Question: 3. Useless Binary Symmetric Channel. Let X ~ Bern(p), p (0,1), be the input to a binary symmetric channel with flipping probability , and let

3. Useless Binary Symmetric Channel. Let X ~

3. Useless Binary Symmetric Channel. Let X ~ Bern(p), p (0,1), be the input to a binary symmetric channel with flipping probability , and let y be the output. That is, px \x(y\x) = Var, y (0,1). The input-output relationship is shown in the figure below: 1/2 0 0 1/2 X Y 1/2 1 1 1/2 (a) (10 points) Carefully show that XIY. (b) (6 points) What is the MAP decoding rule in this case? (C) (4 points) What is the corresponding probability of error (after using MAP)? 3. Useless Binary Symmetric Channel. Let X ~ Bern(p), p (0,1), be the input to a binary symmetric channel with flipping probability , and let y be the output. That is, px \x(y\x) = Var, y (0,1). The input-output relationship is shown in the figure below: 1/2 0 0 1/2 X Y 1/2 1 1 1/2 (a) (10 points) Carefully show that XIY. (b) (6 points) What is the MAP decoding rule in this case? (C) (4 points) What is the corresponding probability of error (after using MAP)

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