The density function of skew normally distributed random variable $X$ is given as
$$f(x)=\frac{2}{\sigma}\phi\left(\frac{x-\mu}{\sigma}\right)\Phi\left(\alpha \left(\frac{x-\mu}{\sigma}\right)\right)$$ where $\phi$ is standard Gaussian density, $\Phi$ is standard Gaussian cumulative distribution function with location $\mu$, scale $\sigma$ and shape $\alpha$ parameters.
What is the mean and variance of the random variable $Y=1/X$?
If $X$ is skew normal, then $Y = 1/X$ does not possess a first moment. Thus mean and variance are not defined.