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Up to now, we have dealt with various foundational aspects of random variables and their distributions. We have occasionally touched on how these variates can arise in practice. In the second part of this book, we start analyzing in more detail how the variates are connected with sampling situations, how we can estimate the parameters of their distributions (which are typically unknown in practice), and how to conduct inference regarding these estimates and their magnitudes. This chapter starts with the first of these three aims. We study the sample mean and variance and their sampling properties, but also the sample's order statistics and extremes. The empirical distribution function (EDF) is defined and analyzed. In the case of multivariate normality, the Wishart distribution arises as a generalization of the chi-squared. We study the properties of matrix Wishart variates. We also show how Hotelling's T² arises as the counterpart of Student's t in the case of multivariate samples. The density of the correlation coefficient is also derived. We introduce rank and sign correlations, known as Spearman's rho and Kendall's tau, respectively.
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