Probability Inequalities for Sums of Bounded Random Variables

Abstract

If S is a random variable with finite rnean and variance, the Bienaymé-Chebyshev inequality states that for x > 0,

$$\Pr \left[ {\left| {S - ES} \right| \geqslant x{{{(\operatorname{var} S)}}^{{1/2}}}} \right] \leqslant {{x}^{{ - 2}}}$$

((1))

If S is the surn of n independent, identically distributed random variables, then, by the central limit theorem*, as n → ∞, the probability on the left approaehes 2Ф( - x), where Ф(x) is the standard normal distribution function. For x large, Ф( - x) behaves as const. x ^-1 exp( - ^x2/2).