Introduction to Probability Theory for Graduate Economics

Up to this point, we have only spoken of RVs as being single-dimensional objects. We will now turn to the case of vector-valued random variables, which are functions mapping a sample space into Rn. Thus, when we talk about the distribution or density of X = (X1, X2, . . . , Xn), we refer to n-dimensional real-valued functions. Alternatively, one could think of the individual elements Xi as distinct univariate random variables jointly sharing some connection through the distribution of X. For this reason, we call the distribution and density of X the “joint” distribution and the “joint” density of the Xis.