Stat 5101 Lecture Notes - School of Statistics

5.1. RANDOM VECTORS 1335.1.7 Linear TransformationsIn this section, we derive the analogs of the formulasE(a + bX) =a+bE(X)var(a + bX) =b 2 var(X)(5.15a)(5.15b)(Corollary 2.2 and Theorem 2.13 in Chapter 2 of these notes) that describethe moments of a linear transformation of a random variable. A general lineartransformation has the formy = a + Bxwhere y and a are m-dimensional vectors, B is an m × n matrix, and x is ann-dimensional vector. The dimensions of each object, considering the vectorsas column vectors (that is, as matrices with just a single column), areym × 1= am × 1+ Bm × nan × 1(5.16)Note that the column dimension of B and the row dimension of x must agree,as in any matrix multiplication. Also note that the dimensions of x and y arenot the same. We are mapping n-dimensional vectors to m-dimensional vectors.Theorem 5.3. If Y = a + BX, where a is a constant vector, B is a constantmatrix, and X is a random vector, thenE(Y) =a+BE(X)var(Y) =Bvar(X)B ′(5.17a)(5.17b)If we write µ X and M X for the mean and variance of X and similarly forY, then (5.17a) and (5.17b) becomeµ Y = a + Bµ X (5.18a)M Y = BM X B ′(5.18b)If we were to add dimension information to (5.18a), it would look much like(5.16). If we add such information to (5.18b) it becomesM Ym × m= Bm × nM Xn × nB ′n × mNote again that, as in any matrix multiplication, the column dimension of theleft hand factor agrees with row dimension of the right hand factor. In particular,the column dimension of B is the row dimension of M X , and the columndimension of M X is the row dimension of B ′ . Indeed, this is the only way thesematrices can be multiplied together to get a result of the appropriate dimension.So merely getting the dimensions right tells you what the formula has to be.