Докажите, что если $\xi \sim N(0,\sigma^2)$, то $$ \mathsf{M}(|\xi|) = \sqrt{\frac{2}{\pi}}\sigma,\qquad \mathsf{D}(|\xi|) = \left(1-\frac{2}{\pi}\right)\sigma^2. $$
(Or, in English:)
Prove that if $\xi \sim N(0,\sigma^2)$, then $$ \mathsf{M}(|\xi|) = \sqrt{\frac{2}{\pi}}\sigma,\qquad \mathsf{D}(|\xi|) = \left(1-\frac{2}{\pi}\right)\sigma^2. $$
I tried to solve this problem with using definition of the mathematical expectation ($\int_{-\infty}^\infty xf(x)\,dx)$, but he is not taken.