Fifth Homework Set — Solutions
Chapter 4
Problem 17
(a)
P
{
X
= 1
}
=
P
{
X
≤
1
} 
P
{
X <
1
}
=
1
2

1
4
=
1
4
P
{
X
= 2
}
=
11
12

3
4
=
1
6
P
{
X
= 3
}
= 1

11
12
=
1
12
(b)
P
braceleftbig
1
2
< X <
3
2
bracerightbig
=
5
8

1
8
=
1
2
.
Problem 19
P
{
X
= 0
}
=
1
2
P
{
X
= 1
}
=
3
5

1
2
=
1
10
P
{
X
= 2
}
=
4
5

3
5
=
1
5
P
{
X
= 3
}
=
9
10

4
5
=
1
10
P
{
X
= 3
.
5
}
= 1

9
10
=
1
10
Problem 21
(a)
E
[
X
] is larger than
E
[
Y
] because the random selection of students
favors larger busloads.
(b)
E
[
X
] =
40
·
40+33
·
33+25
·
25+50
·
50
40+33+25+50
=
5814
148
= 39
.
3,
E
[
Y
] =
148
4
= 37.
Problem 23
(a) Suppose that that you use
x
dollars to buy
x/
2 ounces of the
commodity and keep the rest 1000

x
dollars as cash, and then
sell your commodity at the end of the week. Then the expected
amount of money you have at the end of the week is
1
2
x
2
+
1
2
2
x
+ 1000

x
= 1000 +
x
4
which is an increasing function of
x
. Therefore the best strategy
is to use all your money to buy 500 ounces of the commodity and
then sell at the end of the week.
1
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(b) Suppose that you use
x
dollars to buy
x/
2 ounces of the com
modity at the beginning of the first week and use the remaining
1000

x
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 Probability theory, Order theory, Monotonic function

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