Calculate $E(X^3)$ given value of $E(X)$ and $E(X^2)$

Question

If I am given that for $X$, a random variable, the value of $E(X)=3$ and $E(X^2)=9$, what is the value of $E(X^3)$? I believe I can write each as $$E(X)=\sum xP(X=x)=3$$ $$E(X^2)=\sum x^2P(X=x)=9$$ $$E(X^3)=\sum x^3 P(X=x)$$ What should I do with the $E(X^3)$? I'm pretty sure I can't simply make it $E(X^3)=E(X) \cdot E(X^2)$.

Answer 1

In general you cannot determine $E(X^3)$ given the values of $E(X^2)$ and $E(X)$.

The point of this problem, however, is to recognize a very unusual feature about the values of $E(X)$ and $E(X^2)$, that tells you something surprising about $X$.

We can compute $\mathrm{Var}(X) = E(X^2) - E(X)^2 = 0$, so it turns out that $X$ has no variance! Thus, we have $X=3$ almost surely (that is, $P(X=3) = 1$), and can compute $E(X^3) = E(3^3) = 27$.