the problem area to solve the problem; the problem statement contains everything you need. Some of the data files generate random data (with a fixed seed), which are not necessarily the same for Matlab, Python, and Julia. 1
1. Transforming to a normal distribution. We are given n samples x i R from an unknown distribution. We seek an increasing piecewise-affine function ϕ : R R for which y i = ϕ ( x i ) has a distribution close to N (0 , 1). In other words, the nonlinear transformation x 7→ y = ϕ ( x ) (approximately) transforms the given distribution to a standard normal distribution. You can assume that the samples are distinct and sorted, i.e. , x 1 < x 2 < · · · < x n , and therefore we also have y 1 < y 2 < · · · < y n . The empirical CDF (cumulative distribution function) of y i is the piecewise-constant function F : R R given by F ( z ) = 0 z < y 1 , k/n y k z < y k +1 , k = 1 , . . . , n - 1 , 1 z y n . The Kolmogorov-Smirnov distance between the empirical distribution of y i and the standard normal distribution is given by D = sup z | F ( z ) - Φ( z ) | ,

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