Chapter 10 151
10.47 Try
123
0, 1, 0, 2xxxY
The only possibility is
12 3
1, 0, 1xx x
2
(0,1,0) (1 )f
θθ
10.49
22 22
(1/2) ( ) / ˆ
(/2 )
1/2 /2
11
()
(2 ) (2 )
i
xn
nnn nn
fx x e e
µσ σσ
πσ πσ
Depends only on
22
ˆ
and
σσ
sufficient.
10.50
22
12
ˆ, mm
µµσ
22
21
ˆ()mm
σ
µ
µ
10.54
1
β
1
α
µα
1
1m
α
α
11
mm
αα
1
11
1
ˆ
(1 ) , 1
m
mm m
αα
10.55
1
2
0
2ˆ
() , 3
3
xxdx m
θ
θ
µθ θ
θ
µ
152 Mathematical Statistics, 8E
10.57
1
αβ
22
2
11
()()
βα αβ
10.58
38
µ
012 3
1
01 2 3
3
nnnn
mN
θ
122
23
ˆ
3
nnn
N
θ
10.60
11
(1)
(; ) ()(1)
bx x x
αα
α
αα
α
Γ
ΓΓ
1
() ( )
n
i
Lx
α
αα
ln ( ) ln( ) ( 1) ln
i
Lnn x
αα
1
ln ( ) ln
ln
i
n
i
i
dL n x
d
n
x
α
αα
α
Chapter 10 153
10.62
2
(1/2)[( )/ ]
1
() 2
x
fx e
µ
σ
πσ
22
(1/2 ) ( )
1
() (2 )
x
nn
Le
σµ
σπσ
10.63 (a)
1
1
8m
µ
1
11
ˆ
mx
θ
10.64
2
() 2
x
fx xe
α
α
2
() 2
x
nn
i
Lxe
α
αα
2
ln ( ) ln 2 ln ln
i
Lnn x x
ααα
2
ln ( ) 0
Ln
x
d
α
αα
2
ˆn
x
α
10.66
()/
1
() 8
x
fx e
δθ
(1/ ) ( )
1
(, )
x
n
Le
θδ
θδ θ
154 Mathematical Statistics, 8E
10.67
1
()
fx
β
α
1
(,) ()
n
L
αβ
β
α
To maximize
1
ˆ
y
α
, and
ˆ
n
y
β
10.69
1/
1
() ()
x
fx x e
αβ
α
βα
Γ
(a)
1(1/ )
1
() [()]
i
x
i
nn
Lxe
αβ
α
ββα
Γ
1
ln ( ) ln ln ( ) ( 1) ln
ii
Ln n x x
βαβ αα β
Γ
2
ln ( ) 1
i
dL n x
d
β
α
ββ
β
ˆ
i
xx
n
βαα
(b)
2
21
x
τα
10.70
22
2(1/2) [ ( )] (1/2) [ ( )]
(,) 2
nvw
Le
αβ αβ
αβ π
11
Chapter 10 155
10.71
11
22
Vn
Wn
µ
σ
µ
σ
222 2
12
12 12
(1/2 ) ( ) (1/2 ) ( )
1
2
vw
nn nn
Le
σµσ µ
πσ
µ
µ
10.72 Any value
ˆ
θ
will do so long as
1
11
ˆˆ
and
22
n
yy
θθ
2
11
ˆˆ
+ and
22
n
yy
θθ
1
11
ˆ
22
n
yy
θ
10.73 (a) It is if
11
11 1
()
22 2
nn
YYYY
156 Mathematical Statistics, 8E
10.74
()
x
Ex n
α
θαβ
where
2
00
02
0
(1 ) 1
θσ
αθ σ
0
σ
10.75
40 1
40 40 2
α
µαβ
2
2
40 40 1
324
80 81
1
18
σ
σ
10.76
22
0
10 0
22 22
00
(1 )
n
xxww
nn
σσ
µµ µ
σσ σσ
2
0
22 2
0
2
0
nn
wnn
σ
σσ σ
σ
QED
Chapter 10 157
(b)
()
() 1
x
Ex
αβ
β
Λ
from Theorem 6.3
10.78
25 50
(27.6) (38.1) 34.6
75 75
3
10.81
2
2
ˆ()
nx
xx
α
2
()
ˆxx
nx
β
158 Mathematical Statistics, 8E
5
10.83 The likelihoods are
33
13
4
N
N
N Likelihood N Likelihood
36
13 320
9 0.4762
9126
4
39
13 384
12 0.5091
12 495
4
10.84
1
ˆ3m
θ
0.39x
1
0.39 0.065
6
m
0.39
ˆ3 0.195
6
θ
10.85
2
5524, 2,570,176xx
n = 12
10.86
1
ˆ403y
δ
ˆ460.33 403 57.33
θ
10.87 n = 8
63.1x
2
541.55x
1
63.1 7.8875
8
m
2
541.55 67.69375
m
Chapter 10 159
10.88
ˆ
ˆ4.1 and 11.5
αβ
1
ˆy
α
ˆ
n
y
β
4.35025
4.33244
4.45179
4.42813
4.49693
4.35603
4.36361
66.24567
10.90 n = 3 N = 20
011n
17n
22n
30n
72230 11
ˆ
320 60
θ
67
10.92
107.4v
2116,108v
110n
674w
276,246w
26n
1
1074
ˆ107.4
10
µ
2
674
ˆ112.3
6
µ
2116,108 115,347.6 76,246 75,712.7 1,293.7
ˆ80.86
16 16
σ
160 Mathematical Statistics, 8E
10.94
00.74
θ
00.03
σ
30n
18x
(a)
ˆ0.74
θ
10.95
1715
µ
19.5
σ
712 715 0.32
9.5
z
725 715 1.05
9.5
z
0.1255 0.3531 0.4786p
µ
µ
10.97 (a)
ˆ50 2 100
µ
αβ
(b)
ˆ112x
µ
(c)
1
2(50 112)
ˆ108
3
µµ
10.98
2
22
2
2.575 4.2 467.9.
0.5
z
nE
σ
Rounding up to the next integer, n = 468.
Chapter 10 161
10.100 The sample is more likely to include longer sections than shorter ones; They take more time to
pass the inspection station.