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MATH 4931: TAKE HOME EXAM

DUE APRIL 23RD. LATE EXAMS WILL RECEIVE A GRADE OF ZERO.

DO NOT DISCUSS THIS EXAM WITH OTHERS. You must do this work alone and I will ask you to sign the statement which states that you have not discussed these problems with others or received help on these problems.

In this course, we have learned several methods to (1) generate random variables, several meth- ods of (2) performing Monte Carlo simulations, and several methods of (3) solving optimization problems.

Your goal in the take-home exam will be to compare and contrast the methods via simulation studies designed by you, and to summarize your results in two separate reports.  The studies should be sufficiently extensive to justify being worth 30% of your final grade.

The results of your study should be typed up and presented as a formal report with appropriate plots, and a summary; no more than 4 pages per study.  For each question below you will create a report and a separate Rmarkdown file of your code.  Do not integrate the two. You will also email your  . rmd file to math4931yorku@gmail. com. Don’t forget to include your full name and student number in your PDF file and please include your name (first name is enough) in the name of your  . rmd file. The  . rmd file will not necessarily be graded, but it needs to be emailed before the deadline in case I need to check it.

You must work on your own. You must select your own distribution(s) and examples. Stay away from examples used in class.

1. Compare the following methods of Monte Carlo integration:

. regular Monte Carlo

. importance sampling

. stratified sampling

HINT: come up with appropriate examples of integrals and distribution(s) to fully explore/compare the methods;

2. Compare the following methods of optimization:

. Newton-Raphson method

. basic Monte Carlo optimization (using a uniform distribution)

. Monte Carlo optimization (using something other (better) than a uniform distribution)

. simulated annealing

HINT: come up with appropriate examples of functions and distribution(s) to fully explore/compare the methods; You may use the built-in Newton-Raphson code in R if you wish.

Note that not all methods are always applicable, and there is not necessarily a best overall method - some work better than others in different circumstances. Your study and report should reflect this. You should avoid simply using the examples from class or the text at all costs. You should also find the grading rubric helpful, please look over it carefully before beginning your work.

Each student is also allowed to email me twice with at most three questions each time. However, note that email will only be answered during regular business hours. Also, questions should be more “clarification of expectations” type questions. I will  no longer be explaining methods or helping you code (think of the types of questions that would be acceptable to ask in an typical written exam). Alternatively, students may make an appointment to ask me questions.

You should hand in:

(1)  Report for part 1.

(2)  Rmarkdown PDF of your code for part 1 (you may use smaller sample sizes than in your report if it takes too long to generate).

(3)  Report for part 2.

(4)  Rmarkdown PDF of your code for part 2 (you may use smaller sample sizes than in your report if it takes too long to generate).

(5) Signed and filled out attestation (scanned or hand-written); see next page.

A grading scheme has also been provided for you.  If the signed attestation is missing you will automatically receive a grade of zero.