Let X1,X2,...,XkX_1,X_2,...,X_k be independent random variables with each Xi∈[ai,bi]X_i \in [a_i,b_i]. Let μi=𝔼[Xi]\mu_i=\mathbb{E}[X_i] and μ=∑iμi\mu=\sum_i \mu_i. Then, for any α>0\alpha > 0, S=∑iXiS = \sum_i X_i satisfies: Pr[|S−μ|>α]≤2e−α2∑i=1k(bi−ai)2\mathrm{Pr}[|S-\mu|>\alpha] \leq 2e^{-\frac{\alpha^2}{\sum_{i=1}^k (b_i-a_i)^2}}
Example of Concentration inequality.