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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 $Y=1/X$?

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

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  • $\begingroup$ 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. $\endgroup$ – Seyhmus Güngören Dec 9 '20 at 22:54
  • $\begingroup$ It is certainly possible to say something about the median of $Y$. You should write a new post with your questions. $\endgroup$ – Hans Engler Dec 10 '20 at 2:50
  • $\begingroup$ To say something analytically or just in terms of equation solving? $\endgroup$ – Seyhmus Güngören Dec 10 '20 at 2:54
  • $\begingroup$ Continue as chat? $\endgroup$ – Hans Engler Dec 10 '20 at 2:58
  • $\begingroup$ yes sure. how to do it? $\endgroup$ – Seyhmus Güngören Dec 10 '20 at 3:05

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