Chernoff bound

Chernoff bounds provide exponentially tight concentration inequalities

Chernoff bounds are exponentially decreasing upper bounds on the tail of a random variable, offering tighter concentration inequalities compared to moment-based tail bounds like Markov's or Chebyshev's inequalities.

The Chernoff bound is particularly useful for sums of independent random variables, such as Bernoulli random variables, and it requires no independence condition like Markov's or Chebyshev's inequalities.

Chernoff bounds are related to Bernstein inequalities and are used to prove other inequalities like Hoeffding's, Bennett's, and McDiarmid's inequalities.

Example

Consider a sum of 100 independent Bernoulli random variables with success probability p = 0.5. Using Chernoff bounds, we can tightly estimate the probability that the sum deviates significantly from its expected value.

Remember this

Understanding Chernoff bounds is crucial for accurately estimating probabilities in scenarios involving sums of independent random variables, leading to more precise predictions and risk assessments.

Text adapted from Wikipedia, licensed under CC BY-SA 4.0.

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