# 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 $$Y=1/X$$?

## 1 Answer

If $$X$$ is skew normal, then $$Y = 1/X$$ does not possess a first moment. Thus mean and variance are not defined.

• This is what Mathematica told me. I am wondering still something useful such as the mode of the density of Y. Is it possible? and the corresponding variance using mode as the mean. – Seyhmus Güngören Dec 9 '20 at 22:54
• It is certainly possible to say something about the median of $Y$. You should write a new post with your questions. – Hans Engler Dec 10 '20 at 2:50
• To say something analytically or just in terms of equation solving? – Seyhmus Güngören Dec 10 '20 at 2:54
• Continue as chat? – Hans Engler Dec 10 '20 at 2:58
• yes sure. how to do it? – Seyhmus Güngören Dec 10 '20 at 3:05