What Everyone Should Know: About Univariate Normality and Bivariate Normality, and How They are Co-Related with Correlation and Independence

Abstract

I have collected together 10 results concerning marginal distributions, joint distributions, univariate normality, bivariate normality, correlation and independence. Some of these results are well known, but some are relatively unknown. My experience has been that no single source presents more than a few of these results simultaneously, and whenever I see one or two of these results, I am always left wanting more.

For example, must uncorrelated normally distributed random variables be independent? (No, I give some counterexamples.) Is independence *necessary* for the sum of two normally distributed random variables to be normally distributed? (No, but it is *sufficient*.) Is the weaker assumption of bivariate normality *necessary* for the sum of two normally distributed random variables to be normally distributed? (No, but it is *sufficient*.) If two normally distributed random variables are uncorrelated, are they automatically jointly bivariate normally distributed? (No, I give some counterexamples.) Etcetera.

Keywords: Univariate Normality, Bivariate Normality, Correlation, Independence, Marginal Distributions, Joint Distributions

JEL Classification: A20