Cauchy Schwarz and Absolute Values

Question

I am having trouble with the following:

I know that by CS inequality I have

$$ |E(xy)|\leq \sqrt{ E(x^2)E(y^2)} $$

However, say we have a process like $z_{t}= e_{t}e_{t-1}$ with $e_t$ ~ $iid(0,\sigma^2)$

In class we went through this:

$$ E|z_{t}|^2=E|e_te_{t-1}|^2\leq E(|e_t^2|)E(|e_{t-1}^2|)=\sigma^2\sigma^2 $$

What confuses me is why is $E(|e_t^2|)=\sigma^2$?

Using the CS inequality immidately gives the result of course but I am confused about the absolute values on the $e$ terms.

Answer 1

Since $|x|^2 = x^2$ for any real $x$, we have $\left|e_t^2\right| = e_t^2$. So $\Bbb{E}\left(\left|e_t^2\right|\right)= \Bbb{E}\left(e_t^2\right)= \sigma^2$.