= Solution
By <linearity of expectation> and <independence>,
$$
\mathbb E W=\mu,\qquad\operatorname{Var}(W)=
\frac1{n^2}\sum_{i=1}^n\operatorname{Var}(X_i)=\frac{\sigma^2}{n}.
$$
For $\sigma>0$, apply <Chebyshev's inequality> at threshold $h\sigma$ and take the complementary event:
$$
\boxed{\mathbb P(|W-\mu|\leq h\sigma)
=1-\mathbb P(|W-\mu|>h\sigma)
\geq1-\frac{1}{nh^2}.}
$$
If $\sigma=0$, every $X_i=\mu$ almost surely, so $W=\mu$ almost surely and the bound holds directly. This handles the zero-threshold case without applying <Chebyshev's inequality> at an inadmissible threshold.
Back to article page
