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(20 Points) Let X and Y be the input and output. respectively. of a binary channel. The channel response changes randomly based on the value of a random variable R that is independent of X and has the following pmf: Probl = 0) = pa, Prob(R = 1) = p1, Prob(R = 2) = p2; (pa + pl + pi; = 1) The channel between X and Y is a BSC with a variable error probability a, such that 0, ifR=[} 0:: 1X4, ifR=1 1/2, ifR=2 In other words, the channel is described by: a If R = 0, then 1' = X with probability 1 I I\"? = 1, then ProbO' = 1|X = 1) = Prob' = OIX = 0) = 3/4 and Prob[Y =1IX = 0) = Prob(Y = DIX =1): 1{4 I I\"? = 2. then ProbO' = 1|X = 1) = Prob' = {JIX = 0) = III! and Prob[Y =1|X = 0) = ProMY = DIX =1): 1/2 a) (10 points) Find the capacity of this channel if neither the transmitter nor the receiver has access to the value of the random variable R. b) (10 points) Now suppose that the encoder still has no access to the value of the random variable R, but the receiver knows R for each transmission. In this case the channel can be thought of as having two outputs. Y and R. Find the capacity of the channel in this case by maximizing I (X ; Y, R)