The purpose of the bivariate uniformity test is to check whether the underlying probability distribution, from which a random sample is drawn, differs from the bivariate uniform distribution. Some recently proposed statistical tests for bivariate uniformity can be found in the literature. See, for examples, Justel, Pena and Zamar (1997), Liang, Fang, Hichernell and Li (2001), Berrendero, Cuevas and Vazquez-Grande (2006), and Chen and Hu (2014). Power comparison is always the way to compare performance of different statistical tests. In order to get convincing power comparison results, various alternative distributions should be used when power study is conducted. Chen (2018) proposed a new bivariate distribution, named the pyramidal distribution, with support set [0,1] × [0,1 ] . The proposed distribution is quite flexible so that it can be used to produce different shapes of bivariate distributions, and hence can be used as an alternative distribution in power comparison for bivariate uniformity test. The purpose of this paper is to discuss the properties of the pyramidal distribution. The marginal distributions of the pyramidal distribution are discussed. The mathematical characteristics such as the mean vector and the variance-covariance matrix are discussed as well.