Homework 3

Introduction

You will implement three methods of generating random numbers from a standard normal. You will then compare them on the quality of the values they generated and the efficiency of the methods. Finally, you should analyze and present your findings in a well-written, well-organized report.

Program Specifications

The main components that you have to implement for this assignment are three methods of generating random numbers that follow the standard normal distribution (more details below). Your top-level program should ask the user for a method and a number specifying how many values to generate. As with homework 2, you may develop your program in Java, C, C++, C#, Python, or a language of your choice (pending approval of the TA), and your program should provide some reasonable input prompts for the user, and its output should be stored and/or displayed in some reasonable way. For the core uniform random number generator, use the built-in library function.

You should prepare a README file (separate from the analysis report) for the grader. It should explain how to run your program and provide any other relevant information that might help the grader evaluate your program.

Generation Methods

1. Inverse Transform Method

   Since there is no closed form for the Standard Normal CDF, you must use an approximation. Bowling et al (2009) developed a closed-form approximation of the CDF that you should use:

   Pr(Z <= z) = 1 / [1 + e^(-1.702z)]
2. Accept/Reject Method

   Use the exponential with λ = 1 as your g(x). Keep track and report the average number of rejections in the write-up.

3. The Polar-Coordinate Method

   Use the most efficient version from the notes.

Experiment

To compare the three methods, you should generate a large number of instances for each method (e.g., 10,000). Use the timer function that’s built in to your programming language to profile how long each method took. Build histograms of the generated numbers.

Write-Up / Analysis

Describe your experiment in a report. Your report should include the following:

• Describe the experimental setup. For example, how many numbers did you generate for each method? For each generation method, what implementational decisions did you have to make?
• Prior to running the experiment, how did you think each method would perform in terms of time efficiency? Explain why.
• Plot histograms for the numbers generated by the three methods. Qualitatively, do they look like they are from the Standard Normal Distribution?
• Determine whether the generated numbers might be from the Normal Distribution using the Chi-Square Test. Discuss the results. To get the critical value for the Chi-Square test, use the table available on the daily schedule: prob-table_X2.pdf.
• Report the (average) amount of time it took for each method. Was the experimental result surprising? Discuss.
• In conclusion, which of the three methods would you recommend? Explain.

Submission

The assignment is due Thursday, March 3 by 11:59 pm.

Zip your source code files, your write-up, and your README into one zip file and upload it to CourseWeb, in the Homework 3 location.

For more advice on submitting your assignment, see the Assignments section of the Tips for Success page.