Chapter 4
4.1 (a) 0, 1, 4, 9
4.2 Replace
by
4.3 Replace
by
4.7
432
2
2
4
111648
() ( ) 8 2
2
8832833
yy
EY y y dy
1 56 1 74 37
6
83 83 12
4.9 (a)
1 12 48 64 12 96 192 300 12
() 0 1 2 3 2.4
125 125 125 125 125 125 5
Ex
Chapter 4 47
4.11 (a)
123
22
012
33
()
22 2 2
xxxx
E x dx dx dx
123
32 23
2
012
38
()
22 2 3
xx xx
E x dx dx dx
2
83 11
(53) 5 3
32 6
Ex
4.13
1
2
00
11
22
y
xx
Edxdydy
yy
4.14
1
54
k
for x
1181
(1) (122436)
54 54 3
g
2
(2) 3
g
48 Mathematical Statistics, 8E
4.17
0
0
( ) () () 1x fxdx fxdx
μμ
1
1
( ) () () () 0x fxdx xfxdx fxdx
μμ μ μμ
4.20
3
1
1
3
11 1 31
(3 1)
1
ln3 ln3 (ln3) (ln3)
rr
rr
r
x
xdx rr r
μ
2
ln3
μ
,
2
84
,
2(ln3) ln3
μ
2
22
444(ln31)
ln 3 (ln3) (ln3)
σ
Chapter 4 49
4.23
11
( ) ( ) ( ) 0 exists
x
Ez E Ex
μμμμ
σσ σ
4.24
21
1
2
( ) 2 [ 2 ] 2 exists
1
Ex x dx x
2
1
22ln 1
dx x
x
μ
2
σ
does not exist
μ
μ
μ
4.26 (a)
1(0.05) 2(0.15) 3(0.30) 4(0.30) 5(0.15) 6(0.05) 3.50
μ
222222
2
1 (0.05) 2 (0.15) 3 (0.30) 4 (0.30) 5 (0.15) 6 (0.05) 13.70
μ
(b)
3.5,
μ
2
13.70
μ
,
32
3
1(0.05) 2 (0.20) 6 (0.05) 57.8
μ
50 Mathematical Statistics, 8E
4.27 (a)
0
μ
by symmetry,
2
0
μ
by symmetry
μ
μ
μ
(b)
0
μ
and
3
0
μ
by symmetry
μ
μ
μ
4.29
0
() () () ( )
( ) QED
a
aa
xfxdx xfxdx a fxdx aPx a
Px a
a
μ
μ
4.31 (a)
2
1
10.95
k
,
2
11
0.05 20
k
,
20 4.47k
(b)
2
1
10.99
k
,
2
11
0.01 100
k
,
10k
4.33
0
11
() 0
ttx t
tx
x
ee
Mt edx tt
Chapter 4 51
4.34
1
23
12
() 2 2
33 3
13
t
x
xtt
tx
xt
t
e
ee
Mt e e
e
4.35
1(0)
() ln (), () (), (0)
() (0) 1
x
xxx xx
xx
MM
Rt M t Rt M t R
Mt M
μ
μ
4.36
(0) 0 1
x
M
52 Mathematical Statistics, 8E
4.37
0
0
11
22
tx x tx x
e e dx e e dx
yx
cy dx
4.38
2
2
() 1 2!
x
t
Mt t
(a)
0
μ
,
2
2
σ
4.39 3.
[( )/ ] / / /
()/
( ) ( ) ( ) QED
x a b t at b xt b at b
xa b x
t
M t e fxdx e e fxdx e M b
4.40
1(3)
4
zx
,
3,a
b = 4
z
4.42
( 3, 5), ( 1,1), (1,1), (3,5)
probabilities are ¼
Chapter 4 53
4.43
56 56 8 72
()0 1 2 0.6
120 120 120 120
EX
4.44
3
11 11
2
221232111221
000 00
() ( ) ( )
x
E x x x x e dx dx dx x x x dx dx
3
11
2
23 3 3 1 12 2 1
000
1
21
12
0
() ( )
11 7
()
23412
x
E x x x e dx x x x dx dx
x
xdx
13
77
cov( , ) 1 0
12 12
xx
54 Mathematical Statistics, 8E
4.46 (a) 11 11
( 1,1) , (0,0) , (0,1) 0, (1,0) , (1,1)
46 122
fffff
(b) 1111
(0,0) , g(0) (0) , (0,0) (0) (0)
66424
fh fgh
4.47 (a)
01
22
1111
() ( ) ( ) 0
2323
E U x x dx x x dx
4.48 (a)
11
11
()
()
ii
ii
tx
kk
i
tx
ikk
e f x x dx dx
t
xe f x x dx dx
at
0
i
ts
11
()
ik ki
x f x x dx dx
μ
Chapter 4 55
4.49 (a)
2
2(4) 3(9) 4(3) 7
4(3) 9(7) 16(5) 155
Y
Y
μ
σ
(b)
2
1(4) 2(9) 1(3) 19
1(3) 4(7) 1(5) 36
Z
Z
μ
σ
4.51
12 1
22
00 0
11125
() ( ) (2 2) 1
33339
Ex x xy dydx x x dx
12 1
232 32
00 0
111127
() ( ) (2 2)
3332318
Ex x xy dydx x x dx
2
725635013
18 81 162 162
X
σ
4.53
var( ) var()var()2cov(,)
XY X Y XY
12
1, 1
aa
var( ) var( ) var( ) 2 cov( , )
XY X Y XY
12
1, b 1
b
cov( , ) var( ) var( ) 0 cov( , ) var( ) var( )
XYXY X Y XY X Y
4.54
12
cov( , ) (2)5 (6)(4) 12(7) 7(3) (2)(2)
YY
4.55
cov( , ) 213 327 415 6 42 20 56
YZ
56 Mathematical Statistics, 8E
4.57
12
( 1,2) (11,2) , (21,2)
33
fz
φφ
222
1218
(1,2)1 2 3
3333
Ez
4.59
3
23 2
11
,22
x
fxx x e
23
0 1 and
xx
σ
3
12
23
2
23 2 2 2 3
00
1
22
11 5
1
46 12
x
x
E x x x dx x e dx
Chapter 4 57
a
4.61 (a)
6
(0) (0) 10 0.0001 100
EN
(b)
(0) (0)( )1000.9898
Ep N PpD
(c)
( ) ( ) ( ) 999,900 0.03 29,997EpD NDPpD
4.64 (a)
514.6
1 0.4 $0.77
666
(b)
4117.8
2 0.6 0.8 $1.30
6666
(c)
1113
1.2 0.2 1.6 3 $1.60
6666
(d)
11112
1.6 0.2 1.2 2.6 4 $1.67
66666
(e)
111111
2 0.6 0.8 2.2 3.6 5 $1.50
666666
Expected profit is greatest if he bakes four cakes.
58 Mathematical Statistics, 8E
4.67
2
/3 2 (1/3) (1/3) (1/3)
0
1
() 3 18 54 0
99
154 6 million liters
9
xxxx
x
Ex e dx xe xe e
4.69 p = probability Adam will win
(1 ) , ( ) ,
a
p b pa pa b a p ab
4.71
/4
/4
0
111
14
0
441/164
x
x
e
edx x
μ
2/4
22
0
112
32
44
1
4
x
xe dx
μ
232 16 16
σ
4.73
(0)0.4,.(1)0.3, (2)0.2, (3)0.1gggg
0(0.4) 1(0.3) 2(0.2) 3(0.1) 1
μ
1
μ
2222
20 (0.4) 1 (0.3) 2 (0.2) 3 (0.1) 2
μ
22
21 1
σ
Chapter 4 59
4.74 (a)
41
( 65) 0.631
65
Px
4.75
60 1 63
124, 7.5, (7.5) 60, 8, 1 ,
7.5 64 64
kkp
μσ
at least
63
64
4.77
11.255
4, 4 at least 1 2.25 2.25 9
μσ
By Chebyshev’s theorem probability
(10)Px
is at least 59.
10
(1/4) (1/4)
0
2.5
10
1
0
4
1 1 0.0821 0.9179
xx
edxe
e
4.79 y y
3
x
μ
0.02
x
σ
x
0.3
y
μ
0.005
y
σ
independent
( 2 ) 3 2(0.3) 3.6Ex Y
222
2
(0.02) 4(0.005) 0.0005
xy
σ
0.0005 0.0224
σ
60 Mathematical Statistics, 8E
4.81 (a) X heads
Y getting 6
Z getting ace
4.82
5(0.5) 5(0.45) 4.75
μ
2
5(0.5)(5) 5(0.45)(0.55) 1.25 1.2375 2.4875
σ
1.58
σ
4.83
(0 0) 3 / 10, (1 0) 6 / 10, (2 0) 1/10
φφφ
( ) 0(0.3) 1(0.6) 2(0.10) 0.8EY
4.85
1
1
()
()
xf x dx
N
ED
fx dx