Suppose we are given moments of a random variable $X$. Can we determine based on this if the random variable is continuous or not?
We also assume that the moments of $X$ completely determine the distribution of $X$.
In other words, do moments of continuous random variable behave fundamentally differently than moments of say discrete random variable?
Thanks, looking forward to your ideas.
Edit: It seems like there was some confusion with the questions. Let me demonstrate with an example what I have in mind.
Suppose, we are given moments of some random variable $X$ \begin{align} E[ X^n]=\frac{1}{1+n}, \end{align} for $n \ge 0$.
Can we determine if the distribution of $X$ is continuous or not?
In this example, I took $X$ to be continuous uniform on $(0,1)$.
Some Thoughts: Since we know the moments we can reconstruct the characteristic function of $X$ (I think this can be done, right? If not let as assume this) \begin{align} \phi_X(t) =\sum_{n=0}^\infty \frac{i^n E[X^n]}{n!} t^n \end{align}
We also know that $X$ has a pdf iff $\phi_X(t) \in L_1$.
So it seems it is enough to show that \begin{align} \int_{-\infty}^\infty \left| \sum_{n=0}^\infty \frac{i^n E[X^n]}{n!} t^n \right| dt \end{align} is finite or not. However, I don't think the above approach would work, as we can not switch the integration and summation.