Let's say you are given that $E(X^n)$ = $\frac{n!}{((n+3!)/3!)}$ for a random variable $X$. So the first 4 moments are $\frac{1}{4}, \frac{1}{10}, \frac{1}{20}, \frac{1}{35}$, and so on. Is there any way to back into an approximation of the pdf from this?
It seems like the boundedness of $X$ would suggest uniqueness, but I am having difficulty finding whether we can actually figure out what the random variable might be, or at least learn something more about $X$.
In general, it is not possible to uniquely recover the exact distribution function given the moments. An nice counter example is given on page 48 of Stoyanov's "Counterexamples in Probability and Statistics" (link below). It is possible when moment generating function is smooth and finite around the origin, and the exact answer for the inversion is given by the Mellin transform (http://en.wikipedia.org/wiki/Bromwich_integral). Also see the discussion here: https://stats.stackexchange.com/questions/32706/existence-of-the-moment-generating-function-and-variance. You mentioned in the comments that you determined these moments from the characteristic function. The characteristic function does uniquely determine the probability distribution function by its inverse Fourier transform. Working with the characteristic function itself should be easier.
