What is the relationship between $\mathbb{E}(\|\mathbf{X}\|)$ and $\|\mathbf{Y}\|$?

Question

Given that we have two vectors $\mathbf{X}\in\mathbb{R}^N$ and $\mathbf{Y}\in\mathbb{R}^N$, where $\mathbf{X}$ is a random variable with $\mathbb{E}(\mathbf{X}) = \mathbf{Y}$. Here $\mathbb{E}$ denotes the expectation operator. So, what is the relationship between $\mathbb{E}(\|\mathbf{X}\|)$ and $\|\mathbf{Y}\|$? Here $\|\|$ denotes the $L_2$-norm.

Answer 1

As noted, there is an inequality.
$$ \|\mathbb{E}(\mathbf{X})\| \le \mathbb{E}(\|\mathbf{X}\|) $$

This is a special case of Jensen's inequality: If $\varphi : \mathbb R^N \to \mathbb R$ is a convex function, then

$$ \varphi\big(\mathbb{E}(\mathbf{X})\big) \le \mathbb{E}\big(\varphi(\mathbf{X})\big) $$