Chapter 3 35
3.78 (a)
3
2
12 12
21 3
1
2
()
(,) 1
1
2
2
x
x
xxe xx
fxxx
x
xe
3.79
–
() (,) gx f xy dy
0
() (, ) (, )
x
Gx f xy dy Fx
2
1 0
() (, )
0 elsewhere
x
ex
Gx Fx
(a)
3
3
13
23 11 1 3
13
0 0 or 0
1
(,) ( 1)(1 ) 0 1, 0
2
1 x 1, 0
x
x
xx
Mx x x x e x x
ex
36 Mathematical Statistics, 8E
3.81
11
1
1 0 1
() 2
0 elsewhere
xx
gx
22
2
1 01
() 2
0 elsewhere
xx
hx
3.82 (a)
102, 03
(, ) 6
0elsewhere
xy
gxy
3.83 Heads Tails Probability H-T
0 4 1/16
4
3.84 1 2 3
1 3 4
(a) x 3 4 5 6 7
()fx
1/6 1/6 2/6 1/6 1/6
0 3
x
Chapter 3 37
3.85
2
() 3
PH
3.86
0
1/27
() 7/27
19 / 27
1
Fx
0
01
12
23
3
x
x
x
x
x
720
1 27 27
19 8
1 27 27
(a)
(b)
3.88 (a)
0.20 0.10 0.30
(b)
1 0.70 0.30
3.91 (a)
1(228.65 227.5) 0.23
5
; (b)
1(231.66 229.34) 0.464
5
;
(c)
1(232.5 229.85) 0.53
5
38 Mathematical Statistics, 8E
3.94
322
20,000 20,000 10,000
() 1 1
( 100) 2( 100) ( 100)
Fx dx c
xxx
(a)
2
10,000 1
1 (200) 9
300
F
(b)
10,000 3
(100) 1 40,000 4
f
3.96
/3
/3 /3
0
111 1
() 1 1
991/93 3
xx
xx
e
Fx xe dx c x c c e x
c =1
(a)
22
(6) 1 3 1 3 1 3(0.1353) 0.5491Fee
(b)
3
1 (9) 4 4(0.0498) 0.1992Fe
Chapter 3 39
3.98 (b)
3.99
1
(0,3) 8
f
,
3
(1, 2) 8
f
,
3
(2,1) 8
f
,
1
(3,0) 8
f
1
(0, 3) 8
g
,
3
(1,1) 8
g
,
3
(2,1) 8
g
,
1
(3,3) 8
g
3.101 (a)
0.3 0.3
0.2 2 0.2
0.3 2
20.40.6
0.2
5 5
2
0.3
55
5 0.3038
0.2
22
ps ps
p
p
pe ds dp e dp
e
edp e e
(b)
0.30 1 0.30 0.30
1
5 5 5(1)
0
ps ps p
pe ds dp e dp e dp
40 Mathematical Statistics, 8E
55
3.103 (a)
5
(0) 14
g
,
15
(1) 28
g
and
3
(2) 28
g
(b)
3
(0 0) 10
φ
,
6
(1 0) 10
φ
, and
1
(2 0) 10
φ
11 1 1
(b)
1
0
22
() (4) (2)
55
gx x y dy x
3.105 (a)
48 47 188
(0,0) 52 51 221
f
,
48 4 16
(0,1) 52 51 221
f
448 16
(1, 0) 52 51 221
f
,
48 4 16
(1,1) 52 51 221
f
,
43 1
(1, 2) 52 51 221
f
Chapter 3 41
3.106
(,) 5
ps
fps pe
0.2 0.4p
0s
(a)
50.20.4
5 5 5
00elsewhere
ps
ps ps
p
e
pe ds p e
p
0
3.107 (a)
/2
20 10 20
120 50
25 0 elsewhere
x
x
xx
xdy
x
3.108
2
(, ) (2 3)
5
fxy x y
2
1
23 23
() 2 2
0
5252
y
gx xy x
42 Mathematical Statistics, 8E
3.109 (a)
3
123
333
123 223
(20,000) 0, 0, 0
( , , )
( 100) ( 100) ( 100)
0 elsewhere
xxx
fxx x xxx
3.110 (a) 5 9 4 5 7 9 9 8
6 1 3 5 0 2 1 7 0 8 4 5 2 0 2 1 3 1
3.111 *=Station 105 o = Station 107
3.115 Class Limits Frequency
40.0 44.9 5
45.0 49.9 7
Chapter 3 43
3.117 The class boundaries are: 39.95, 44.95, 49.95, 54.95, 59.95, 64.95, 69.95, 79.95;
the class interval is 5;
the class marks are: 42.45, 47.45, 52.45, 57.45, 62.45, 67.45, 72.45, 77.45.
3.118 The class boundaries are: 2.95, 4.95, 6.95, 8.95, 10.95, 12.95, 14.95;
3.116 Class Limits Frequency
3.0 4.9 15
5.0 6.9 25
3.119 Class Limits Frequency Class Boundary Class Mark
0 1 12 0.5 1.5 0.5
2 3 7 1.5 3.5 2.5
3.120 Class Limits Frequency Percentage
3.0 4.9 15 18.75%
5.0 6.9 25 31.25
3.121 Class Limits Frequency Percentage
40.0 44.9 5 5.0%
45.0 49.9 7 7.0
44 Mathematical Statistics, 8E
3.122 Percentage
Class Limits
Shipping
Department
Security
Department
0 1 43.3% 45.0%
100.0 100.0
The patterns seem comparable for the two departments.
3.125 Cumulative Percentage
Class Limits
Shipping
Department
Security
Department
1.5 43.3% 45.0%
3.123
Upper Class Boundary
Frequency
Cumulative
Frequency
44.95 5 5
49.95 7 12
3.124
Upper Class Boundary
Frequency
Cumulative
Frequency
4.95 15 15
6.95 25 40
Chapter 3 45
3.126 (a) Class Limits Frequency
0 1 12
(b) No. The class interval of the last class is
greater than that of the others.
2 3 7
3.127 (a) Class Limits Frequency Class Marks
0 99 4 49.5
(b) Yes, [see part (a)..
100 199 3 149.5
3.130 The class marks are found from the class boundaries by averaging them; thus, the first class
mark is (2.95 + 4.95)/2 = 3.95, and so forth.
3.135 The MINITAB output is:
3.136 The MINITAB output is:
MIDDLE OF
INTERVAL
NUMBER OF
OBSERVATIONS
6.0 2 **
6.5 5 *****
MIDDLE OF
INTERVAL
NUMBER OF
OBSERVATIONS
40 1 *
45 7 *******