Calculare the mean of $X^2$

Question

If I know that the mean of $X$ is $\mu$, is it true that the mean of $X^2$ is $\mu ^2$?

Answer 1

Consider Normal random variable $X \sim N(\mu,\sigma^2)$ as an example, but the following formula is true for every random variable.

$$ \mathbb{E}[x^2] = Var(x) + \mathbb{E}^2[x]=Var(x)+\mu^2\neq \mu^2 $$

Answer 2

By the definition of variance, the mean of $X^2$ is actually $\mu^2+\sigma^2$. Your guess only works for constant variables.