Stat 430/Math468 – Notes #4
Chapter 2
Random Vectors and Matrices
A
random vector
(or
matrix)
is a vector (or matrix) whose elements are random
variables.
Suppose
1
(
,...,
)'
p
X
X
=
X
is a random vector with joint PDF/PMF
1
( ,.
..,
)
p
f
xx
and each
element
has marginal PDF/PMF
i
X
()
ii
f
x
with
marginal mean
( )
,
if
is a continuous r.v.
( ),
if
is a discrete r.v.
i
ii i
i
i
i
all x
xf x dx
x
EX
xf x
x
μ
⎧
⎪
==
⎨
⎪
⎩
∫
∑
and
marginal variance
2
22
2
(
)
( )
if
is a continuous r.v.
() [
]
(
)
if
is a discrete r.v.
i
i
i
i
i
i
i
i
i
i
i
i
all x
xf
x
d
x
x
Var X
E X
x
x
σσ
⎧
−
⎪
=
−
=
⎨
−
⎪
⎩
∫
∑
The
covariance
between any pair of random variables,
,
i
X
j
X
measures their the linear
relationship.
It is calculated through their joint PDF/PMF (, )
ij
i
j
f xx
.
(
)
(
,
),
if
,
are continuous r.v.'s
(, ) [
(
)
(
)
]
(
)
(
,
)
,
if
,
are di
ij
j
j
i
j
i
j
i
j
ij
i
j
i
i
j
j
j j
i
j
i
j
all x all x
xxf
x
x
d
x
d
x
Cov X
X
E X
X
x
x
μμ
σμ
−−
−
−
=
∫∫
∑∑
screte r.v.'s
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
The
correlation coefficient
between
and
i
X
j
X
is defined as
ij
ij
ii
jj
σ
ρ
=
.
The p random variables
1
,...,
p
X
X
are said to be (mutually statistically)
independent
, if
their joint PDF/PMF can be written as the product of their marginal PDF/PMF’s, that is
12
1122
( ,
,...,
)
( )
(
)...
(
)
pp
p
fxx
x
fx f x
f x
=
, for all p-tuples (
,
,...,
p
x
).
Specially, two random variables
,
i
X
j
X
are said to be (mutually statistically)
independent
, if (, )
() ( )
ij
i
j
i
i
j
j
=
, for all pairs (
,
x
x
).
Two random variables
,
i
X
j
X
are said to be (mutually statistically)
uncorrelated
, if
, which is equivalent to
(, )0
ij
i
j
Cov X
X
)
(
)
(
i
j
EXX
EX EX
=
.
In general, independent
→
uncorrelated, but uncorrelated
→
independent.
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