Does $|x| \leq |y|$ imply $|\mathsf E[x]| \leq |\mathsf E[y]|$?

Question

Does $|x| \leq |y|$ imply $|\mathsf{E}[x]| \leq |\mathsf{E}[y]|$?

Answer 1

No, $|x| \leq |y|$ doesn't necessarily imply $|\mathsf E[x]| \leq |\mathsf{E}[y]|$.

One counterexample is as follow:

• $x$: $\mathsf P\{x=-1\} = \frac13$, $\mathsf P\{x=+1\} = \frac23$;
• $y$: $\mathsf P\{y=-2\} = \frac12$, $\mathsf P\{y=+2\} = \frac12$.

$\mathsf{E}[x] = (-1) \cdot \mathsf P\{x=-1\} + (+1) \cdot \mathsf P\{x=+1\} = -\frac13+\frac23 = \frac13$

$\mathsf{E}[y] = (-2) \cdot \mathsf P\{y=-2\} + (+2) \cdot \mathsf P\{y=+2\} = -1+1 = 0$

In this case, $1=|x|< |y|=2$, yet $\frac13 = |\mathsf{E}[x]| > |\mathsf{E}[y]| =0$.