Some New Results on the Beta Skew-Normal Distribution Valentina Mameli and Monica Musio Abstract In this paper we study the Beta skew-normal distribution introduced by Mameli and Musio (2013). Some new properties of this distribution are derived including formulae for moments in particular cases and bi-modality properties. Further- more, we provide expansions for its distribution and density functions. Bounds for the moments and the variance of the Beta skew-normal are derived. Some of the results presented in this work can be extended to the entire family of the Beta-generated distribution introduced by Jones (Test 13(1):1–43, 2004). 1 Introduction The literature related to the skew-normal distribution (SN), introduced in [1], has grown rapidly in recent years. Mameli and Musio [12] proposed a generalization of the skew-normal called Beta skew-normal (BSN). This distribution arises quite naturally if we consider the distribution function of the order statistics of the skew- normal distribution and it can also be seen as a special case of the Beta-generated family proposed by Jones [8]. In [12] the authors studied some properties of the Beta skew-normal distribution. In particular, they derived the moment generating function, recurrence relations for moments and two methods for simulating. The main aim of this paper is to study some new properties of the Beta skew- normal distribution. Particularly, inspired by Gupta and Nadarajah [6], we obtain general expressions for the moments of the BSN. Expansions for its distribution and density functions are also provided. Moreover, motivated by the well-known bounds V. Mameli () • M. Musio University of Cagliari, via Ospedale 72, Cagliari, Italy e-mail: mameli.valentina@virgilio.it; mmusio@unica.it © Springer International Publishing Switzerland 2016 G. Alleva, A. Giommi (eds.), Topics in Theoretical and Applied Statistics, Studies in Theoretical and Applied Statistics, DOI 10.1007/978-3-319-27274-0_3 25 mameli.valentina@virgilio.it