ECE 5510: Random Processes
Lecture Notes
Fall 2008
Lecture 9
Today: (1) Conditional Distribution Review (2) Joint distribu
tions: Intro
•
Exam is a week from today in class (Sept. 30)
•
Thursday Sept 25 is a review session. Bring questions to go
over.
•
HW 4 due Sept 25 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 34pm.
•
Appl Assignment 2 is due Sept 25 at 5pm.
1
Conditional Distribution Review
W
is Gaussian with mean 0 and variance
σ
2
= 16. Given the event
C
=
{
W >
0
}
,
1. What is
f
W

C
(
w
)? First, since
W
is zeromean and
f
W
is
symmetric,
P
[
C
] = 1
/
2. Then,
f
W

C
(
w
) =
b
2
1
√
32
π
e

w
2
/
(32)
, w >
0
0
,
o.w.
2. What is
E
W

C
[
W

C
]?
E
W

C
[
W

C
] =
i
∞
0
2
w
1
√
32
π
e

w
2
/
32
Making the substitution
v
=
w
2
/
32,
dv
= 2
w/
32,
E
W

C
[
W

C
] =
32
√
32
π
i
∞
0
e

v
=
r
32
π
3. What is Var
W

C
[
W

C
]?
E
W

C
B
W
2

C
±
=
i
∞
0
2
w
2
1
√
32
π
e

w
2
/
32
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2
Knowing that
2
w
2
√
32
π
e

w
2
/
32
is an even expression, the integral
for
w >
0 is the same as the integral for
w <
0 – half the total.
E
W

C
b
W
2

C
B
=
i
∞
∞
w
2
1
√
32
π
e

w
2
/
32
= Var
W
[
W
] = 16
.
The variance is then
Var
W

C
[
W

C
] =
E
W

C
b
W
2

C
B
−
(
E
W

C
[
W

C
])
2
= 16
−
32
π
2
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 di±erent
sensors to determine what to do to keep the drive safe, stable,
and comfortable.
4. etc.
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 Fall '08
 Chen,R
 Probability theory, probability density function, Cumulative distribution function, CDF

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