Question #1:
Let (X1, X2, ..., Xn) be the random sample of Bernoulli RV X with pmf given by:
f(x,p) = p^x(1-p)^(1-x)
where x = 0,1; and 0 ? p ? 1 is unknown.
Assume that p is uniform RV over (0,1).
Find the Bayes' estimator of p
Question #2:
Let Y = X^2, and X is a uniform RV over (-1,1).
Find the mean square estimator of Y in terms of X and its mean square error.
Question #3:
\hat{Y} is the estimator of Y, given that:
\hat{Y} = g(X) = aX + b
show that the minimum mean square error e_{MSE} = \sigma_Y^2(1 - \rho_{XY}^2),
where \sigma_{XY} = cov(X, Y) and \rho_{XY} is the correlation coefficient of X and Y.
Question #4:
Y = X^2, and let X be a uniform RV over (-1,1),
Find the Linear Mean Square Estimator of Y in terms of X and its mean square error.
