Summary: Linear transformations and Gaussian random vectors.
Remember, n-vectors are the same as n × 1 matrices.
Let X a random n-vector. We let
be the n-vector whose i-th entry is E(Xi). If Y is a random n-vector we let
be the n × n matrix whose i, j entry is Cov(Xi, Yj) and we let
Var(X) = Cov(X, X).
We have already proved the simple
Proposition. Suppose X is a random n-vector, A is an n × n matrix, b is an n-vector and
Y = AX + b.
E(Y) = AE(X) + b
Var(Y) = ACov(X)AT
Definition. We say the random vector X is standard normal if its components X1, . . . , Xn are indepen-
dent and standard normal. Evidently, this is the case if and only if X is continuous and
fX(x) = (2)-n/2