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Problem 3.3 Let X, Y, Z be random variables with finite second moments and suppose that X is to be estimated. For each of the following, if true, give a brief explanation. If false, give a counter example. Note that the notation E[X|Y] refers to the Linear Least Mean Square estimator, as opposed to E[X|Y], which is the conditional mean. (a) E[(X – E[X|Y])²] ≤ E[(X – Ê[X|Y, Y²])²]. - (b) E[(X – E[X|Y])²] = E[(X — Ê[X|Y, Y²])²] if X and Y are jointly Gaussian. (c) E[(X – E[E[X|Z]|Y])²] ≤ E[(X – E[X|Y])²]. (d) If E[(X – E[X|Y])²] = Var(X), then X and Y are independent.

Problem 3.3 Let X, Y, Z be random variables with finite second moments and suppose that X is to be estimated. For each of the following, if true, give a brief explanation. If false, give a counter example. Note that the notation E[X|Y] refers to the Linear Least Mean Square estimator, as opposed to E[X|Y], which is the conditional mean. (a) E[(X – E[X|Y])²] ≤ E[(X – Ê[X|Y, Y²])²]. - (b) E[(X – E[X|Y])²] = E[(X — Ê[X|Y, Y²])²] if X and Y are jointly Gaussian. (c) E[(X – E[E[X|Z]|Y])²] ≤ E[(X – E[X|Y])²]. (d) If E[(X – E[X|Y])²] = Var(X), then X and Y are independent.

Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018
Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018
18th Edition
ISBN:9780079039897
Author:Carter
Publisher:Carter
Chapter10: Statistics
Section10.1: Measures Of Center
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Problem 3.3 Let X, Y, Z be random variables with finite second moments and suppose that X is to be estimated. For each of the following, if true, give a brief explanation. If false, give a counter example. Note that the notation E[X|Y] refers to the Linear Least Mean Square estimator, as opposed to E[X|Y], which is the conditional mean. (a) E[(X – E[X|Y])²] < E[(X – Ê[X|Y,Y²))²]. - (b) E[(X – E[X|Y])°] = E[(X – Ê[X|Y,Y²))²] if X and Y are jointly Gaussian. (c) E[(X – E[E[X|Z]|Y])*] < E[(X – E[X|Y])°J. (d) If E[(X – E[X|Y])²] = Var(X), then X and Y are independent.
Transcribed Image Text:Problem 3.3 Let X, Y, Z be random variables with finite second moments and suppose that X is to be estimated. For each of the following, if true, give a brief explanation. If false, give a counter example. Note that the notation E[X|Y] refers to the Linear Least Mean Square estimator, as opposed to E[X|Y], which is the conditional mean. (a) E[(X – E[X|Y])²] < E[(X – Ê[X|Y,Y²))²]. - (b) E[(X – E[X|Y])°] = E[(X – Ê[X|Y,Y²))²] if X and Y are jointly Gaussian. (c) E[(X – E[E[X|Z]|Y])*] < E[(X – E[X|Y])°J. (d) If E[(X – E[X|Y])²] = Var(X), then X and Y are independent.
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