Expectations of absolute values

Question

$X_1 , X_2, ..., X_n$ is a random sample from a $\mathcal{N} (0,1)$ distribution.

$Y_{1} = |1/n\sum_{i=1}^nX_{i}|$

$Y_{2} = 1/n\sum_{i=1}^n|X_{i}|$

What is value of $E(Y_1)$ and $E(Y_2)$ ?

Answer 1

Hints:

• The first is a half-normal distribution with a suitable scale while the second is the sum of $n$ half-normal distributions with suitable scales

• The expectation of a half-normal distribution with scale $\sigma$ is $\sqrt{\dfrac2{\pi}}\sigma$