1
Kyung Hee University
Department of Electronics and Radio Engineering
C1002900
Random Processing
Midterm Exam Solutions
Spring 2010
Professor Hyundong Shin
April 26, 2010
19:00 – 21:30 (150 minutes)
This is an inclass closed book exam, but one A4size sheet of notes (single side) is allowed.
A correct answer does not guarantee full credit, and a wrong answer does not guarantee
loss of credit. You should clearly but concisely expose your reasoning and show
all rele
vant work
. Your grade on each problem will be based on our best assessment of your level
of understanding as reflected by what you have written in the answer sheets.
Please be neat
―
we cannot grade what we cannot decipher!
Problem 1
(10 points)
Arrivals of certain events at points in time are known to constitute a Poisson random variable,
but it is unknown which of two possible values of the average arrival rate
describes the ran
dom variable. Our
a priori
estimate is that
2 or
4 with equal probability. We observe the
random variable for
t
units of time and observe exactly
k
arrivals. Given this information, de
termine the conditional probability that
2 . Check to see whether or not your answer is rea
sonable for some simple limiting values for
k
and
t
.
Solution:
We have a Poisson random variable with an average arrival rate
, which is equally likely to be
either 2 or 4 . Thus,
.
1
24
2
We observe the variable for
t
time units and observe
k
arrivals. The conditional probability
that
2 is, by definition
.
kt
2 and
arrivals in time
2
arrivals in time
arrivals in time
Now, we know that
.
!
k
t
te
k
2
2 and
arrivals in time
arrivals in time
2
2
2
1
2
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Similarly,
.
!
k
t
te
kt
k
4
4
1
4 and arrivals in time
2
Hence,
!
!!
.
k
t
kk
tt
k
t
k
e
2
24
2
2
2
2
2
arrivals in time
22
2
1
12
To check whether this answer is reasonable, suppose
t
is large and
2 (observed arrival
rate equals 2 ). Then,
2
arrivals in time
approaches 1 as
t
goes to
. Similarly, if
t
is large and
4 (observed arrival rate equals 4 ), then
2
arrivals in time
approach
es 0 as
t
goes to
.
Problem 2
(15 points)
(a)
(5 points) Let
,
X
01
and
,
Y
be statistical independent. Consider a complex
random variable
Z
Xj
Y
. Find the probability that the instantaneous power
Z
2
of
Z
is less than or equal to its average power.
(b)
(5 points) Let
,
X
and
,
Y
be statistical independent.
1
Consider random
variables
Z XY
and
WXY
. Are
Z
and
W
independent?
Uncorrelated?
(c)
(5 points) Let
U
and
V
have the joint PDF
,
,,
,
,o
t
h
e
r
w
i
s
e
.
UV
uv
u
v
fu
v
0
Let
XU
2
and
YU V
1
. Find the joint PDF
,
,
XY
f
x y
of
X
and
Y
.
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 Spring '10
 HyungdongShin
 Probability theory

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