GATE EC 2025 Set 1 Q31 — Probability and Statistics

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 $\alpha$ ?