Obtaining the expectation of function of random variable

Question

Let $X$ follows $N(0,1)$ and let $Y=X+|X|$. Then, it is required to obtain the $E(Y^3)$.

My approach

The function $Y$ can be written as:

$\begin{cases} 0 & \text{ if } x<0 \\ 2X & \text{ if } x\geq0 \end{cases}$

My question is can I use the following expression to obtain the expectation:

$E(y^3)=E(8x^3)=\int_{0}^{\infty}8x^3f_x(x)dx$.

Instead of obtaining the density of $y$, is the above mentioned approach is correct?.

Answer 1

$E(y^3)=E(8x^3)=\int_{0}^{\infty}8x^3f_x(x)dx$.

That is indeed the correct approach.

More correctly: $\displaystyle~\mathsf E(Y^3)~{=\mathsf E(8X^3~[X\geqslant 0]) \\= \int_0^\infty 8x^3 f_X(x)\mathsf d x \\= \int_0^\infty (8x^3 e^{-x^2/2}/\sqrt{2\pi})~\mathsf d x\\~~\vdots}$