True or false? An expectation inequality

Question

Is the inequality $$ \mathbb{E} \left[ |X| \right] \leq 1 + \mathbb{E}(X^2) $$ True or false? I have been stuck trying to prove it with Jensen's inequality, in particular by noting that

$$ |\mathbb{E}[X]| \leq \mathbb{E}[|X|] $$ However, I'm missing how to put the pieces together. Hope that somebody can help! :)

Answer 1

The inequality is true and is strict, and you can find it using the linear property of expectation by completing the square and then reversing the argument

• $0 \le \mathbb E\left[\left(\frac12-|X|\right)^2\right]$
• $0 \le \mathbb E\left[\frac14\right] -\mathbb E\left[|X|\right] + \mathbb E\left[|X|^2\right]$
• $\mathbb E\left[|X|\right] \le \frac14 + \mathbb E\left[X^2\right] $
• $\mathbb E\left[|X|\right] \lt 1 + \mathbb E\left[X^2\right]$

Answer 2

It's true, as Henry showed. An alternate proof: \begin{align*} \mathbb E[|X|] &= \mathbb E \left[ |X| \cdot 1_{|X|\leq 1} + |X| \cdot 1_{|X| > 1}\right] \\ &<\mathbb E \left[1 + |X|^2 \right] \end{align*} since $|X| \leq 1$ on the event $\{|X| \leq 1\}$ tautologically, and $|X|^2 > |X|$ on the event $\{|X| > 1\}$.

(I suppose the strict inequality is a tad more obnoxious to show in the case where $\{|X| > 1\} = \emptyset$, but it's not hard to see that it should hold there too.)