Suppose that X-N(0,1). Let Y be independent of X with P(Y = 1) = P(Y = -1) = 1/2. Define the random variable Z be setting Z = XY. a) Compute Cov(X,Z). b) Show that P(Z>= 1) = P(X>= 1). Use this fact to conclude that Z and X are NOT independent. c) Generalize part (b) to show that P(Z >= x) = P(X >= x) for every xeR. This implies that Z~N(0,1)

Suppose that X-N(0,1). Let Y be independent of X with P(Y = 1) = P(Y = -1) = 1/2. Define the random variable Z be setting Z = XY. a) Compute Cov(X,Z). b) Show that P(Z>= 1) = P(X>= 1). Use this fact to conclude that Z and X are NOT independent. c) Generalize part (b) to show that P(Z >= x) = P(X >= x) for every xeR. This implies that Z~N(0,1)

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter5: Orthogonality

Section5.3: The Gram-schmidt Process And The Qr Factorization

Problem 11AEXP

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Question

Suppose that X-N (0,1). Let Y be independent of X with P(Y = 1) = P(Y = -1) = 1/2.
Define the random variable Z be setting Z = XY.
a) Compute Cov(X,Z).
b) Show that P(Z >= 1) = P(X >= 1). Use this fact to conclude that Z and X are NOT independent.
c) Generalize part (b) to show that P(Z >= x) = P(X >= x) for every xeR. This implies that Z~N(0,1)

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