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Let's reverse the previous problem: let Ui and U2 be independent random variables uniformly distributed on [0, 1]. Then let Y1 = -2 1n(U1) and Y2 = 2«U2. (a) Find the distributions of Y1 and Y2. Can you name them? (b) Let Z1 = VYi cos Y2 and Z2 = VYı sin Y2. Show that Z1 and Z2 are independent standard normal variables. This shows that if we can generate independent uniform values, we can relatively easily generate independent normal values. This process is called the Box-Muller transformation, and was very popular before software could deal with generating normal values directly.

Let's reverse the previous problem: let Ui and U2 be independent random variables uniformly distributed on [0, 1]. Then let Y1 = -2 1n(U1) and Y2 = 2«U2. (a) Find the distributions of Y1 and Y2. Can you name them? (b) Let Z1 = VYi cos Y2 and Z2 = VYı sin Y2. Show that Z1 and Z2 are independent standard normal variables. This shows that if we can generate independent uniform values, we can relatively easily generate independent normal values. This process is called the Box-Muller transformation, and was very popular before software could deal with generating normal values directly.

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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7. Let's reverse the previous problem: let U1 and U2 be independent random variables uniformly distributed on [0, 1]. Then let Y1 = -2 In(U1) and Y2 = 2¬U2. (a) Find the distributions of Y1 and Y2. Can you name them? (b) Let Z1 = VYi cos Y2 and Z2 = VYı sin Y2. Show that Z1 and Z2 are independent standard normal variables. This shows that if we can generate independent uniform values, we can relatively easily generate independent normal values. This process is called the Box-Muller transformation, and was very popular before software could deal with generating normal values directly.
Transcribed Image Text:7. Let's reverse the previous problem: let U1 and U2 be independent random variables uniformly distributed on [0, 1]. Then let Y1 = -2 In(U1) and Y2 = 2¬U2. (a) Find the distributions of Y1 and Y2. Can you name them? (b) Let Z1 = VYi cos Y2 and Z2 = VYı sin Y2. Show that Z1 and Z2 are independent standard normal variables. This shows that if we can generate independent uniform values, we can relatively easily generate independent normal values. This process is called the Box-Muller transformation, and was very popular before software could deal with generating normal values directly.
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