Generate random variable from series of its expected values E[X], E[X^2], E[X^3], ...?

Question

Given a series of all the expected values of a random variable, can we find the random variable itself ?

Answer 1

Let $\varphi(t)=\mathbb{E}[e^{itX}]$. Then: $$i^n\,\mathbb{E}[X^n] = \varphi^{(n)}(0)$$ hence the problem of finding the distribution of $X$ given its moments is equivalent to finding an inverse Fourier transform for the characteristic function $\varphi(t)$.

Answer 2

You can't find the random variable itself, but you may be able to find its distribution. Look up "moment generating function" or "characteristic function".

Answer 3

Take $x_1 = 1_{[0,{1 \over 2})}$, $x_2 = 1_{[{1 \over 2},1]}$ on $[0,1]$. Then $E[x_1^k] = E[x_2^k] = {1 \over 2}$ for all $k$, but clearly $x_1 \neq x_2$.