TIFR ECE 2017 | Question: 12

Question

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.

• A. Only $\text{(i)}$
• B. Only $\text{(ii)}$
• C. Only $\text{(iii)}$
• D. Only $\text{(i)}$ and $\text{(ii)}$
• E. Only $\text{(i)}$ and $\text{(iii)}$