most recent 30 from http://mathoverflow.net 2013-05-20T04:01:31Z http://mathoverflow.net/feeds/question/52563 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/52563/approximating-expectation unradis 2011-01-19T23:50:05Z 2011-01-20T01:30:53Z

<p>if we are given a finite number N of points drawn from a probability distribution, expectation can be approximated as a finite sum over these points: E[f]=(1/N)(summation of f(x) over these N points).</p> <p>comparing this to the actual calculation of E[f]=summation of p(x)f(x), won't the difference between the actual value and approximate value be a lot in cases where p(x) varies a lot?</p>

http://mathoverflow.net/questions/52563/approximating-expectation/52565#52565 Alex R. 2011-01-20T00:19:46Z 2011-01-20T01:30:53Z

<p>The Strong Law of Large Numbers guarantees almost sure convergence of the sample mean to the population mean. If your distribution has large variance then yes the convergence is slower. However, the probability of being away from the population mean is bounded by:</p> <p>$P(|s_n-\mu|>\epsilon)&lt;\frac{\sigma^2}{n\epsilon^2}$</p> <p>Where $\mu$ and $\sigma$ are true mean and standard deviation and $s_n$ is the sample mean from $n$ points.</p>