This preview shows pages 1–3. Sign up to view the full content.
1
Kyung Hee University
Department of Electronics and Radio Engineering
C1002900
Random Processing
Homework 1
Spring 2010
Professor Hyundong Shin
Issued:
March 22, 2010
Due:
March 31, 2010
Reading:
Course textbook Chapters 1–6
HW 1.1
A random variable
X
has a cumulative distribution function (CDF)
,.
x
X
Fx
e
x
2
10
(a)
Calculate the following probabilities:
.
PX
1
2
2
(b)
Find the probability density function (PDF)
X
f
x
of
X
.
(c)
Let
Y
be a random variable obtained from
X
as follows
:
,
X
Y
X
02
12
Find the PDF
Y
fy
of
Y
.
HW 1.2
Let
X
and
Y
be independent identically distributed (i.i.d.) random variables with common
density function
,
,
f
1
0o
t
h
e
r
w
i
s
e
.
Let
SXY
.
(a)
Find and sketch
S
f
s
.
(b)
Find and sketch
XS
f
xs
versus
x
with
s
viewed as a known parameter.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document 2
(c)
The conditional mean of
X
given
Ss
is
XS
E XS s
x
f
xsdx
.
Find
.
EXS
05 .
(d)
The conditional mean of
X
given
S
is
E XS
x
f
xSdx
.
Since
X S
is a function of the random variable
S
, it is also a random variable. Find the
density function of
X S
.
HW 1.3
Wanting to browse the net, Bob uses his highspeed 300 baud modem to connect through his
Internet Service Provider. The modem transmits bits in such a fashion that
1 is sent if a given
bit is zero and
1 is sent if a given bit is one. The telephone line has an additive zeromean
Gaussian noise with variance
2
(so, the receiver on the other end gets a signal which is the
sum of the transmitted signal and the channel noise). The value of the noise is assumed to be
independent of the encoded signal value.
,
2
0
We assume that the probability of the modem sending
1 is
p
and the probability of sending
p
1
.
(a)
Suppose we conclude that an encoded signal
1 was sent when the value received on the
other end of the line is less than
a
(where
a
11
), and conclude
1 was sent when the
value is more than
a
. What is the probability of making an error?
(b)
Answer part (a) assuming that
/
p
25 ,
/
a
12 ,
and
/
2
14 .
HW 1.4
You take a safari trip to the Porobilati game reserve. A highlight of the game reserve is the Po
seni river where one can watch deer and elephants coming to drink water. Deer come to the
river according to a Poisson distribution with arrival rate
d
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