What are the mean and variance of $\sqrt{|X|}$ using transformation of random variables?

Question

So I was trying to find the pdf of $\sqrt{|X|}$ since I know the pdf of $|X|$ if $X$ is normally distributed, using transformation of RV which states that

$$f_{Y}(y) = f_{X}(g^{-1}(y))\left|\frac {d}{dy} g^{-1}(y)\right|$$

for the case of when $g(x) = y = \sqrt{x}$, for $x$ is a folded normal distribution, I was able to obtain $f_{Y}(y)$ but I'm not sure how I can calculate expectation from that.

Answer 1

If $X\sim N(0,1)$, $$ \mathsf{E}Y=\mathsf{E}\sqrt{|X|}=\frac{2}{\sqrt{2\pi}}\int_0^\infty\sqrt{x}e^{-x^2/2}dx=\frac{2^{1/4}\Gamma(3/4)}{\sqrt{\pi}} $$ and $$ \mathsf{E}Y^2=\mathsf{E}|X|=\frac{2}{\sqrt{2\pi}}\int_0^\infty xe^{-x^2/2}dx=\sqrt{\frac{2}{\pi}}. $$