The First Step in The Proof of Chernoff Bounds
The Chernoff bound for a random variable X, which is the sum of n independent random variables, is obtained by applying etX for some well-chosen value of t. This method was first applied by Sergei Bernstein to prove the related Bernstein inequalities.
From Markov's inequality and using independence we can derive the following useful inequality:
For any t > 0,
In particular optimizing over t and using independence we obtain,
-
(1)
Similarly,
and so,
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