1 Vectors of Random Variables
1.1
NOTATION
Matrices and vectors are denoted by boldface letters A and a, respectively, and scalars by italics. Random variables are represented by capital letters and their values by lowercase letters (e.g., Y and y, respectively). This use of capitals for random variables, which seems to be widely accepted, is particularly useful in regression when distinguishing between fixed and random regressor (independent) variables. However, it does cause problems because a vector of random variables, Y" say, then looks like a matrix. Occasionally, because of a shortage of letters, aboldface lowercase letter represents a vector of random variables. If X and Yare randomvariables, then the symbols E[Y), var[Y], cov[X, Y), and E[XIY = y) (or, more briefly, E[XIY)) represent expectation, variance, covariance, and conditional expectation, respectively. The n x n matrix with diagonal elements d 1 , d2 , •.• ,dn and zeros elsewhere is denoted by diag( d 1 , d 2 , •.. , dn ), and when all the di's are unity we have the identity Il].atrix In. If a is an n x 1 column vector with elements al, a2, . .. , an, we write a = (ai), and the length or norm of a is denoted by Iiali. Thus
lIall = Va'a = (a~ + a~ + ... + a~y/2. The vector with elements all equal to unity is represented by In, and the set of all vectors having n elements is denoted by lR n . If the m x n matrix A has elements aij, we write A = (aij), and the sum of the diagonal elements, called the trace of A, is denoted by tr(A) (= a11 + a22 + ... + akk, where k is the smaller of m and n). The transpose 1