5. Binary erasure channel (15 points). A binary erasure channel has binary input X e {0, 1} and ternary output Y E {0, 1, 2} (see the probability transition diagram below). The erasure, represented by 2, c0rreSponds to the case when the receiver cannot deter mine whether the received signal is a 0 or a 1. When a 0 is sent, a 0 is received with probability 0.9, i.e., 10le (0|0) = 0.9, and a 2 is received with probability 0.1. When a 1 is sent, a 1 is received with probability 0.8 and a 2 is received with probability 0.2. In answering the following questions, assume that X m Bern(1 / 3). 0 0.9 0 0.1 X 2 Y 0.2 1 0.8 1 (a) Find PX,Y(33: y), Jib/(y): and Ike/(35W) for y = 091:2- (1)) Find the error probability PE, = P{X 75 Y}. (c) Now, suppose we use the MAP rule to estimate X by X E {0, 1}. What is the probability of decoding error, i.e., P{X 7 X}? Compare this probability of error to that without decoding