Univariate, Bivariate, and Multivariate Normal Distributions

Many multivariate analysis procedures are based on the assumption that the responses have a multivariate Normal distribution. In studying multivariate statistical analysis, therefore it’s helpful for us to have some familiarity with the multivariate Normal distribution; specifically, the form of the distribution, the parameters it contains, and what those parameters represent. In this lecture we investigate the univariate, bivariate, and general multivariate Normal distributions. We start off by looking at the univariate Normal distribution. Next, we step up one dimension and look at the bivariate Normal distribution. Finally, we generalize to the multivariate Normal distribution of any dimension, p. For each of these classes of Normal distributions we look at the form of their pdf, and identify and interpret their various distributional parameters. We will see that these pdfs are all similar in structure, the differences being related to the dimensions of the parameters. Next lecture in this series:    • Estimating the Parameters of the Univariat...   Previous lecture in this series:    • Useful Results from Distribution Theory   First lecture in this series: "A Course on Multivariate Analysis" playlist:    • A Course in Multivariate Analysis   0:00 Intro music 0:20 Introduction 0:46 Outline of lecture video 1:34 Univariate Normal distribution 1:41 Form of the univariate Normal distribution 4:04 The Standard Normal distribution 5:03 Effects of the mean and standard deviation on the Normal distribution 8:02 Dynamic demonstration of the effects of the mean and standard deviation on the Normal distribution 12:37 Bivariate Normal distribution 12:45 Form of the Bivariate Normal pdf 13:31 Parameters of the Bivariate Normal distribution 14:17 Matrix representation of the parameters of the Bivariate Normal distribution 19:11 Review of the determinant and inverse of a general 2x2 matrix 21:43 Form of the determinant and inverse of the covariance matrix of a Bivariate Normal distribution 25:01 Pdf of the Bivariate Normal distribution 26:34 Multivariate Normal distribution 29:33 Summary and Outro