Consider a communication scheme where the binary values signal X satisfies P{X = 1} = 0.75 and P{X = -1} = 0.25. The received signal Y = X + Z, where Z is Gaussian random variable with zero mean and variance σ². The received signal Y is fed to the threshold detector. The output of the threshold detector \[{{\rm{\hat X}}}\] is
\[{\rm{\hat X}} = \left\{ \begin{array}{l} + 1,\,\,\,{\rm{Y}} > \tau \\ - 1,\,\,\,{\rm{Y}} \le \tau \end{array} \right.\]
To achieve a minimum probability of error \[{\rm{P}}\left\{ {{\rm{\hat X}} \ne {\rm{X}}} \right\}{\rm{,}}\]   the threshold τ should be