Chernoff's inequality

In probability theory, Chernoff's inequality, named after Herman Chernoff, states the following. Let


 * $$X_1,X_2,\dots,X_n$$

be independent random variables, such that


 * $$E[X_i]=0 \,$$

and


 * $$\left|X_i\right|\leq 1\, $$ for all i.

Let


 * $$X=\sum_{i=1}^n X_i$$

and let &sigma;2 be the variance of X. Then


 * $$P(\left|X\right|\geq k\sigma)\leq 2e^{-k^2/4}$$

for any


 * $$0 \leq k \leq 2 \sigma.\,$$