Expected value of the norm of a centered vector.

Question

Let $X$ be a random vector in $\mathbb{R}^n$ such that $\mathbb{E}[X]=0$. Is it true that $\mathbb{E}[ \|X\|]=0$ ? (Euclidean norm).

Answer 1

HINT Not true because $$\left\|\vec{X}\right\| = 0 \iff \vec{X} = 0$$

In other words, for example, let the coordinates of $\vec{X}$ be distributed symmetrically, e.g. $\mathcal{U}(-1,1)$. Then what is the expected value?

But the Euclidean norm is $$ \sqrt{\sum_{k=1}^n X_k^2} $$ and $X_k^2 > 0$ a.s.

Answer 2

Maybe I can take in $\mathbb{R}$ the random variable $X$ in $\{-1,1\}$ with probability 1/2 each. then $|X|=1$ always.