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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.

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