November 2012
Intermediate to advanced
794 pages
22h 9m
English
book
Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications
by John J. Shynk
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F.4 CHERNOFF'S INEQUALITY
Theorem F.9 (Chernoff's inequality). For random variable X and every 
:
(F.26) 
where 
is the moment generating function of X and 
.
Proof. From the definition of the moment generating function:
(F.27) 
The first inequality is obtained because 
, and the second inequality follows because the smallest value 
has replaced exp(tx) in the integrand. Rearranging the final expression completes the proof.
Since Chernoff's bound must hold for all 
, the tightest bound is
(F.28) 
which depends on δ and the parameters of the specific moment generating function.
Example F.2. Consider ...
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