How can the var(|X|) be defined?

Question

I know that $var(|X|) = E[|X|^2] - (E[|X|])^2 = E[X^2] - (E[|X|])^2$.

However, I don’t know if (E[|X|])^2 can be simplified in terms of E[X] or something similar that has no absolute value.

Answer 1

Generally speaking, you have $$ \begin{split} \mathbb{E}[X] &= \int_\mathbb{R} x f(x) dx\\ \mathbb{E}[|X|] &= \int_\mathbb{R} |x| f(x) dx = \int_0^\infty x f(x) dx - \int_{-\infty}^0 x f(x) dx\\ &= \mathbb{E}[X] - 2\int_{-\infty}^0 x f(x) dx \end{split} $$

Answer 2

Let $X_+=\max(X,0)=X{\bf 1}_{\{X>0\}},~X_-=\max(-X,0)=-X{\bf 1}_{\{X<0\}}$.

Then $X=X_+-X_-$, and $|X|=X_++X_-$. Also, $X_+,X_-\ge 0$ and $X_+X_-=0$.

Using these,

$$\mbox{Var}(|X|) = \mbox{Var}(X)-4 E[X_+]E[X_-]=\mbox{Var}(X) + 4 E[X{\bf 1}_{\{X>0\}}]E[X{\bf 1}_{\{X<0\}}].$$

So Variance of $|X|$ is always less than or equal to variance of $X$ with equality if and only if $|X|=X$ with probability $1$.