Determine Random Variable

Question

We have a random variable $X$. Given the values for $E(X), E(X^2), E(X^3), ...$, is it possible to determine the distribution of the random variable X?

PS: Here $E(X)$ is the expected value of the random variable $X$.

Answer 1

In some cases, e.g. Cauchy distribution, expectation does not even exist. Even if they exist, they are not sufficient to determine the distribution. The classical example is the lognormal distribution $$ f(x) =cx^{-1}e^{-\log^2(x)/2}, x>0 $$ for some appropriate constant $c>0$ and $$g(x) =f(x)(1+a \sin (2\pi \log x)) $$ have the same $n$th moments $e^{n^2}/2$.

A soft criterion to moments to determine distribution is if the moment generating function $E(e^{\lambda X})$ is finite for values of $\lambda$ in some non-trivial interval around $0$. A more involved criterion for moments to determine distribution is : $\limsup \frac{m_{2k}^{1/{2k}}}{k} < \infty$ where $m_n$ is the $n$th moment.