Chapter 9
9.2 Let
ij
a
be element in ith row and jth column. Since saddle point is minimum of row and
maximum of column
j l
9.3 If we let
0x
for n heads,
1x
at least one tail
9.4
2
1
0
1
(, ) ( )
Rd ckx d
θ
θθθ
θ
9.5
2
2
2
0
2
()
k
xk
px k dx
θ
θ
1
θ
2
θ
1
θ
0 C
2
θ
C 0
Probability
2
2
1
k
θ
2
2
1
1k
θ
2
2
2
k
θ
2
2
2
1k
θ
Chapter 9 131
9.6 Maximizing
(, )
Rd
θ
with respect to
θ
yields
9.7
11
22
11
() , ( )
23
E x dx E x dxΘΘ
9.8
()
x
x
gx e d e
θ
θ
for
0
x
() 0
gx
elsewhere
9.9 (a)
1
(, ) (1 )
x
gx
θθ θ
1, 2, 3, …
x
1
(, ) (1 ) 1
x
fx
θθ θ
1, 2, 3, …
x
01
θ
Beta distribution with
2,
ax
β
132 Mathematical Statistics, 8E
(b)
1
21
10
() (1 ) ( 1)
x
x
cdx xx d
θθ θ θ
1
1
2() (1 ) ( 1)
x
cdx xxd
θθ θ θ
9.10 expand wait
Good times
164,000
80,000
0.4 4/11
Recession 40,000
8,000
0.6 7/11
(a)
(0.4)( 164,000) (0.6)(40,000) 41,600
E
(0.4)( 80,000) (0.6)( 8,000) 36,800
E
Manufacturer should expand now.
9.11 (a) expand wait
Good times
200,000
80,000
1/3
Recession 40,000
8,000
2/3
Chapter 9 133
(b) expand wait
good times
164,00080,000
2/5
recession 60,000
8,000
3/5
9.12 Reservation at
x Y (a) (b)
x 65 68.40 3/4 5/6
Y 72 62.40 1/4 1/6
9.13 go to
27 33 (a) (b) (c)
27 45 1/6 1/3 1/4
27
should go to
33 39 33 5/6 2/3 3/4
(a)
15
(27) (39) 37
66
ED
9.14 (a) If he goes to x worst cost is 72.00; if he goes to Y worst cost is 68.40. Worst cost is
minimized if he chooses Y.
134 Mathematical Statistics, 8E
9.15 (a) If he expands now, maximum gain is 164,000; if he waits maximum gain is 80,000.
Maximum gain is maximized if he expands now.
9.16 (a) opportunity losses are
0 84,000
9.17 (a) opportunity losses are
0 2.40
9.60 0
Maximum opportunity losses are 9.60 and 2.40; they are
minimized if she chooses Hotel Y.
(b) opportunity losses are
9.18 Expected losses with perfect information =
12
( 164,000) ( 8,000) 60,000
33
60,000 exceeds 28,000 and 32,000 by more than 15,000
Hiring the forecaster is worthwhile.
9.19 (a) Cross out first row, cross out second column, optimum strategies I and 2; value = 5
(b) Cross out first column, cross out second row, optimum strategies II and 1; value = 11
9.20 (a) Mimima of rows are
2, 0, 4
; only second is largest of its column. Saddle point
corresponds to I and 2; value = 0.
Chapter 9 135
9.21 (a) no glasses glasses
no
9.22
p 8
5
82(1)56(1)pppp
9.23 x
1
x
y 3
4
1
y
3
1
(a)
3 4(1 ) 3 (1 )
xxxx
11 5
x
5
11
x
9.24 x
1
x
66 68.40
66 68.40(1 ) 72 62.40(1 )
xxxx
72 62.40
6(1 ) 6
xx
1
xx
1
2
x
136 Mathematical Statistics, 8E
9.25 enemy attacks
y 2
1
y
10
x
212 2
12 10(1 ) 2 12(1 )
xxxx
country
9.26 (a) first person
1 4
second 0
1
2
9.27 first
lowers not
9.28 (a) 0 1/2 1
0 0 50 100
1/2 50 0 50
1 100 50 0
2
Chapter 9 137
(c) The risk functions are
1
d
2
d
3
d
4
d
5
d
6
d
7
d
8
d
9
d
0 0 0 0 50 50 50 100 100 100
(d) Bayes risks are
2
111
0 25 50 25
333
d
9.29 (a) 1/4 1/2
1/4 0 160
1/2 160 0
(b)
111 2 22
11 1 11 1
(0), (1); (2), (0); (1), (2);
44 4 44 2
dddddd
(c) The risk functions are
1
d
2
d
3
d
4
d
5
d
6
d
7
d
8
d
1/4 0 10 60 70 60 100 150 160
1/2 160 120 80 40 120 80 40 0
138 Mathematical Statistics, 8E
(d) The maxima corresponds to
1234 8
, , , , and dddd d
are 160, 120, 80, 70, and 160. So the
minimax criterion yields
4
d
.
9.30 (a) no inspection inspect 1 inspect 2
0 0
β
2
β
repeat 1
2
αβφ
2
2
α
β
φ
2
β
φ
2
22
αβφ
22
βφ
22
βφ
β
β
β
β
β
β
β
β
β
β
β
9.31
12
( ) ( , ) ( , ) (1,000 2,000)[ (1;10, ) (0;10, )]Rd Rd B B
δθ θ θ θ θ θ
As in the example, the first term always negative, and the second term is always positive; thus,
()
δθ
is always negative. As before,
1
d
dominates
2
d
and it is preferred.
9.32
( ) ( )[ (2; , ) (1; ; )].
wd
Cn C B n Bn
δθ θ θ θ
w
nC