how to show $E[|X|]= \sigma$ where $X \sim N(0, \sigma^2)$ let $X \sim N(0, \sigma^2)$ I want to show $$E[|X|]= \sigma$$
thanks for help

Hint: $$E[|X|]=\int_{-\infty}^{\infty} |x| \frac{1}{\sigma\sqrt{2\pi}}\, e^{-\frac{x ^2}{2 \sigma^2}} dx=2\int_{0}^{\infty} x \frac{1}{\sigma\sqrt{2\pi}}\, e^{-\frac{x^2}{2 \sigma^2}} dx$$ which is not a difficult integral: try the substitution $y=x^2$.
