Is it a typo in Jensen's inequality?

Question

I am reading a book where it is written:

For any $n \geq 1$, $E(|x|^n) \geq (E(|x|))^n$

I understand that it is simple apllication of Jensen' inequality for function $f(x) = |x|^n$, but I have some question about right side of the equation. Shouldn't it be $(|E(x)|)^n$ instead of $(E(|x|))^n$? Apparently these are quite different.

@gammatester If $x = 1,-1$ with equal chance, then $|E(x)| = 0$, while $E(|x|) = 1$ — Markoff Chainz, Oct 31 '18 at 12:15

score 1 · Answer 1 · answered Oct 31 '18 at 12:08

1

I believe that it is an application of Jensen's inequality to the convex function $\varphi(t)=t^n$ and the integrand $x \mapsto |x|$: $$ \left( \int |x| \, d\mu(x) \right)^n \leq \int |x|^n \, d\mu(x). $$

answered Oct 31 '18 at 12:08

Siminore

35,136

In the book it is written $|t|^n$ – Markoff Chainz Oct 31 '18 at 12:11
It is pretty much the same. You apply the convex function to $|\cdot|$, so the integral is non-negative. – Siminore Oct 31 '18 at 12:13
@Siminoire, If $x = 1,-1$ with equal chance, then $|E(x)| = 0$, while $E(|x|) = 1$. I dont understand what shall I use? – Markoff Chainz Oct 31 '18 at 12:16
In the statement, there is only the expectation of $|x|$. Now apply the convex function $\varphi(t)=t^n$ to $|x|$. Voilà. – Siminore Oct 31 '18 at 12:18
It seems like you dont understand what I am trying to say. 1) $E(|x|^n) \geq (E(|x|))^n.$ 2) $E(|x|^n) \geq (|E(x)|)^n$. 1 and 2 are different, which one is correct? 1 or 2 – Markoff Chainz Oct 31 '18 at 12:27
What do you believe? Notice that $E(|x|) \geq |E(x)|$. – Siminore Oct 31 '18 at 12:42
I believe in 2, because this is what comes out of direct application of Jensen inequality. Author writes stronger inequality as you have mentioned – Markoff Chainz Oct 31 '18 at 12:47
Inequality 2 follows from inequality 1. Since 1 is true, they both are true. – Siminore Oct 31 '18 at 13:27

Is it a typo in Jensen's inequality?

1 Answers1