Let X, Y, Z be random variables with finite second moments and suppose that X is to beestimated. For each of the following, if true, give a brief explanation. If false, give a counter example. Notethat the notation Ê̟X\Y] refers to the Linear Least Mean Square estimator, as opposed to E[X|Y], whichis the conditional mean.(a) E((X – E\X|Y])ª] < E[(X – Ê{X|Y, Y²l)°].(b) E[(X – E[X|Y])ª] = E[(X – Ê{X|Y, Y²)ª] if X and Y are jointly Gaussian.(c) E[(X – E\E[X\Z]|Y])°] < E(X – E[X|Y])*].(d) If E[(X – E[X|Y])*] = Var(X), then X and Y are independent.

The given problem has random variables X, Y, and Z with finite second moments. The task is to…

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-EX Y])²] ≤ E[(X - Ê[XY, Y²])²]. (b) E(X - E[X|Y])²] = E(X-Ê[X|Y, Y²])²] if X and Y are jointly Gaussian. (e) E(X-ELE[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.

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-EX Y])²] ≤ E[(X - Ê[XY, Y²])²]. (b) E(X - E[X|Y])²] = E(X-Ê[X|Y, Y²])²] if X and Y are jointly Gaussian. (e) E(X-ELE[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.

Calculus For The Life Sciences

2nd Edition

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Chapter1: Functions

Section1.2: The Least Square Line

Problem 5E

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