Let $X$ be a random variable which takes values $1$ and $-1$ with probability $1 / 2$ each. Suppose $Y=X+N$, where $N$ is a random variable independent of $X$ with the following probability density function (p.d.f.):
$$f_{N}(n)=\left\{\begin{array}{ll}
c\left(1-\frac{1}{2} n\right) & 0 \leq n \leq 2 \\
c\left(1+\frac{1}{2} n\right) & -2 \leq n<0 \\
0 & \text { otherwise }
\end{array}\right.$$
where $c$ is such that the above is a p.d.f. Now consider a "detector" which tries to guess $X$ based on observing $Y$. Suppose the detector makes a decision
$$\hat{X}=\left\{\begin{aligned}
1 & \text { if } Y \geq \lambda \\
-1 & \text { if } Y<\lambda
\end{aligned}\right.$$
where $\lambda$ is chosen such that the probability of making an incorrect decision, i.e., $\operatorname{Pr}(\hat{X} \neq X)$, is minimized. What is this minimum probability of incorrect decision?
- $0$
- $1 / 8$
- $1 / 4$
- $1 / 2$
- None of the above