# What is the mean and variance of $1/X$ if $X$ is skew normal distributed?

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.
• It is certainly possible to say something about the median of $Y$. You should write a new post with your questions. Dec 10, 2020 at 2:50