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Consider a signal $X$ that can take two values, $-1$ with probability $p$ and $+1$ with probability $1-p$. Let $Y=X+N$, where $N$ is mean zero random noise that has probability density function symmetric about $0.$ Given $p$ and on observing $Y$, the detection problem is to decide on a value for $X$ from $-1$ and $+1$. Let $\hat{X}$ denote the decision, then error is said to happen if $\hat{X}$ is not the true $X$. Consider the following statements about the optimal detector that minimizes the probability of error.

  1. If $p=1 / 2$, then choosing $\hat{X}=+1$ if $Y>0$ and $\hat{X}=-1$ if $Y<0$ minimizes the probability of error.
  2. The probability of error of the optimal detector for $p=1 / 3$ is larger in comparison to the probability of error of the optimal detector for $p=1 / 2$.
  3. If $p=0$, then choosing $\hat{X}=+1$ for any $Y$ minimizes the probability of error.

Which of the above statements is/are $\text{TRUE?}$ Choose from the following options.

  1. Only $\text{(i)}$
  2. Only $\text{(ii)}$
  3. Only $\text{(iii)}$
  4. Only $\text{(i)}$ and $\text{(ii)}$
  5. Only $\text{(i)}$ and $\text{(iii)}$
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