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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