Suppose you had random variables $X_1, X_2…X_n$ that are iid and uniformly distributed on $[-1,1]$. I am trying to find an upper bound on the probability that the sum $S_n = X_1 + X_2 … + X_n$ exceeds some value. By Chebyshev's inequality,
$P(|S_n| \geq n\sigma_{S_n}) < 1/n^2$
However, I am interested in $P(S_n \geq value)$, not $P(|S_n|\geq value)$. Expanding the previous expression doesn't allow me to clearly separate out $P(S_n\geq value)$:
$P(|S_n| \geq value) = P(S_n \geq value)+P(S_n\leq -value) $
Is there something obvious to do next that I am missing?
Best Answer
Use Chernoff or Hoeffding here, they will be tighter than Chebyshev. And things are symmetric, so $\mathbb{P}\{ S_n \geq x\} = \mathbb{P}\{ S_n \leq -x\}$: $$ \forall x\in [0,1], \qquad \mathbb{P}\{ S_n \geq x\} = \frac{1}{2}\mathbb{P}\{ \lvert S_n\rvert \geq x\} $$