Absolute moment of normal distributed random variable.

Question

Let $u=(u_n)\in l^2(\mathbb{N})$ and $(Z_n)$ be an i.i.d. sequence of standard normal distributed random variables. Define $Z(u)= \sum_{n\geq 1} u_nZ_n$ and note that $Z(u)\sim \mathcal{N}(0,\|u\|_2^2)$. I'm reading a proof which simply states that $E|Z(u)|=\|u\|_2 E(|Z_1|) \, $ - How is this seen?

Answer 1

The expectation of the absolute value of $\mathcal{N}(0,\sigma^{2})$ is $\sigma \sqrt{\dfrac{2}{\pi}}$, because it follows a half-normal distribution. Then $E|Z(u)|=\|u\|_2 \sqrt{\dfrac{2}{\pi}}$ and $E|Z_1|=\sqrt{\dfrac{2}{\pi}}$.