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I have some random variable with an expected value of $n/8$. I want to use Chernoff bounds to show that with some high probability, the actual value is at least $(1 - \epsilon)n/24$. How could I go about this?

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It's not really the Chernoff bound that applies here but Markov's inequality ( The Chernoff bound is for a sum of independent random variables. – Akhil Mathew Apr 11 '11 at 17:51
He never said that the r.v. is non-negative. In any case, to apply Markov here you need an upper bound. – Yuval Filmus Apr 11 '11 at 19:00
Please see my answer to…. It's about the same, with different numbers. You do need to know, however, that your random variable is a sum of 0/1 Poisson trials which are either independent or negatively correlated. – leif Apr 12 '11 at 5:50
I would advise some caution when using the answer leif refers to, for reasons explained in the comments. – Did Apr 12 '11 at 13:07

1 Answer 1

The result isn't true. Take the random variable which is $0$ w.p. $1-\epsilon$ and $n/8\epsilon$ w.p. $\epsilon$. Its expectation is $n/8$ yet it is not concentrated at all around its expectation.

In order to use a concentration bound you need to know more about your random variable, e.g. its variance (then you can use Chebyshev), boundedness (then you can use Markov) or it being a sum of a lot of "small" variables (then you can use Chernoff).

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