Home

About

Advanced Search

Browse by Discipline

Scientific Societies

E-print Alerts

Add E-prints

E-print Network
FAQHELPSITE MAPCONTACT US


  Advanced Search  

 
Linear transformations and Gaussian random vectors. Remember, n-vectors are the same as n 1 matrices.
 

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
E(X)
be the n-vector whose i-th entry is E(Xi). If Y is a random n-vector 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 n-vector, A is an n n matrix, b is an n-vector 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
e-|x|2

  

Source: Allard, William K. - Department of Mathematics, Duke University

 

Collections: Mathematics