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Estiation Theory

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Let the random variables [x11, ...., xN] satisfy the model
xN = theta + w_n; n = 1, ...., N

where {w_m} is i.i.d. zero-mean Gaussian of unit variance. Assume now that you cannot access the sequence [xN] but instead you observe the following function of xN

yN = sign(xN - T); n = 1, ...., N,

where T a is a known constant. Using [y1,..., yN} compute th Maximum Likelihood Estimator (MLE) of theta.

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This solution uses estiation theory to solve the problems.

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Gaussian probability density of unit variance is:

p(w_n)dw_n = 1/(sqrt(2pi)) e^(-(w_n)^2 / 2) dw_n,

We see that y_n can take only two possible values:
One is y_n = +1, when x_n > y, and the probability of that event is

p_+ = 1/(sqrt(2pi)) Integral (w^2/e^2 dw)

Another is y_n = -1, ...

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