TITLE: The expectation of absolute value of random variables
QUESTION [21 upvotes]: I need some help with the following problem: 

Let $X_1,...,X_n$ be a random sample from Normal$(0,1)$ population. Define $$Y_1=| {{1 \over n}\sum_{i=1}^{n}X_i}|, \ Y_2={1 \over n}\sum_{i=1}^{n}|X_i|.$$ Calculate $E[Y_1]$ and $E[Y_2]$, and establish the inequality between them. 

I may feel this should not be a very hard problem but I did get stuck somewhere. And I know it is $E[Y_1]\le E[Y_2]$ and I could prove this. But can anyone help me with how to exact find $E[Y_1]$ and $E[Y_2]$? 
Thanks in advance.

REPLY [4 votes]: Regarding the general form of the mean of the absolute value of a Normal$(\mu,\sigma^2)$:
Following @Henry's link to Wikipedia's article on the half-normal, I found that there is now a page for the so-called "Folded normal distribution", which is an extension of the half-normal for $\mu\neq 0$. They give the expression for the mean, which coincides with the one given by @Henry for $\mu=0$
$$
\sigma \sqrt{\frac{2}{\pi}} e^{-\frac{\mu^2}{2\sigma^2}} + \mu \left(1 - 2\Phi\left(\frac{-\mu}{\sigma}\right) \right)
$$