Consider a random variable which is given by an orthogonal polynomial expansion in one parameter (or polynomial chaos expansion PCE), i.e. , $$ f(\alpha) = \sum\limits_{n=0}^{\infty} \hat{f} (n) \psi _n (\alpha) \, , $$

where $\alpha$ is uniformly distributed in $[-1,1]$, $\psi _n (\alpha)$ are the orthogonal Legendre polynomials $$\hat{f} = \frac{1}{2}(f,\psi _n )_{L^2[-1,1]} \, .$$

My question: is there a closed form formula or an approximation for the cumulative distribution function (CDF) or the probability distribution function (PDF) of $f$ in a closed form, using only $\psi _n$ and $\hat{f} (n)$?

This is somewhat related to this post, and was asked by me before on MO here

  • $\begingroup$ I believe whether or not a closed form is available depends on whether or not you can find a closed form expression for $f^{-1}(\alpha)$. If you go through the logic of computing a PDF for a transformed variable, all the derivatives are fairly easy to compute in terms of $\alpha$; the hardest part is re-expressing everything in terms of $f$ instead of $\alpha$. $\endgroup$ – nukeguy Feb 9 '17 at 22:52
  • $\begingroup$ @nukeguy from one hand, this is a natural point of view, and this is the hurdle I'm struggling with. However, the gPC expansion allows one to compute every moment in terms of $\hat{f} (n)$, so there's room to think that you can go from moments to distribution. $\endgroup$ – Amir Sagiv Feb 10 '17 at 8:13

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