A random vector XX is an ordered sequence of random
variables
X=
X
1
…
X
L
T
X
X
1
…
X
L
. The density function of a random vector is defined
in a manner similar to that for pairs of random variables
considered previously. The expected value of a random vector is
the vector of expected values.
EX=∫-∞∞xpXxdx=E
X
1
⋮E
X
L
X
x
x
p
X
x
X
1
⋮
X
L
(1)
The
covariance matrix
K X
K X is an
L
×
L
L
×
L
matrix consisting of all possible covariances among
the random vector's components.
∀i,j,i∧j∈1…L:
K
X
ij=cov
X
i
X
j
=E
X
i
X
j
¯-E
X
i
E
X
j
¯
i
j
i
j
1
…
L
K
X
i
j
cov
X
i
X
j
X
i
X
j
X
i
X
j
(2)
Using matrix notation, the covariance matrix can be written as
K
X
=EX-EXX-EXT
K
X
X
X
X
X
. Using this expression, the covariance matrix is seen
to be a symmetric matrix and, when the random vector has no
zero-variance component, its covariance matrix is
positive-definite. Note in particular that when the random
variables are real-valued, the diagonal elements of a covariance
matrix equal the variances of the components:
K
X
ii=
σ
X
i
2
K
X
i
i
σ
X
i
2
.
Circular random vectors are
complex-valued with uncorrelated, identically distributed, real
and imaginary parts. In this case,
E|
X
i
|2=2
σ
X
i
2
X
i
2
2
σ
X
i
2
, and
E
X
i
2=0
X
i
2
0
. By convention,
σ
X
i
2
σ
X
i
2
denotes the variance of the real (or imaginary) parts. The
characteristic function of a real-valued random vector is
defined to be
Φ
X
ⅈν=EⅇⅈνTX
Φ
X
ν
ν
X
(3)
The maximum of a random vector is a random variables whose
probability density is usually quite different from the
distributions of the vector's components. The probability that
the maximum is less than some number
μμ is equal to the probability
that all of the components are less than
μμ.
Prmax{X}<μ=PXμ…μ
X
μ
P
X
μ
…
μ
(4)
Assuming that the components of
XX are statistically independent,
this expression becomes
Prmax{X}<μ=∏i=1dimXPXiμ
X
μ
i
1
dim
X
P
X
i
μ
(5)
