98
Chapter 7
Chapter 7
Problems
1.
Let
X
= 1 if the coin toss lands heads, and let it equal 0 otherwise.
Also, let
Y
denote the
value that shows up on the die.
Then, with
p
(
i
,
j
) =
P
{
X
=
i
,
Y
=
j
}
E
[return] =
∑
∑
=
=
+
6
1
6
1
)
,
0
(
2
)
,
1
(
2
j
j
j
p
j
j
jp
=
)
5
.
10
42
(
12
1
+
= 52.5/12
2.
(a) 6
⋅
6
⋅
9 = 324
(b)
X
= (6
−
S
)(6
−
W
)(9
−
R
)
(c)
E
[
X
] = 6(6)(6)
P
{
S
= 0,
W
= 0,
R
= 3} + 6(3)(9)
P
{
S
= 0,
W
= 3,
R
= 0}
+ 3(6)(9)
P
{
S
= 3,
W
= 0,
R
= 0} + 6(5)(7)
P
{
S
= 0,
W
= 1,
R
= 2}
+ 5(6)(7)
P
{
S
= 1,
W
= 0,
R
= 2} + 6(4)(8)
P
{
S
= 0,
W
= 2,
R
= 1}
+ 4(6)(8)
P
{
S
= 2,
W
= 0,
R
= 1} + 5(4)(9)
P
{
S
= 1,
W
= 2,
R
= 0}
+ 4(5)(9)
P
{
S
= 2,
W
= 1,
R
= 0} + 5(5)(8)
P
{
S
= 1,
W
= 1,
R
= 1}
=
+
+
+
⋅
+
+
)
9
)(
6
)(
6
(
200
6
2
6
360
9
2
6
384
2
9
6
420
3
6
324
3
9
216
3
21
1
≈
198.8
3.
]
[
a
Y
X
E
−
=
dydx
y
x
a
∫∫
−
1
0
1
0
.
Now
∫
∫
∫
−
+
−
=
−
1
0
1
0
)
(
)
(
x
a
x
a
a
dy
x
y
dy
y
x
dy
y
x
−
+
=
xx
a
a
du
u
du
u
0
1
0
)
1
/(
]
)
1
(
[
1
1
+
−
+
=
+
+
a
x
x
a
a
Hence,
]
[
a
Y
X
E
−
=
∫
+
+
−
+
+
1
0
1
1
]
)
1
(
[
1
1
dx
x
x
a
a
a
=
)
2
)(
1
(
2
+
+
a
a