ECE302 Spring 2006
HW1 Solutions
January 16, 2006
1
Solutions to HW1
Note: These solutions were generated by R. D. Yates and D. J. Goodman, the authors of our
textbook. I have added comments in italics where I thought more detail was appropriate.
Problem 1.2.1
•
A fax transmission can take place at any of three speeds depending on the condition of the
phone connection between the two fax machines. The speeds are high (
h
) at 14
,
400 b/s,
medium (
m
) at 9600 b/s, and low (
l
) at 4800 b/s. In response to requests for information, a
company sends either short faxes of two (
t
) pages, or long faxes of four (
f
) pages. Consider
the experiment of monitoring a fax transmission and observing the transmission speed and
length. An observation is a twoletter word, for example, a highspeed, twopage fax is
ht
.
(a) What is the sample space of the experiment?
(b) Let
A
1
be the event “mediumspeed fax.” What are the outcomes in
A
1
?
(c) Let
A
2
be the event “short (twopage) fax.” What are the outcomes in
A
2
?
(d) Let
A
3
be the event “highspeed fax or lowspeed fax.”
What are the outcomes in
A
3
?
(e) Are
A
1
,
A
2
, and
A
3
mutually exclusive?
(f) Are
A
1
,
A
2
, and
A
3
collectively exhaustive?
Problem 1.2.1 Solution
(a) An outcome specifies whether the fax is high (
h
), medium (
m
), or low (
l
) speed, and
whether the fax has two (
t
) pages or four (
f
) pages. The sample space is
S
=
{
ht, hf, mt, mf, lt, lf
}
.
(1)
(b) The event that the fax is medium speed is
A
1
=
{
mt, mf
}
.
(c) The event that a fax has two pages is
A
2
=
{
ht, mt, lt
}
.
(d) The event that a fax is either high speed or low speed is
A
3
=
{
ht, hf, lt, lf
}
.
(e) Since
A
1
∩
A
2
=
{
mt
}
and is not empty,
A
1
,
A
2
, and
A
3
are not mutually exclusive.
(Equivalently, since
A
2
∩
A
3
negationslash
=
∅
,
A
1
,
A
2
, and
A
3
are not mutually exclusive.)
(f) Since
A
1
∪
A
2
∪
A
3
=
{
ht, hf, mt, mf, lt, lf
}
=
S,
(2)
the collection
A
1
,
A
2
,
A
3
is collectively exhaustive.
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ECE302 Spring 2006
HW1 Solutions
January 16, 2006
2
Problem 1.2.2
•
An integrated circuit factory has three machines
X
,
Y
, and
Z
. Test one integrated circuit
produced by each machine. Either a circuit is acceptable (
a
) or it fails (
f
). An observation
is a sequence of three test results corresponding to the circuits from machines
X
,
Y
, and
Z
, respectively. For example,
aaf
is the observation that the circuits from
X
and
Y
pass
the test and the circuit from
Z
fails the test.
(a) What are the elements of the sample space of this experiment?
(b) What are the elements of the sets
Z
F
=
{
circuit from
Z
fails
}
,
X
A
=
{
circuit from
X
is acceptable
}
.
(c) Are
Z
F
and
X
A
mutually exclusive?
(d) Are
Z
F
and
X
A
collectively exhaustive?
(e) What are the elements of the sets
C
=
{
more than one circuit acceptable
}
,
D
=
{
at least two circuits fail
}
.
(f) Are
C
and
D
mutually exclusive?
(g) Are
C
and
D
collectively exhaustive?
Problem 1.2.2 Solution
(a) The sample space of the experiment is
S
=
{
aaa, aaf, afa, faa, ffa, faf, aff, fff
}
.
(1)
(b) The event that the circuit from
Z
fails is
Z
F
=
{
aaf, aff, faf, fff
}
.
(2)
The event that the circuit from
X
is acceptable is
X
A
=
{
aaa, aaf, afa, aff
}
.
(3)
(c) Since
Z
F
∩
X
A
=
{
aaf, aff
} negationslash
=
φ
,
Z
F
and
X
A
are not mutually exclusive.
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 Conditional Probability, Probability, Probability theory

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