$X$ is random variable for which it is valid $E(X^{2})=E(X^{3})=E(X^{4})<\infty$. Prove that X is discrete random variable with values in {0,1}

Question

I have problem with this task. Is it perhaps good to assume the opposite? Any idea will be helpful, thanks in advance.

Answer 1

Here's a (detailed) hint: first, if $\mathbb{E}[X^2] = 0$, you are done. (Why?)

If not, suppose you can find a polynomial of the form $P(X)$ with the following properties:

• $P$ has degree 4 and no degree-1 or degree-0 term: $P(x) = ax^2 + bx^3 + cx^4$
• $P(x) \geq 0$ for all $x \in \mathbb{R}$
• the only roots of $P$ are $0$ and $1$ (possibly multiple)
• $a+b+c=0$

Then if you can find such a polynomial $P$, then looking at $\mathbb{E}[P(X)]$ will allow you to conclude. Can you see why?

Note: If that helps finding it, without loss of generality you can assume $c=1$ (why?)