Editing Chernoff bound (section)

== Sums of independent bounded random variables ==
{{main|Hoeffding's inequality}}

Chernoff bounds may also be applied to general sums of independent, bounded random variables, regardless of their distribution; this is known as [[Hoeffding's inequality]]. The proof follows a similar approach to the other Chernoff bounds, but applying [[Hoeffding's lemma]] to bound the moment generating functions (see [[Hoeffding's inequality]]).  
:'''[[Hoeffding's inequality]].''' Suppose {{math|''X''<sub>1</sub>, ..., ''X<sub>n</sub>''}} are [[Statistical independence|independent]] random variables taking values in {{math|[a,b].}} Let {{mvar|X}} denote their sum and let {{math|''μ'' {{=}} E[''X'']}} denote the sum's expected value. Then for any <math>t>0</math>,
::<math>\Pr (X \le \mu-t) < e^{-2t^2/(n(b-a)^2)},</math>
::<math>\Pr (X \ge \mu+t) < e^{-2t^2/(n(b-a)^2)}.</math>