I have a random variable, $X$, and am able to find $\mathbf{E}(e^{-\zeta X})$ for many $\zeta$ (through the Laplace transform solving an ODE as this actually evolves over time)

Is there any way I can approximate power moments of $X$: i.e. $\mathbf{E}(X^{\alpha})$ where $\alpha$ is not necessarily an integer? (I realize that this will not always be convergent for higher moments, or necessarily very accurate, especially if $\alpha < 0$ and hence the function is not analytic. See notes in When do Taylor series approximations to expectations of (entire) functions converge?)

I tried to find some transformation of a taylor series, but couldn't figure it out. What I want to do is avoid solving an ODE in transformed $\log(X)$ variable if I can.

  • $\begingroup$ I think If $X$ has some specific distributions, such as Gamma distribuiton then $E(X^{\alpha})$ can be analytically expressed but that only for $\alpha$ is a positive integer $\endgroup$
    – Deep North
    Aug 15, 2015 at 9:20
  • $\begingroup$ Thanks. Alas, no chance on an analytic distribution. $\endgroup$
    – jlperla
    Aug 15, 2015 at 15:16


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