Determining distribution based on first three even moments Given that $E(X^2) = 1, E(X^4) = 3, E(X^6) = 9$, determine all possible distributions of X.
I'm sure there's some insight I'm missing here. I noted that X is mesokurtic, so it's possibly normal... but I calculated the even moments of the standard normal, and I don't think a purely scale transformation of Z fits: $E(Z^2)= 1, E(Z^4) = 3, E(Z^6) = 15$.
A hint towards the correct answer would be appreciated.

Take a linear combination of $X^2,X^4$ and $X^6$: $$g(X)=aX^6+bX^4+cX^2$$
Choose scalars $a,b,c$ such that $E\left[g(X)\right]=0$ and $g(X)\ge 0$ almost surely. This would imply $g(X)=0$ almost surely (see e.g. https://math.stackexchange.com/q/897876/321264), based on which one can determine the distribution of $X$ from the given information.
Observe that $0\le (X^3-3X)^2=X^6-6X^4+9X^2$ almost surely, and $E\left[X^6-6X^4+9X^2\right]=0$, so one such choice of $(a,b,c)$ is $(1,-6,9)$. I am not sure if this is the only choice.
Therefore, the distribution of $X$ must satisfy $$(X^3-3X)^2=X^2(X^2-3)^2=0 \quad,\text{ a.s. }$$
