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A number of recent papers have investigated optimum reception of signals transmitted through random channels. Some have been so general as to be primarily of academic interest while others have made use of models which, although simple, are rather poor approximations. The treatment in this paper lies between these two extremes. After formulating the problem in the usual way, the investigation proceeds to the point where it is possible to determine the error probability for binary signals. Next, two particular cases are considered, viz., the case where the variances are equal and the case of exact estimation of parameters. Curves for both of these cases are presented.