Chapter 6
6.1
()
11
[()]
p
dx p p
a
αβα
α
αβαα
ββα
6.2
22
11 1 1
()()
22 2( ) 2
xdx
a
β
α
β
ααβ
μβαβα
βα β βα βα
μ
6.3
1
()
x
x
Fx dx
α
α
β
αβα
0
( )
1
x
x
fx x
x
α
ααβ
βα
β
6.4
11
[2 ( )]
2()2
r
r
rr
xdx x dx
ββ
αβ
μαβ
βα βα
6.5
4
22
4
12 34
1( ) ( ) 1 ) 1
0, , 0, ( )
34 12 52 80
βα βα βα
μ
μμμβα
6.6 Intergals do not exist.
Chapter 6 81
6.7
1
()
x
xedx
α
α
Γ
1
ux
α
6.8
2
1
2
yz
dy z dz
2
2
1
2
1(1/2)
00
121(1/2)
0
()
2
2
yz
z
z
y e dy e z dz
zer dz
α
α
αα
α
Γ
6.10 (a)
/3 /3
44
/3
4/3 4/3
11
2, 3, 4,
91 9
11 777
1 (0.2645) 0.6171
91/9 3 3 3 3
xx
x
x p xe dx xe dx
exe e
αβ
(b)
2/4 2/4
44
11
3, 4, 0.7818
64 2 128
xx
p xedx xedx
αβ
82 Mathematical Statistics, 8E
6.14
3233
3
3233
(1)(2) 3(1) 2
[( 1)( 2) 3 ( 1) 2 ] [2] 2
μα α βααβαβαβ
αβ α α α α α αβ αβ
2
323/2
22
()
αβ
ααβ α
4322244
4
4234
(1)(2)(3) 4(1)(2) 6(1) 3
2[(1)(2)(3)4(1)(2)6(1)3]
μ
αα α α β αα α β αβ αα β αβ αβ
βα α α αα α αα α αβ
4
4
24
(3 6) 6
3
αβ α
αα
αβ
θ
6.18
22 33 44
22 33 44
1(1)
2! 3! 4!
ttt
ttttt
θθθ
θθθθθ
Chapter 6 83
6.19
, 2
2
ν
αβ
See 6.11
From 6.11
(1)2 1 2
2
x
ν
βα ν
02
ν
function
when 0
x
2
ν
function has absolute maximum at x = 0
6.21
1
1
r
r
xdx
α
μα
exists only if
11
2
r
r
α
α
6.22
1
1
1
1
11
x
xdx
α
α
α
μα α αα
6.24 (a)
/
/
//
0
/
/
1
( ) ( ) 1 1
1
() 1
1()
t
xutt
t
t
fx e Ft e du e e
e
ft
Ft e
θ
θθθ
θ
θ
θ
θ
θ
84 Mathematical Statistics, 8E
6.25 (a)
1245
33
0
1
(6) 3
(1 ) 20 0
(2) (4) 2 4 5
131 1
20 1 20 1
245 20
xxx
xxdx x
Γ
ΓΓ
(b)
1
22
0
(6) 111 1
(1 ) 30 30 1
(3) (3) 3 2 5 30
xxdx
Γ
ΓΓ
6.28
1
11
2
0
() (1 )
()()
( )(2)() (1)
()() ( 2) ( 1)( )
xxdx
αβ
αβ
μαβ
αβ α β α α
αβ αβ αβ αβ
Γ
ΓΓ
ΓΓΓ
ΓΓ Γ
Chapter 6 85
6.30 (a)
1() 1
() 5
df x d x d
f x dx bx bx
/
(1/ )
/
1
ln ( ) ln
() 1
ln , ( )
bd b x
bd
d
fx x x c
bb
fx xfx kx e
b
x
6.31
2
2
(1/2)[( )/ ]
11
( ) ln ( ) ln 2 2
2
x
x
fx e fx
μσ
μ
πσ σ
πσ
(a)
2
1
ln ( ) 2
1()1 1
0
()
x
fx k
df x x x x
fx dx
μ
σ
μμ
μ
σσ σσ
μ
6.32
2
1() ()
ln ( )
() 2
df x d x d x
fx c
fx dx a a
1/2 2
( ) ( ) QED
a
fx ke x d
86 Mathematical Statistics, 8E
6.33
22 2 2 2 2
2222
[()]()[2()]
( )[( ) 3 ]
Mt tMMt
Mt t
μ
σ σ μσ μσ σ
μσ μσ σ
μ
μ
μ
6.35
34
4
0 and 3
σ
αα
σ
6.36
22
(1/2)
()
tt
x
Mt e
μσ
22 2
( / ) ( / ) (1/2) ( / ) (1/2)
()/
tt t t
x
Mee e
μσ μ σ σ σ
μσ
6.37
222 33 2
( ) , ( ) , ( ) 3Ex Ex Ex
μ
σμ μ μσ
23 2 22 2
cov( , ) ( 3 ) ( ) 2xx
μ
μσ μ σ μ μσ
Chapter 6 87
6.39
234
12 3 4
234
12 3 4
( ) ( ) ln ( )
() 1 2! 3! 4!
ln ( ) ln 1 2! 3! 4!
t
xx x x
x
x
MeMt Kt tMt
ttt
Mt t
ttt
Mt t
μ
μ
μ
μμ μ μ
μμ μ μ
6.40
22
(1/2)
()
tttt
xx
MeMte
μ
μμ σ
μ
22
22
1
ln ( ) 2
1
( ) 2
x
x
Mt t
Kt t
μ
σ
σ
2
12
0, ; 0 for 2
r
KK K r
σ
6.41
(1)
( ) ,
t
e
Mt e
λ
μ
λσ λ
88 Mathematical Statistics, 8E
6.42
( ) (1 ) ,
x
Mt t
α
β
μαβσβα
6.43 (a) Constant terms of
12
11
( ) and ( ) are and
22
gx hy
σπ σπ
Constant term of
2
22
1
(, )
21
fxy
p
πσ σ
If independent then
2
212
12
111
11, 0
22
21
pp
p
σπσ π
πσ σ
(b) Substitute p = 0 into f(x, y) and it becomes product of g(x) and h(y).
6.44 Substitute y = a + bx into f(x, y)
μ
6.46 Equating coefficients of
22
, , and xxy y
with those of bivariate normal density
22
ρ
ρ
ρ
27 (1 )
ρ
σ
multiply first and third and divide by square of second
Chapter 6 89
6.47
1212
2
2, 5, 3, 6,
3
p
μμσσ
6.48
, UXYVXY
12 1 2
() , ()EU EV
μ
μμμ
22
12
22 22
1 2 12 1 2 12
22
12
2
22 222
12 12
22
4
σσ
ρ
σσ ρσσσσ ρσσ
σσ
ρ
σσ ρσσ
6.49 (a)
22 22
1 1 2 2 1 1 1 212 2 2
(1/2)[ 2 ]
12
(, )
tt tptt t Q
Mt t e e
μμ σ σσ σ
2
111 122 112
1
() at 0
Q
tte tt
t
μσ ρσσ μ
6.50 (a)
0.003 (0.002) 0.005 1 ;
0.03 0.030 6
(b)
2(0.1) 2
0.03 3
90 Mathematical Statistics, 8E
6.51
()
22
aa
xax a
6.52
3, 2
αβ
6.53
80 2 160nn
μαβ
160
160 8 8 0 100
2
dE
Enn n
dn n
6.55 (a)
(1/40) /40 1/2
20
10.6065
20
40
xx
edxe e
(b)
30
(1/40) /40 3/4
0
30
11 1 0.4724 0.5276
0
40
xx
edxe e
Chapter 6 91
6.59
0.5 0.5 1.5
3
0.5 0.2231
3
tt
B
edt e e
λ
6.61
0.025, 0.5
αβ
(a)
2
2
2
(0.025) (3) 3200 hours
(0.025)
μ
Γ
(b)
1
4000
x
xe dx
β
βα
αβ
1
0.025 4000 1.58
yx y
dy x dx
β
β
α
αβ
1.58
1.58
0.2060
y
edy e
(d)
0.4713 0.1700 0.6413
6.63 (a)
0.5 0.3729 0.1271
(b)
0.5 0.1406 0.6406
(c)
0.1772 0.359 0.1413
(d)
0.2190 0.3686 0.5876
92 Mathematical Statistics, 8E
6.65 (a) z = 1.92
6.66 (a) 2(0.3413) = 0.6826
6.67 (a)
0.05
1.645z
0.4500
6.68 (a) Using MINITAB and entering
2.159
and 0.5670 into C1, then giving the commands
(b)
1
2.159 1.786 0.958
1.0416
z
2
0.5670 1.786 2.25
1.0416
z
The corresponding cumulative probabilities are obtained from Table II (with
interpolation) to be 0.3602 and 0.9881. Thus the required probability is
0.9881 0.3602 0.6279
6.69 (a) Using MINITAB and entering 8.626 into C1,
MTB> CDF C1;
(b) z =
8.625 5.853 2.0367; 0.5 0.47915 0.02085
1.361 p
6.70 (a)
44.5 37.6 1.5 0.5 0.4332 0.0668
4.6
z
Chapter 6 93
6.71 (a)
16 15.40 1.25 0.5 0.3944 0.1056
0.48
z
6.72
82.5 0.92
10
82.5 9.2
73.3
58.3 73.3 1.5
10
0.5 0.4332 0.9332
z
μ
μ
μ
μ
6.74 (a)
3.2, (1 ) 15.68, Nonn
θθ
(b)
6.5, (1 ) 58.5, Yesnn
θθ
(c)
117.6, (1 ) 2.4, Nonn
θθ
94 Mathematical Statistics, 8E
6.76
12
1 6.5 7 7.5 7
14, 7, , 0.27, 0.27
2 1.871 1.871
nx z z
θ
2(0.1064) 0.2128 Table yields 0.2095
ρ
6.79
225, 0.2, 45, 6n
θμσ
40.5 45 0.75
6
0.5 0.2734 0.2266
z