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5. We know that the MMSE of Y given X is given by g(X) = E[Y|X]. We also know that the LLSE (Linear Least Square Estimate) of Y given X, denoted by L[Y|X], is shown as follows: Cov(X, Y) Var (X) (X - E(X)). L[Y|X] = E(Y) + Now we wish to estimate the probability of landing heads, denoted by 0, of a biased coin. We model as the value of a random variable with a known prior with PDF fe Unif(0, 1).. We consider n independent tosses and let X be the number of heads observed. (a) (b) Show that E[(- E[O[X])h(X)] = 0 for any real function h(.). Find the MMSE E[X] and the LLSE L[X]. (Eve's law: Var(Y) = E[Var(Y|X)] + Var[E(Y|X)].) ~

5. We know that the MMSE of Y given X is given by g(X) = E[Y|X]. We also know that the LLSE (Linear Least Square Estimate) of Y given X, denoted by L[Y|X], is shown as follows: Cov(X, Y) Var (X) (X - E(X)). L[Y|X] = E(Y) + Now we wish to estimate the probability of landing heads, denoted by 0, of a biased coin. We model as the value of a random variable with a known prior with PDF fe Unif(0, 1).. We consider n independent tosses and let X be the number of heads observed. (a) (b) Show that E[(- E[O[X])h(X)] = 0 for any real function h(.). Find the MMSE E[X] and the LLSE L[X]. (Eve's law: Var(Y) = E[Var(Y|X)] + Var[E(Y|X)].) ~

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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5. We know that the MMSE of Y given X is given by g(X) = E[Y|X]. We also know that the LLSE (Linear Least Square Estimate) of Y given X, denoted by L[Y|X], is shown as follows: Cov(X, Y) Var (X) (X - E(X)). L[Y|X] = E(Y) + Now we wish to estimate the probability of landing heads, denoted by 0, of a biased coin. We model as the value of a random variable with a known prior with PDF fe Unif(0, 1).. We consider n independent tosses and let X be the number of heads observed. (a) (b) Show that E[(- E[O[X])h(X)] = 0 for any real function h(.). Find the MMSE E[X] and the LLSE L[X]. (Eve's law: Var(Y) = E[Var(Y|X)] + Var[E(Y|X)].) ~
Transcribed Image Text:5. We know that the MMSE of Y given X is given by g(X) = E[Y|X]. We also know that the LLSE (Linear Least Square Estimate) of Y given X, denoted by L[Y|X], is shown as follows: Cov(X, Y) Var (X) (X - E(X)). L[Y|X] = E(Y) + Now we wish to estimate the probability of landing heads, denoted by 0, of a biased coin. We model as the value of a random variable with a known prior with PDF fe Unif(0, 1).. We consider n independent tosses and let X be the number of heads observed. (a) (b) Show that E[(- E[O[X])h(X)] = 0 for any real function h(.). Find the MMSE E[X] and the LLSE L[X]. (Eve's law: Var(Y) = E[Var(Y|X)] + Var[E(Y|X)].) ~
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