Editing Chernoff bound (section)

== Sums of independent random variables ==
When {{mvar|X}} is the sum of {{mvar|n}} independent random variables {{math|''X''<sub>1</sub>, ..., ''X<sub>n</sub>''}}, the moment generating function of {{mvar|X}} is the product of the individual moment generating functions, giving that:

{{NumBlk|:|<math>\Pr(X \geq a) \leq \inf_{t > 0} \frac{\operatorname E \left [\prod_i e^{t\cdot X_i}\right]}{e^{t\cdot a}} = \inf_{t > 0} e^{-t\cdot a}\prod_i\operatorname E\left[ e^{t\cdot X_i}\right].</math>|{{EquationRef|1}}}}

and:

: <math> \Pr (X \leq a) \leq \inf_{t < 0} e^{-ta} \prod_i \operatorname E \left[e^{t X_i} \right ]</math>

Specific Chernoff bounds are attained by calculating the moment-generating function <math>\operatorname E \left[e^{-t\cdot X_i} \right ]</math> for specific instances of the random variables <math>X_i</math>.

When the random variables are also ''identically distributed'' ([[Independent and identically distributed random variables|iid]]), the Chernoff bound for the sum reduces to a simple rescaling of the single-variable Chernoff bound. That is, the Chernoff bound for the ''average'' of ''n'' iid variables is equivalent to the ''n''th power of the Chernoff bound on a single variable (see [[Cramér's theorem (large deviations)|Cramér's theorem]]).