GATE EC 2014 Set 4 Q49 — Probability and Statistics

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