Chernoff Bounds for Independent Bernoulli Sums

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Thread starter WMDhamnekar
Start date Jul 20, 2023

In summary, Chernoff Bounds for Independent Bernoulli Sums are probabilistic inequalities that provide upper and lower bounds for the probability of deviation from expected value. They are important for estimating rare events and analyzing algorithms in computer science and engineering. These bounds are calculated using the moment generating function, and can be applied to non-Bernoulli distributions. However, they are only approximations and may be inaccurate for large systems.

Jul 20, 2023

WMDhamnekar

MHB

TL;DR Summary: What is wrong with this proof? Can you notice that? or I am wrong. In my opinion, in the R.H.S. of inequality (3.2), the index of 'e' must be positive if we use the proof. I also want to know how to derive the proof of inequality(3.3)? Author said it is similar to that of (3.2). But I don't understand that.

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Jul 21, 2023

WMDhamnekar

MHB

I cleared my doubt taking suitable guidelines from other statistician on Internet.

What are Chernoff Bounds for Independent Bernoulli Sums?

Chernoff Bounds for Independent Bernoulli Sums are a mathematical tool used to estimate the probability of the sum of independent Bernoulli random variables exceeding a certain threshold.

How are Chernoff Bounds calculated?

Chernoff Bounds are calculated using the moment generating function (MGF) of the sum of independent Bernoulli random variables. The MGF is then used to determine the upper and lower bounds for the probability of the sum exceeding a certain threshold.

What is the significance of Chernoff Bounds in probability theory?

Chernoff Bounds are significant in probability theory because they provide a way to estimate the probability of rare events occurring. They are also useful in analyzing the performance of algorithms and systems that involve the sum of independent Bernoulli random variables.

What are the assumptions for using Chernoff Bounds?

The main assumption for using Chernoff Bounds is that the random variables are independent and identically distributed (i.i.d.). Additionally, the MGF of the sum of the random variables must exist and be finite in a certain interval.

Can Chernoff Bounds be used for non-Bernoulli random variables?

While Chernoff Bounds are specifically designed for independent Bernoulli random variables, they can also be applied to other types of random variables with some modifications. However, the results may not be as accurate as when used for Bernoulli random variables.