Stat 5101 Lecture Notes - School of Statistics

128 Stat 5101 (Geyer) Course Notesconsider vectors as special cases of matrices. A column vector is an n×1 matrix⎛ ⎞x 1x 2x = ⎜ ⎟(5.2)⎝ . ⎠x nand a row vector isa1×nmatrixx ′ = ( x 1 x 2 ···)x n(5.3)Note that (5.2) is indeed the transpose of (5.3) as the notation x and x ′ indicates.Note that even when we consider vectors as special matrices we still use boldfacelower case letters for nonrandom vectors, as we always have, rather than theboldface capital letters we use for matrices.5.1.2 Random VectorsA random vector is just a vector whose components are random scalars.We have always denoted random vectors using boldface capital letters X =(X 1 ,...,X n ), which conflicts with the new convention that matrices are boldfacecapital letters. So when you see a boldface capital letter, you must decidewhether this indicates a random vector or a constant (nonrandom) matrix. Onehint is that we usually use letters like X, Y and Z for random vectors, and wewill usually use letters earlier in the alphabet for matrices. If you are not surewhat is meant by this notation (or any notation), look at the context, it shouldbe defined nearby.The expectation or mean of a random vector X =(X 1 ,...,X n ) is definedcomponentwise. The mean of X is the vectorµ X = E(X) = ( E(X 1 ),...,E(X n ) )having components that are the expectations of the corresponding componentsof X.5.1.3 Random MatricesSimilarly, we define random matrix to be a matrix whose components arerandom scalars. Let X denote a random matrix with elements X ij . We can seethat the boldface and capital letter conventions have now pooped out. There isno “double bold” or “double capital” type face to indicate the difference betweena random vector and a random matrix. 1 The reader will just have to rememberin this section X is a matrix not a vector.1 This is one reason to avoid the “vectors are bold” and “random objects are capitals”conventions. They violate “mathematics is invariant under changes of notation.” The typeface conventions work in simple situations, but in complicated situations they are part ofthe problem rather than part of the solution. That’s why modern advanced mathematicsdoesn’t use the “vectors are bold” convention. It’s nineteenth century notation still survivingin statistics.