Probability and Stochastic Processes
Homework 1 and 2 Solutions
October 4, 2001
Problem Solutions
: Yates and Goodman,1.2.1 1.2.2 1.2.3 1.2.4 1.2.5 1.2.6 1.3.1 1.3.2 1.4.1
1.4.2 and 1.4.3
Please contact the TA if you have any questions.
Problem 1.2.1
(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
,
h f
,
mt
,
m f
,
lt
,
l f
}
(b) The event that the fax is medium speed is
A
1
=
{
mt
,
m f
}
.
(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
,
h f
,
lt
,
l f
}
.
(e) Since
A
1
∩
A
2
=
{
mt
}
and is not empty,
A
1
,
A
2
, and
A
3
are not mutually exclusive.
(f) Since
A
1
∪
A
2
∪
A
3
=
{
ht
,
h f
,
mt
,
m f
,
lt
,
l f
}
=
S
,
the collection
A
1
,
A
2
,
A
3
is collectively exhaustive.
Problem 1.2.2
(a) The sample space of the experiment is
S
=
{
aaa
,
aa f
,
a f a
,
f aa
,
f f a
,
f a f
,
a f f
,
f f f
}
(b) The event that the circuit from
Z
fails is
Z
F
=
{
aa f
,
a f f
,
f a f
,
f f f
}
The event that the circuit from
X
is acceptable is
X
A
=
{
aaa
,
aa f
,
a f a
,
a f f
}
(c) Since
Z
F
∩
X
A
=
{
aa f
,
a f f
} 6
=
φ
,
Z
F
and
X
A
are not mutually exclusive.
(d) Since
Z
F
∪
X
A
=
{
aaa
,
aa f
,
a f a
,
a f f
,
f a f
,
f f f
} 6
=
S
,
Z
F
and
X
A
are not collectively exhaus-
tive.
1