This preview shows pages 1–4. Sign up to view the full content.
Kyung Hee University
Random Processing
Department of Electronics and Radio Engineering
Prof. Hyundong Shin
Communications and Coding Theory Laboratory (CCTLAB)
II1
C1002900 RP Lecture Handout II:
Random Variables and Their Distributions
Reading:
Chapter 4
2.1
Random Variable
Formally, a random variable
X
is a real valued function on the sample space
.
Definition 2.1
(Random Variable):
Let a probability space
,,
P
be given. A random
variable is a function
X
from
to the real line
, which is
measurable, meaning
that for any number
x
,
:
Xx
.
If
is finite or countable infinite, then
can be the set of all subsets of
, in which
case real valued function on
is a random variable. If
P
is given as in the uni
form phase example with
equal to the Borel subsets of
,
0 2
, then the random va
riables on
P
are called the Borel measurable function on
.
2.2.
Distribution and Density Function
Definition 2.2
(CDF):
The cumulative distribution function (CDF) of a random varia
ble
X
, denoted by
X
F
, is defined by
:
(for shorthand notation).
X
Fx P X
x
PX
x
This distribution completely characterizes the random variable.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document Kyung Hee University
Random Processing
Department of Electronics and Radio Engineering
Prof. Hyundong Shin
Communications and Coding Theory Laboratory (CCTLAB)
II2
If
X
denotes the outcome of rolling a fair die and if
Y
is uniformly distributed on the
interval
,
01,
then
X
F
and
Y
F
are shown in Figure 2.1.
X
F
1
1
0
2
3
4
5
6
1
1
Y
F
0
Figure 2.1: Examples of CDFs.
Example 2.1:
1
0
1
X
F
x
x
1)
What is the random variable
X
?
2)
?
PX
0
Proposition 2.1:
A function
F
is the CDF of some random variable if and only if it has
the following three properties:
F.1
F
is nondecreasing.
F.2
lim
x
Fx
1 and
lim
x
0.
F.3
F
is right continuous, that is
XX
F
xF
x
.
Kyung Hee University
Random Processing
Department of Electronics and Radio Engineering
Prof. Hyundong Shin
Communications and Coding Theory Laboratory (CCTLAB)
II3
: definedtobethesizeofthejumpof
at .
XX
X
X
PX x PX x
PX x
Fx Fx
F
xF
x
The CDF of a random variable
X
determines
P
Xx
for any real number
x
. How
ever, what about
P
and
P
? Let
,,
xx
12
be a monotone nondecreas
ing sequence that converges to
x
from the left. This means
ij
xxx
for
and
lim
jj
. Then, the events
j
are nested:
i
j
for
, and
the union of all such events is the event
. Therefore,
lim
lim
.
i
i
Xi
i
X
P
PXx
Fx
Hence,
X
P
Xx Fx Fx
where
X
F
x
is defined to be the size of the jump of
X
F
at
x
. For example, if
X
has
the CDF shown in Example 2.1, then
/
PX
0
1 2 . The requirement that
X
F
is right
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 06/10/2010 for the course ELECTRONIC C1002900 taught by Professor Hyungdongshin during the Spring '10 term at Kyung Hee.
 Spring '10
 HyungdongShin

Click to edit the document details