Past exam of the year 1 of the computer science course of the University of Oxford / 2014 / Discrete mathematics, continuous mathematics and probability / 9 / b / i / Solution

By linearity of expectation and independence,For $\sigma >0$, apply Chebyshev's inequality at threshold $h\sigma$ and take the complementary event: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.