# zbMATH — the first resource for mathematics

A probabilistic representation of the ‘skew-normal’ distribution. (English) Zbl 0648.62016
A. Azzalini, ibid. 12, 171-178 (1985; Zbl 0581.62014), introduced the class $${\mathcal S}{\mathcal N}=\{{\mathcal S}{\mathcal N}(\lambda):\lambda\in {\mathbb{R}}\}$$ of skew-normal probability distributions and studied its main properties. The salient features of the class $${\mathcal S}{\mathcal N}$$ are mathematical tractability and strict inclusion of the normal law (for $$\lambda =0)$$. The shape parameter $$\lambda$$, to some extent, controls the index of skewness.
It is the purpose of this note to give a probabilistic representation of the distribution $${\mathcal S}{\mathcal N}(\lambda)$$ in terms of normal and truncated normal laws. This representation reveals the “structure” of the class $${\mathcal S}{\mathcal N}$$ and indicates the kind of departure from normality. The moments of a random variable $$Z_{\lambda}$$ with the distribution $${\mathcal S}{\mathcal N}(\lambda)$$ are explicitly determined, and an efficient method for the Monte Carlo generation of $$Z_{\lambda}$$ is shown.

##### MSC:
 62E10 Characterization and structure theory of statistical distributions 60E05 Probability distributions: general theory 65C10 Random number generation in numerical analysis