Can we find the expected value of $Z$, where $Z=X^p$ with $p>0$ and $X\sim\mathcal{N}(\mu_X,\sigma_X)$ ?
With the help of Gaussian distribution raised to a power, I think the pdf of Z should be, $$f_Z(z)=pz^{p-1}f_X(x).$$ So, expected value should be, $$\mathrm{E}(Z) = \int_{-\infty}^{\infty}{pz^pf_X(x)\mathrm{d}z}.$$
I am stuck with, $$\mathrm{E}(Z) = \int_{-\infty}^{\infty}{(\mu_X+\sigma_{X} y)^{1/p} \frac{1}{\sqrt{2\pi}} \exp\left( -\frac{y^2}{2} \right)\mathrm{d}y},$$ where $y=\frac{x-\mu_X}{\sigma_X}$
Can we solve this?
