In a binary PCM system, symbols 0 and 1 have a priori probabilities p₀ and p₁, respectively. The conditional probability density function of the random variable Y (with sample value y) obtained by sampling the matched filter output in the receiver of Figure at end of a signaling interval, given that symbol 0 was transmitted, is denoted by ?_Y (y|0). Similarly, ?_Y (y|1) denotes the conditional probability density function of Y, given symbol 1 was transmitted. Let ? denote the threshold used in the receiver, so that if the sample value y exceeds ?, the receiver decides in favor of symbol 1; otherwise, it decide in favor of symbol 0. Show that the optimum threshold ?_opi, for which the average probability of error is a minimum, is given by the solution of

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