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Azzalini’s skew-normal class of distributions are a family of distributions that includes the normal distribution as a special case. They have an extra parameter (λ) to regulate skewness [1].

These one parameter distributions are defined by the probability density function (PDF):

Where:

- Φ(x) and φ(x) denote the PDF and cumulative density function (CDF) of the standard normal distribution.
- The parameter λ varies in (-∞, ∞); λ = 0 gives the unit normal distribution.

Location and scale parameters are an optional addition to the PDF.

While there are many references to the skew-normal in the literature, it was Azzalini [2] who gave a systematic treatment of the distribution. Therefore, it is occasionally referred to as the Azzalini distribution (e.g., Johnson et al. [3].)

The distribution originates from Azzalini’s note [2] that if X and Y are two independent random variables with individual PDFs that are symmetric about zero, then for any A

Consequently, 2pY(y)FX(λy) is a PDF. If we take X and Y to be unit normal variables, we get Azzalini’s distribution.

## Usefulness of the Azzalini Distribution

The Azzalini distribution has a number of useful properties which approximate the normal distribution, justifying the “skew-normal” name. For example, if X has a Azzalini PDF, then X2 follows a chi-squared distribution with one degree of freedom for all values of X. In applied statistics, the distribution can be used to analyze skewed data from a unimodal empirical distribution, which occurs frequently in practical problems.

## References

[1] Azzalini, A. & Valles, D. (1996). The multivariate skew-normal distribution. Biometrika, 83, 4, pp. 715-726.

[2] Azzalini, A. (1985). A class of distributions which includes the normal ones, Scandinavian Journal of Statistics, 12, 171-178.

[3] Johnson, Kotz, and Balakrishnan, (1994), Continuous Univariate Distributions, Volumes I and II, 2nd. Ed., John Wiley and Sons.

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