I am studying about the skew-normal distribution and stumbled this work, where a skew-normal RV has the form [Eq. (3)]:
$X = \delta |Z_1| + \sqrt{1-\delta^2} Z_2$,
where $Z_1$ and $Z_2$ are independent standard normal RVs and $\delta$ is some constant.
I am really curious that what is the distribution of $X$ when $Z_1$ is not standard normal RV? Specifically, when $Z_1 \sim {\cal N}(\mu, \sigma^2)$? Is $X$ still called skew-normal?