GATE 2025 EC – Question 41
A source transmits symbol $S$ that takes values uniformly at random from the set $\{-2,0,2\}$. The receiver obtains $Y=S+N$, where $N$ is a zero-mean Gaussian random variable independent of $S$. The receiver uses the maximum likelihood decoder to estimate the transmitted symbol $S$.
Suppose the probability of symbol estimation error $P_e$ is expressed as follows:
$$P_e=\alpha\,P(N>1),$$
where $P(N>1)$ denotes the probability that $N$ exceeds 1.
What is the value of $\alpha$?
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Show answer and explanation
Correct answer: (D) $\frac43$
Explanation
The ML decision thresholds are the midpoints $-1$ and $+1$. For $S=0$ an error occurs if $|N|>1$: probability $2P(N>1)$. For $S=2$ an error occurs if $N<-1$, probability $P(N>1)$ by symmetry; likewise for $S=-2$. So $P_e=\frac13\left[2P(N>1)+P(N>1)+P(N>1)\right]=\frac43P(N>1)$, giving $\alpha=4/3$.