Harmonic mean of random variables - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T23:05:39Z http://mathoverflow.net/feeds/question/6112 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/6112/harmonic-mean-of-random-variables Harmonic mean of random variables John D. Cook 2009-11-19T15:30:37Z 2009-11-29T03:54:51Z <p>The arithmetic mean of normal random variables is normal. The geometric mean of log-normal random variables is log-normal. But is there a common distribution family closed under taking harmonic means?</p> http://mathoverflow.net/questions/6112/harmonic-mean-of-random-variables/6115#6115 Answer by Michael Lugo for Harmonic mean of random variables Michael Lugo 2009-11-19T15:44:24Z 2009-11-19T15:44:24Z <p>Perhaps variables of the form 1/X, where X is normal? (We can ignore the problem of division by zero because X is zero only with probability zero.) Of course this isn't exactly <i>common</i>. </p> http://mathoverflow.net/questions/6112/harmonic-mean-of-random-variables/6133#6133 Answer by John D. Cook for Harmonic mean of random variables John D. Cook 2009-11-19T17:02:17Z 2009-11-19T17:02:17Z <p>It just occurred to me that since Cauchy random variables are closed under reciprocals and closed under sums, they're closed under harmonic means.</p> http://mathoverflow.net/questions/6112/harmonic-mean-of-random-variables/6259#6259 Answer by Mark Meckes for Harmonic mean of random variables Mark Meckes 2009-11-20T13:02:24Z 2009-11-20T13:02:24Z <p>Any class of distributions which is closed under independent sums and almost surely nonzero will work here (and of course will also give an example for geometric means corresponding to the log-normal) the same way as in Michael's example. So besides normal (p=2) and Cauchy (p=1) random variables, the reciprocals of any p-stable random variables work. Of course, only Cauchy have the property of being distributed the same as their reciprocals, so John's answer doesn't generalize this far.</p> http://mathoverflow.net/questions/6112/harmonic-mean-of-random-variables/7108#7108 Answer by omer angel for Harmonic mean of random variables omer angel 2009-11-29T03:54:51Z 2009-11-29T03:54:51Z <p>In general you want to take \$X=1/Y\$, where \$Y\$ is a stable random variable. A surprising example is \$X=N(0,1)^2\$, since \$N(0,1)^{-2}\$ is stable of index \$1/2\$.</p>