Multivariate Normal And Related Distributions

Multivariate normal distribution is the natural extension of the bivariate normal to the case of several jointly distributed random variables. Dating back to the works of Galton, Karl Pearson, Edgeworth, and later Ronald Fisher, the multivariate normal distribution has occupied the central place in modeling jointly distributed continuous random variables. There are several reasons for its special status. Its mathematical properties show a remarkable amount of intrinsic structure; the properties are extremely well studied; statistical methodologies in common use often have their best or optimal performance when the variables are distributed as multivariate normal; and, there is the multidimensional central limit theorem and its various consequences which imply that many kinds of functions of independent random variables are approximately normally distributed, in some suitable sense. We present some of the multivariate normal theory and facts with examples in this chapter.