Expectation inequality involving absolute values

Question

Is the below result true for any random variable $X$?

$|\mathbb{E}(X)|\leq \mathbb{E}(|X|)$

Below is my attempt so far..

For any $X$, we have $X\leq|X|$ and taking expectation on both sides yields, $\mathbb{E}(X)\leq \mathbb{E}(|X|)$ , the quantity on the right is always positive but the quantity on the left could be both positive or negative.

The main result comes from taking absolute values on both sides of the expression $\mathbb{E}(X)\leq \mathbb{E}(|X|)$ which might not always be true (e.g., $-5<2$ but $|-5|>2$).

Can anyone help me with the correct reasoning behind the inequality $|\mathbb{E}(X)|\leq \mathbb{E}(|X|)$?

Answer 1

You can use the triangle inequality. Sketch:

$$\vert E(X)\vert = \vert \sum k \Pr(X=k) \vert \leq \sum \vert k \vert \Pr(X=k) =E(\vert X\vert)$$

Answer 2

$|EX|=EX$ or $|EX|=-EX$. In the first case use the fact that $X \leq |X|$ and in the second case use the fact that $-X \leq |X|$.