0
$\begingroup$

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?

  • 1
    It's not really the Chernoff bound that applies here but Markov's inequality (http://en.wikipedia.org/wiki/Markov_inequality). The Chernoff bound is for a sum of independent random variables.2011-04-11
  • 0
    He never said that the r.v. is non-negative. In any case, to apply Markov here you need an *upper* bound.2011-04-11
  • 0
    Please see my answer to http://math.stackexchange.com/questions/32457/reviewing-for-exam-chernoff-bounds-for-a-series-of-random-variables. 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.2011-04-12
  • 0
    I would advise some caution when using the answer leif refers to, for reasons explained in the comments.2011-04-12

1 Answers 1

2

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).