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Assume that we have a continuous random variable $X$ with bounded support $[a,b]$ such that $0 and two implicit functions of $X$, $g:R^+ \rightarrow R^+$ and $g:R^+ \rightarrow R^+$. We don't have the closed form solutions for the functions but only know their domain and range information. Are there any conditions over these functions or the random variable $X$ that can be specified such that $E[\frac{1}{g(X)^2}] \leq E[\frac{1}{h(X)^2}]$ would imply $E[g(X)] \geq E[h(X)]$?

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Here are some conditions... not very interesting ones.

  1. $X$ attains only one value.
  2. $g\ge h$ everywhere.
  3. $g\le h$ everywhere.

Only the third one merits an explanation: if $g\le h$ and $E[g(X)^{-2}]\le E[h(X)^{-2}]$, then the essential range of $X$ is contained in the set $g=h$, which implies $E[g(X)]=E[h(X)]$.

If you allow $X$ to attain two values, one can construct two functions $g,h\colon [0,\infty)\to [0,\infty)$ that will break the implication. (Just make $h$ huge at one of the two values).

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    Thanks for the answer but X should be a non-degenerate continuous random variable. I added some assumptions accordingly.2012-06-07