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Exercise 12. Let X, Y, Z~ N(0, 1) be independent. Determine the distribution, expectation and variance of the following random variables: a) X+Y+Z

Exercise 12. Let X, Y, Z~ N(0, 1) be independent. Determine the distribution, expectation and variance of the following random variables: a) X+Y+Z

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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c) Write clearly and legibly, and leave margins for the marker to write comments in. d) Write complete sentences, including correct punctuation. e) Explain how you obtained your solution. Just giving the final answer is not enough, all intermediate steps are also required. Exercise 12. Let X, Y, Z ~ N(0, 1) be independent. Determine the distribution, expectation and variance of the following random variables: a) X+Y+Z b) X² + y² + Z² Exercise 13. Let X₁,..., Xn ~ N(x, o²) i.i.d. and Y₁,..., Yn ~ N(µy, 0²) i.i.d. with known 2. We assume that for each i the random variables X; and Y; are correlated, with Corr(X₁,Y₂) = 1/2. Define D₁ = X₁ - Yį for all i = {1,...,n}. Finally, let X, Y and D be the averages of the Xi, Yi and Di, respectively. a) Show that Cov(X₁, Yi) = 0²/2. b) Determine E(D;) and Var(D₂). c) Consider the test for Ho: x=y which uses the test statistic Z= √nD. σ and which rejects Ho, if and only if |Z| > 1.96. Show that P(type I error) = 5%. d) In two or three sentences, discuss the differences between the test from part (c) on the one hand, and the non-paired test for comparing the means of two populations on the other hand. 2
Transcribed Image Text:c) Write clearly and legibly, and leave margins for the marker to write comments in. d) Write complete sentences, including correct punctuation. e) Explain how you obtained your solution. Just giving the final answer is not enough, all intermediate steps are also required. Exercise 12. Let X, Y, Z ~ N(0, 1) be independent. Determine the distribution, expectation and variance of the following random variables: a) X+Y+Z b) X² + y² + Z² Exercise 13. Let X₁,..., Xn ~ N(x, o²) i.i.d. and Y₁,..., Yn ~ N(µy, 0²) i.i.d. with known 2. We assume that for each i the random variables X; and Y; are correlated, with Corr(X₁,Y₂) = 1/2. Define D₁ = X₁ - Yį for all i = {1,...,n}. Finally, let X, Y and D be the averages of the Xi, Yi and Di, respectively. a) Show that Cov(X₁, Yi) = 0²/2. b) Determine E(D;) and Var(D₂). c) Consider the test for Ho: x=y which uses the test statistic Z= √nD. σ and which rejects Ho, if and only if |Z| > 1.96. Show that P(type I error) = 5%. d) In two or three sentences, discuss the differences between the test from part (c) on the one hand, and the non-paired test for comparing the means of two populations on the other hand. 2
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