= Solution
The probability generating function of each $X_i$ is $\exp(z-1)$. Independence makes the generating function of $S_n$ equal to $\exp(n(z-1))$, so the <addition of independent Poisson random variables> gives $S_n\sim\operatorname{Pois}(n)$.
The <Poisson distribution> has mean and variance $n$, hence $\mathbb E Y_n=0$ and $\mathbb E[Y_n^2]=1$. Since $Y_n^-\leq|Y_n|$, <Chebyshev inequality> gives
$$
\mathbb P(Y_n^-\geq a)
\leq\mathbb P(|Y_n|\geq a)
\leq\frac1{a^2}.
$$
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