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
F
x
e
x
2
1
0
(a)
Calculate the following probabilities:
.
P
X
P
X
P
X
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
0
2
1
2
Find the PDF
Y
f
y
of
Y
.
HW 1.2
Let
X
and
Y
be independent identically distributed (i.i.d.) random variables with common
density function
,
,
f
1
0
1
0
otherwise.
Let
S
X
Y
.
(a)
Find and sketch
S
f
s
.
(b)
Find and sketch
X S
f
x s
versus
x
with
s
viewed as a known parameter.
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(c)
The conditional mean of
X
given
S
s
is
X S
E
X S
s
xf
x s dx
.
Find
.
E
X S
0 5
.
(d)
The conditional mean of
X
given
S
is
X S
X S
E
X S
xf
x S dx
.
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
1
is
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
1
1
), 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
2
5
,
/
a
1
2
,
and
/
2
1
4
.
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 Spring '10
 HyungdongShin
 Normal Distribution, Probability theory, Poseni river

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