II- Probability

A binary signal X takes values - 1 and 1 with probabilities 1 / 3 and 2 / 3, respectively. The signal is transmitted over a communication channel and is received as Y = X + Z , where the noise Z is independent of the signal value X and has Laplace pdf (probability density function)

f Z ( z ) = e - 2 |z| , -8 < z < 8

(i) Determine the conditional distribution of Y given each of the possible values of X .

(ii) Give an expression for the (unconditional) probability density function of Y .

(iii) The receiver declares that X = 1 was sent if the received value Y is greater than a threshold t ; and that X = - 1 was sent otherwise. Assuming that - 1 = t = 1, derive an expression for the probability that the receiver decodes Y incorrectly. Denote this error probability by P e ( t ).

(iv) Determine the minimum value of P e ( t ) over all choices of threshold t [ - 1 , +1].