6. A certain transmission system has 2 transmission modes. In Mode 1, the transmitted signal X is equally likely to be +1 or 1. In Mode 2, the transmitted signal X is 0. The 2 transmission modes are equally likely to be used. Note that X 6 {1,0, +1}. Let the received (observed) signal be Y = X + Z where the noise Z N U (1, +1) is independent of X. (a) Determine the conditional pdf fy'MCylM : 1), i.e. the conditional pdf of Y given Mode 1. (b) Determine the conditional pdf fY'M(y|M : 2), i.e. the conditional pdf of Y given Mode 2. (c) Determine the optimal (i.e. minimum probability of error) decoding rule for choosing the mode of the system, i.e. for a given value 3;, specify whether we should choose Mode 1 or Mode 2. Your answer should take the following form: choose Mode 1 if y is a value in this (or these) interval(s); otherwise, choose Mode 2. (d) Determine the probability of error for the decoder in Part (c)