Question

For each of the following statements, explain whether it is true or false. If you think it is true, then prove why you think it is always true. If you believe it is false, then giving a counterexample is sufficient. (a) If A, B are independent events and P(B) > 0, then P(A ? B|B) = P(A ? B). (b) If Var(X1 + X2) = Var(X1) + Var(X2), then X1 and X2 are independent. (c) E[E[X|Z]|Y] = E[X|Z]. (d) Let X1, ..., Xn be a collection of normal (independence is not guaranteed) random variable with mean ?i and variance ?i^2 > 0, and Y = a1X1 + ... + anXn + b be a linear function of Xi's where ai > 0 for all i = 1, ..., n. If Y is normal then Xi's are independent. (e) If the posterior distribution is symmetric then the MAP and LMS estimators coincide.

          For each of the following statements, explain whether it is true or false. If you think it is true, then prove why you think it is always true. If you believe it is false, then giving a counterexample is sufficient.
(a) If A, B are independent events and P(B) > 0, then P(A ? B|B) = P(A ? B).
(b) If Var(X1 + X2) = Var(X1) + Var(X2), then X1 and X2 are independent.
(c) E[E[X|Z]|Y] = E[X|Z].
(d) Let X1, ..., Xn be a collection of normal (independence is not guaranteed) random variable with mean ?i and variance ?i^2 > 0, and Y = a1X1 + ... + anXn + b be a linear function of Xi's where ai > 0 for all i = 1, ..., n. If Y is normal then Xi's are independent.
(e) If the posterior distribution is symmetric then the MAP and LMS estimators coincide.

For each of the following statements, explain whether it is true or false. If you think it is true, then prove why you think it is always true. If you believe it is false, then giving a counterexample is sufficient.
(a) If A, B are independent events and P(B) > 0, then P(A ? B|B) = P(A ? B).
(b) If Var(X1 + X2) = Var(X1) + Var(X2), then X1 and X2 are independent.
(c) E[E[X|Z]|Y] = E[X|Z].
(d) Let X1, ..., Xn be a collection of normal (independence is not guaranteed) random variable with mean ?i and variance ?i^2 > 0, and Y = a1X1 + ... + anXn + b be a linear function of Xi's where ai > 0 for all i = 1, ..., n. If Y is normal then Xi's are independent.
(e) If the posterior distribution is symmetric then the MAP and LMS estimators coincide.

Added by Lawrence F.

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