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Consider a communication scheme where the binary valued signal $X$ satisfies $P\{X=+1\}=0.75$ and $P\{X=-1 \}=0.25$. The received signal $Y=X+Z$, where $Z$ is a Gaussian random variable with zero mean and variance $\sigma ^2$. The received signal $Y$ is fed to the threshold detector. The output of the threshold detector $\hat{X}$ is: $$ \hat{X} = \begin{cases} +1, & Y>\tau \\ -1, &Y \leq \tau \end{cases}.$$ To achieve a minimum probability of error $P\{\hat{X} \neq X \}$, the threshold $\tau$ should be

  1. strictly positive
  2. zero
  3. strictly negative
  4. strictly positive, zero, or strictly negative depending on the nonzero value of $\sigma ^2$
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