, and is invariant under translation: . The Chernoff bound is exact if only if is a single concentrated mass (degenerate distribution). The bound is tight only at or beyond the extremes of a bounded random variable, where the infima are attained for infinite . For unbounded random variables the bound is nowhere tight, though it is asymptotically tight up to sub-exponential factors ("exponentially tight"). Individual moments can provide tighter bounds, at the cost of greater analytical complexity. In practice, the exact Chernoff bound may be unwieldy or difficult to evaluate analytically, in which case a suitable upper bound on the moment (or cumulant) generating function may be used instead (e.g. a sub-parabolic CGF giving a sub-Gaussian Chernoff bound). Sums of independent random variables When is the sum of independent random variables , the moment generating function of is the product of the individual moment generating functions, giving that: and: Specific Chernoff bounds are attained by calculating the moment-generating function for specific instances of the random variables . When the random variables are also identically distributed (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 nth power of the Chernoff bound on a single variable (see Cramér's theorem). Sums of independent bounded random variables 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 are independent random variables taking values in Let denote their sum and let denote the sum's expected value. Then for any , Sums of independent Bernoulli random variables The bounds in the following sections for Bernoulli random variables are derived by using that, for a Bernoulli random variable with probability p of being equal to 1, One can encounter many flavors of Chernoff bounds: the original additive form (which gives a bound on the absolute error) or the more practical multiplicative form (which bounds the error relative to the mean). Multiplicative form (relative error) Multiplicative Chernoff bound. Suppose are independent random variables taking values in Let denote their sum and let denote the sum's expected value. Then for any , A