GATE 2019 EC – Question 57
A random variable $X$ takes values $-1$ and $+1$ with probabilities 0.2 and 0.8, respectively. It is transmitted across a channel which adds noise $N$, so that the random variable at the channel output is $Y=X+N$. The noise $N$ is independent of $X$, and is uniformly distributed over the interval $[-2,2]$. The receiver makes a decision
$$\hat X=\begin{cases}-1,&\text{if }Y\le\theta\\+1,&\text{if }Y>\theta\end{cases}$$
where the threshold $\theta\in[-1,1]$ is chosen so as to minimize the probability of error $\Pr[\hat X\ne X]$. The minimum probability of error, rounded off to 1 decimal place, is ________.
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Show answer and explanation
Correct answer: 0.09 to 0.11
Explanation
For $X=-1$, $Y$ is uniform on $[-3,1]$ and for $X=+1$ it is uniform on $[-1,3]$, each with density $\frac14$. For $\theta\in[-1,1]$, $P_e=0.2\cdot\frac{1-\theta}{4}+0.8\cdot\frac{\theta+1}{4}=0.25+0.15\theta$, which is smallest at $\theta=-1$, giving $P_e=0.1$.