How to find the distribution of X from the moments of X?

Question

For example the moments of X is defined by E(x^n)=0.7, for $n\in[1,\infty]$. How to get the distribution of X?

Answer 1

Depending on the domain, look up Hamburger moment problem, Stieltjes moment problem, or Hausdorff moment problem.

Your example is easy. Hint: $0$ and $1$ satisfy $x^n = x$ for all $n \ge 1$.

Somewhat more generally, if the moment generating function $E[e^{tX}] = \sum_{n=0}^\infty \dfrac{t^n}{n!} E[X^n]$ is a linear combination of exponentials, then you can read off a linear combination of point masses that gives those moments.