ECE 5510: Random Processes
Lecture Notes
Fall 2009
Lecture 9
Today: (1) Joint distributions: Intro
•
Exam is a week from today in class (Sept. 29)
•
Thursday Sept 24 is a review session. Bring questions to go
over.
•
HW 4 due Sept 24 at 10:45 am (at start of lecture). No late
HW is accepted, as we will hand out solutions in class. OH
are today until 1:15. Extra OH tomorrow 9:3011am.
•
Appl Assignment 2 is due Sept 24 at midnight. Discussion?
•
Option:
Record lectures into 5 minute video segments, for
watching prior to lecture. Then, the extra time in class would
be used to answer questions and do examples.
Would you
watch?
1
Joint distributions: Intro (Multiple Ran
dom Variables)
Often engineering problems can’t be described with just one random
variable. And random variables are often related to each other. For
example:
1. ICs are made up of resistances, capacitances, inductances, and
transistor characteristics, all of which are random, dependent
on the outcome of the manufacturing process. A voltage read
ing at some point in the IC may depend on many of these
parameters.
2. A chemical reaction may depend on the concentration of mul
tiple reactants, which may change randomly over time.
3. Control systems for vehicles may measure from many different
sensors to determine what to do to control the vehicle.
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ECE 5510 Fall 2009
2
(a)
X
2
X
1
x
2
x
1
(b)
X
2
X
1
c
b
d
a
Figure 1: (a) A 2D joint CDF gives the probability of (
X
1
,X
2
) in
the area shown. (b) The smaller area shown can also be calculated
from the joint CDF.
1.1
Event Space and Multiple Random Variables
Def’n:
Multiple Random Variables
A set of
n
random variables which result from the same experiment
or measurement are a mapping from the sample space
S
to
R
n
Example: Two dice are rolled.
An outcome
s
∈
S
is the result of the experiment of rolling two
dice. Let
X
1
(
s
) be the number on die 1, and
X
2
(
s
) be the number
on die 2. The coordinate (
X
1
,X
2
) is a function of
s
and lies in a
twodimensional space
R
2
or more specifically,
{
1
,
2
,
3
,
4
,
5
,
6
}
2
.
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 Fall '08
 Chen,R
 Probability theory, probability density function, Cumulative distribution function, CDF

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