Suppose X follows a standard normal distribution, does E(x^3)=E(x)E(x^2) ??

Question

Suppose X follows a standard normal distribution,does $E(x^3)=E(x)E(x^2)$ ? and Dose$ E(x^3)=3E(x)$?

If so, the equation $E(x^3)=E(x)E(x^2)$ is not right, but i read from a book that it is right.

Answer 1

Well technically both equations are right since $E[X]=0$ and $E[X^3]=0$, both by symmetry arguments (both $X$ and $X^3$ are symmetric about $0$). But in general, the equation $E[X^3]=E[X]E[X^2]$ does not necessarily hold for an arbitrary random variable.

Answer 2

It is not right. Suppose that $X$ takes value $0$ or $1$ with probability $1/2$ each. Then $X^2$ and $X^3$ do the same. This means $E[X]=E[X^2]=E[X^3]=1/2$, so $1/2=E[X^3]\neq E[X]E[X^2]=1/4$.

Edit: With your modification that $X$ follows a standard normal distribution, $E[X]=E[X^3]=0$. So the equation is trivially true.