Taylor approximation of a function of a random variable - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T03:28:30Z http://mathoverflow.net/feeds/question/73015 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/73015/taylor-approximation-of-a-function-of-a-random-variable Taylor approximation of a function of a random variable madison54 2011-08-16T21:36:12Z 2011-08-16T21:36:12Z <p>Suppose we have a random variable \$X\$ and a smooth function \$g\$. We want to calculate the expectation value \$\mathbb{E}(g(X))\$. To be able to write down at least an approximate solution, we perform a Taylor expansion \$g_T\$ of \$g\$ up to second order around the mean of \$X\$ and use \$\mathbb{E}(g_T(X))\$ as the approximation.</p> <p>Now, what would be a reasonable estimate of how good this approximation is? Or an estimate that tells me for which \$X\$ the truncation after second order is justified? </p> <p>Whereas for a real valued function this is clear, in this case I am looking for an adequate statistical dispersion measure. I have simulation data, so can I could actually calculate the measure and then want to make an appropriate approximation to get an analytical insight into my system.</p>