 
Summary: Linear transformations and Gaussian random vectors.
Remember, nvectors are the same as n × 1 matrices.
Let X a random nvector. We let
E(X)
be the nvector whose ith entry is E(Xi). If Y is a random nvector we let
Cov(X, Y)
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 nvector, A is an n × n matrix, b is an nvector and
Y = AX + b.
Then
E(Y) = AE(X) + b
and
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
ex2
