The Chernoff bound is an exponential upper bound on the tail probability of a real random
variable 
. Let ![D_+={t>0:E[e^(tX)]<infty}](/images/equations/ChernoffBound/Inline2.svg)
be nonempty. For any real
number 
, Markov's inequality
applied to 
gives
![P(X>=a)<=inf_(t in D_+){e^(-ta)E[e^(tX)]},](/images/equations/ChernoffBound/NumberedEquation1.svg) |
where ![E[Y]](/images/equations/ChernoffBound/Inline5.svg)
denotes the expectation value of 
(Durrett 2019, p. 106). Taking the infimum
gives the best bound within this family, even if no minimizing parameter exists.
Only a finite positive exponential moment is needed. Finiteness on a two-sided neighborhood of zero, used for existence of a moment-generating
function, is a stronger requirement.
Applying the same argument to 
gives a lower tail probability
bound when the corresponding negative exponential moment is finite. For a sum
of independent random variables with the
required exponential moments finite, the exponential expectation
value factors into a product. If the variables have
finite expectation values, the same argument
applied to their centered versions bounds deviations from the expectation
value of the sum.
