Random Variables: Multivariate Case

One can naturally expect that the bivariate distribution (in the form of cdf, joint density, or probability mass function, as the case might be) contains more information than the univariate distributions of X and Y separately. The notions of marginal and conditional distributions and densities remain very much the same as in the bivariate case, except that the marginal distributions may now be themselves multivariate, and the same applies to conditional distribution, with the additional feature that the conditioning event may involve several random variables. This chapter extends the concepts introduced in the case of bivariate distributions to the case of multivariate (or multidimensional) distributions. The motivation for these concepts lies in the frequency of practical situations when the analysis concerns many random variables simultaneously. The relation between cdf and the probability mass function in the discrete case, or the density function in the continuous case, is similar to those for univariate random variables.