Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Average of function vs. function evaluated at average

  1. Aug 22, 2011 #1
    Dear everyone,
    consider a function f(x) of a random variable x with average m and probability distribution p(x).

    I would like to know under which conditions the average of f(x) is greater than f(m), i.e., under which conditions is the average of the function greater than the function evaluated at the average value of the random variable.

    \int p(x) f(x) dx > f(m) ?

    Does anyone know about theorems that might help me? I suppose it depends on the concaveness of the function f and shape of the distribution p.

    I would be very grateful for help.
  2. jcsd
  3. Aug 22, 2011 #2
    If the function is concave, it does not depend on tha shape of p. See http://en.wikipedia.org/wiki/Jensen%27s_inequality" [Broken]
    Last edited by a moderator: May 5, 2017
  4. Aug 23, 2011 #3
    In fact, that property is often used as the very definition of concave, convex, and linear.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook