This preview shows pages 1–3. Sign up to view the full content.
Stochastic Signals and Systems
Random Variables
Virginia Tech
Fall 2008
Deﬁnition
The
probability density function of
X
(pdf)
, if it exists, is
deﬁned as the derivative of
F
X
(
x
)
:
f
X
(
x
) =
dF
x
(
x
)
dx
.
Interpretation of f
X
(
x
)
.
P
[
x
<
X
≤
x
+ Δ
x
] =
F
X
(
x
+ Δ
x
)

F
X
(
x
)
.
If
F
X
(
x
)
is
continuous in its ﬁrst derivative then, for sufﬁciently small
Δ
x
,
F
X
(
x
+ Δ
x
)

F
X
(
x
) =
Z
x
+Δ
x
x
f
(
ξ
)
d
ξ
’
f
X
(
x
)Δ
x
Hence for small
Δ
x
:
P
[
x
<
X
≤
x
+ Δ
x
]
’
f
X
(
x
)Δ
x
Thus
f
x
(
x
)
represents the “density” of probability at the point
x
in the sense that the probability that
X
is in a small interval in
the vicinity of
x
is approximately
f
X
(
x
)Δ
x
.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
1
f
x
(
x
)
≥
0.
2
P
[
a
<
X
≤
b
] =
F
X
(
b
)

F
X
(
a
) =
Z
b
∞
f
X
(
ξ
)
d
ξ

Z
a
∞
f
X
(
ξ
)
d
ξ
=
Z
b
a
f
X
(
ξ
)
d
ξ
3
F
X
(
x
) =
R
x
∞
f
X
(
ξ
)
d
ξ
4
R
∞
∞
f
X
(
ξ
)
d
ξ
=
F
X
(
∞
)

F
X
(
∞
) =
1
.
Example
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 05/05/2010 for the course ECECS 5605 taught by Professor Dasilver during the Fall '08 term at Virginia Tech.
 Fall '08
 DASILVER

Click to edit the document details