Exercise 2 pls Exercise 12. Let X, Y, Z ~ N(0,1) be independent. Determine the distribution, expectationand variance of the following random variables:a) X+Y+Zb) x² + y² + Z²Exercise 13. Let X₁,..., X₁~ N(2,02) i.i.d. and Y₁,...,Y₁ ~ N(Hy, ²) i.i.d. with known o².We assume that for each i the random variables X, and Y; are correlated, with Corr(X₁,Y)= 1/2.Define D₂ = X₁ - Y; for all i € {1,...,n}. Finally, let X, Y and D be the averages of the X₁, Y;and D₁, respectively.a) Show that Cov(X₁,Y₁) = 0²/2.b) Determine E(D;) and Var(D₂).c) Consider the test for Ho: z = Hy which uses the test statisticZ=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 onehand, and the non-paired test for comparing the means of two populations on the otherhand.2Exercise 14. Let X₁, X2, X3 be independent Bernoulli trials, i.e. the X, are i.i.d. random variableswith P(X; = 1) = p and P(X; = 0) = 1-p. We want to make inference about p, from observations#1, #2, #3 € {0, 1}.Assume we already know that either p = 0.3 or p = 0.8, and other values of p are not possible.In this case we want to decide whether the hypothesis Ho: p = 0.3 or the alternative H₁: p = 0.8is more likely. Consider the test which rejects Ho if and only if s = ₁1₁ ≥ 2. For this test,determine the probabilities of type I and type II errors.

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Answered: Exercise 12. Let X, Y, Z N(0,1) be…

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

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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Question

Exercise 2 pls

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,..., X₁~ N(2,02) i.i.d. and Y₁,...,Y₁ ~ N(Hy, ²) i.i.d. with known o².
We assume that for each i the random variables X, and Y; are correlated, with Corr(X₁,Y)= 1/2.
Define D₂ = X₁ - Y; for all i € {1,...,n}. Finally, let X, Y and D be the averages of the X₁, Y;
and D₁, respectively.
a) Show that Cov(X₁,Y₁) = 0²/2.
b) Determine E(D;) and Var(D₂).
c) Consider the test for Ho: z = Hy which uses the test statistic
Z=
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
Exercise 14. Let X₁, X2, X3 be independent Bernoulli trials, i.e. the X, are i.i.d. random variables
with P(X; = 1) = p and P(X; = 0) = 1-p. We want to make inference about p, from observations
#1, #2, #3 € {0, 1}.
Assume we already know that either p = 0.3 or p = 0.8, and other values of p are not possible.
In this case we want to decide whether the hypothesis Ho: p = 0.3 or the alternative H₁: p = 0.8
is more likely. Consider the test which rejects Ho if and only if s = ₁1₁ ≥ 2. For this test,
determine the probabilities of type I and type II errors.

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