Chapter 3
3.1 (a) No, because f(4) is negative; (b) Yes; (c) No, because f(1) + f(2) + f(3) + f(4) =
18
19
is
less than 1.
3.4 (a)
(1 2 3 5) 1;c
thus
1
15
C
(b)
555
511
234
c
; thus,
12
137
c
14
3.5 For
() (1 )
x
fx kk
to converge to 1, 0 < k < 1.
24 Mathematical Statistics, 8E
3.10
41
(0) ;
20 5
f
26 12 3
(1) ,
20 20 5
f
41
(2) 20 5
F
0
1/5
0
01
x
x
3.12
()Fx
0
1/15
3/15
6/15
10 /15
1
1
12
23
34
45
5
x
x
x
x
x
x
25
()Fx
0
3/25
7/25
12 / 25
18 / 25
1
1
12
23
34
45
5
x
x
x
x
x
x
Chapter 3 25
3.18 (a)
() 0,fx
0,x
and
0
00
() 1
x
fxdx e dx e
(c)
1
1
(1)
x
Px e dx e
3.20 (a)
3.2
2
3.2
11 1
( 1) (8.32 4) 0.54
2
882 8
z
y
ydy y
(b)
3.2
2.9
3.2
11 1
( 1) (8.32 7.105) 0.1519
2.9
882 8
x
y
ydy y
26 Mathematical Statistics, 8E
(b)
2
12.9
(3.2) (2.9) 0.54 2.9 4 0.54 0.3881 0.1519
82
FF
3.23
1
() 2
Fx x
0
1
() 2
1
Fx x
0
04
4
x
x
x
1111
4224
F
and 1 =
11
(1) 1 22
F
3.26
23
1/4
1315
(3 2 ) 0
4163232
Px x x
1
23
1/2
1
1311
6(1 ) (3 2 ) 1
1/2
2442
Px x xdx x x
Chapter 3 27
3.28
2
() 2
x
x
Fx xdx
0 to 1
3.29
0
11
() 33
x
Fx dx x
0 to 1
1
() 3
Fx
1 to 2
0 0
1
0 1
3
x
xx
3.30 (a)
11.2 22
0.8 1
11.2
11
(2 ) 2 0.32) 2.4 0.72 2 0.36
0.8 1
222 2
xx
xdx xdx x
3.31
0x
() 0Fx
28 Mathematical Statistics, 8E
3.32 (a)
11311
;
22442
FF
(3) (2) 1 1 0FF
3.34 (a)
916
(5) 1 25 25
F
(b)
99
1(8)11
64 64
F
3.37
2
( 2) (2) 1 3 1 3(0.1353) 1 0.4074 0.5926Px F e
3.38
x
dF xe
dx
for > 0; 0 elsewhere
3.39 (a) for
0x
() 0Fx
(b) for
00.5x
1
() 2
Fx x
Chapter 3 29
3.42 (a)
1;
20
(b)
113
;
488
(c)
11 1 1
;
6412 2
(d)
11 1 28 7
6 24 40 120 30
3.45 (a)
129
(10 9 10)
89 89
; (b)
15
(1 4)
89 89
(c)
155
(9 5 10 13 18)
89 89
3.47
3.48 (a)
(, )0Px y
(b)
(, )1Px y
(c)
( , ) ( , ) three probabilitiesFbc Fac
(,) (,)Fbc Fac
30 Mathematical Statistics, 8E
3.51 (a)
1
2
(b)
122 4 5
12 1
233 9 9
(c)
121 111 3 1
2233 233 9 3
(, ) 2
Fxy xy
for
0, 0, 1
xyxy
3.54
22 22 22
()
22 4 4
xy xy xy
Fxe ye xy e xye
yx
0, 0
xy
and
(, ) 0
fxy
elsewhere
3.56
2
xxy xy
FF
ee e
xxy
0, 0
xy
= 0 elsewhere
Chapter 3 31
3.59 a = 1, b = 3, c = 1, d = 2
32 12 31 11
(3,2) (1,2) (3,1) (1,1)
(1 )(1 ) (1 )(1 ) (1 )(1 ) (1 )(1 )
FFFF
ee ee ee ee
3.60
44 14 14 11
(2,2) (1,2) (2,1) (1,1)
(1 )(1 ) (1 )(1 ) (1 )(1 ) (1 )(1 )
F FFF
ee ee ee ee
3.61
(3,3) (2,3) (3,2) (2,2)
FFFF
3.62 x = 1, 2
3.63 (a)
11
(1 2)
54 18
(b)
1147
(8 6)
54 54 27
3.64 (a)
191
(1 2 2 4)
54 54 6
; (b) 0; (c) 1
32 Mathematical Statistics, 8E
3.66
1
1/ 2 1/2
00 0
144 (1 ) 0.15625
xy
x
xy z dz dy dx
3.69 (a) 13
(1) , (1)
44
gg
(b) 51 1
(1) , (0) , (1)
84 8
hh h
(c) 1/8 1
(1 1) 1/8 1/2 5
f
; 1/2 4
(1 1) 1/8 1/2 5
f
(b) 111 7
(0) 12624 24
h; 11 1 21
(1) 4440 40
h
11 7
(2) 820 40
h ; 1
(3) 120
h
Chapter 3 33
3.71 (a) ( , ) (1 2) for 1,2,3
108 36
xy xy
mxy x
; 1, 2, 3y
3.72 (a) 5
(0) 12
g, 1
(1) 2
g; 1
(2) 12
g
0
5/12
() 11/ 12
1
Gx
0
01
12
2
x
x
x
x
3.73 (a)
1
() 2
fx
for
1, 1x
;
1
() 2
gy
for
1, 1y
;
11 1
22 4
, independent
3.74 (a)
22
0
2
11 11
(2 ) 2 (4 2) (2 1)
0
44242
y
xydy xy x x
for
01x
= 0 elsewhere
34 Mathematical Statistics, 8E
3.75 (a)
1
2
0
1
111
(2 ) ( ) (1 )
0
444
xydx x xy y
for
02y
= 0 elsewhere
3.76 (a)
123
2
0
223
33 3
1
( ) 24 ( ) 24 0
22 3
12(1)12(1)8(1)
12(1 ) 8(1 ) 4(1 )
x
x
yxy y
fx y xy y dy
xxx x
xx x
3
4(1 ) 0 1
() 0 elsewhere
xx
fx
3.77