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)].) ~
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Chapter1: Combinatorial Analysis
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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)].)
~](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8e7463e4-4687-4985-b43b-49d313d2d2f5%2F9dfaa81e-d134-49bd-ac58-33e91228c500%2Fw82rgx_processed.png&w=3840&q=75)
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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