Digital Signal Processing Reference
In-Depth Information
that is,
F
X
(
α
)
measures the probability that
X
takes values less than or equal
to
.The
probability density function
(pdf ) is related to the cdf (assuming
that
F
X
is differentiable) as
α
l
g
r
,
y
i
d
.
,
©
,
L
s
dF
X
(
α
)
d
f
X
(
α
)=
,
α
∈
α
and thus
α
F
X
(
α
)=
f
X
(
x
)
dx
,
α
∈
−∞
Expectation and Second Order Statistics.
For random variables, a
fundamental concept is that of
expectation,
defined as follows:
∞
E
[
X
]=
xf
X
(
x
)
dx
−∞
The expectation operator is linear; given two random variables
X
and
Y
,we
have
E
[
aX
+
bY
]=
a
E
[
X
]+
b
E
[
Y
]
Furthermare, given a function
g
:
→
,wehave
∞
E
g
)
=
(
X
g
(
x
)
f
X
(
x
)
dx
−∞
The expectation of a random variable is called its
mean
, and we will indicate
it by
m
X
. The expectation of the product of two random variables defines
their
correlation
:
R
XY
=
E
[
XY
]
The variables are uncorrelated if
E
[
XY
]=
E
[
X
]
E
[
Y
]
Sometimes, the “centralized” correlation, or
covariance
,isused,namely
E
(
)
K
XY
=
X
−
m
X
)(
Y
−
m
Y
=
E
[
XY
]
−
E
[
X
]
E
[
Y
]
Again, the two variables are uncorrelated if and only if their covariance is
zero. Note that if two random variables are independent, then they are also
