chapter 3 · 3.54 f 22 4 4xe ye xy e xyexy xy xy22 22 22() yx xy 0, 0 and fxy(, ) 0 elsewhere 3.55...

23 Copyright © 2014 Pearson Education, Inc.

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. 3.2 (a) No, because f(1) is negative; (b) Yes; (c) No, because f(0) + f(1) + f(2) + f(3) + f(4) +

f(5) is greater than 1. 3.3 ( ) 0f x for each value of x and

1

2 2 ( 1)( ) (1 2 ) 1

( 1) ( 1) 2

k

x

k kf x k

k k k k

3.4 (a) (1 2 3 5) 1;c thus1

15C

(b) 5 5 5

5 1 12 3 4

c

; thus, 12

137c

(c) 2

1 1

( ) ( ,2)k k

x x

f x c x cS k

From Theorem A.1 we obtain 1

( ,2) ( 1)(2 1)6

S k k k k

Thus, for ( )f x to be a distribution function, 6

, 0( 1)(2 1)

c kk k k

.

(d) 1 1

1( )

4

x

x x

f x c

The right-hand sum is a geometric progression with a = 1 and r = 1/4. For x = 1 to n, this sum equals

11

1/ 4 141 3 / 4 314

n

nS

as n . Therefore, 3c .

3.5 For ( ) (1 ) xf x k k to converge to 1, 0 < k < 1. 3.6 For c > 0, f(x) diverges. For c = 0, f(x) = 0 for all x, and it cannot be a density function 3.9 (a) No, because (4) 1;F (b) No, because (2) (1);F F (c) Yes.

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24 Mathematical Statistics, 8E

Copyright © 2014 Pearson Education, Inc.

3.10 4 1

(0) ;20 5

f 2 6 12 3

(1) ,20 20 5

f

4 1(2)

20 5F

( )F x

0

1/ 5

4 / 5

1

0

0 1

1 2

2

x

x

x

x

3.11 (a) 5 1 1

6 3 2 ; (b)

1 1 1

2 3 6 ; (c)

1(1)

3f ,

1(4)

6f ,

1(6)

3f and

1(10) ,

6f

0 elsewhere. 3.12 ( )F x 0

1/15

3 /15

6 /15

10 /15

1

1

1 2

2 3

3 4

4 5

5

x

x

x

x

x

x

3.13 (a) 3

4 (b)

3 1 1

4 2 4 (c)

1

2 (d)

1 31

4 4

(e) 3 1 1

4 4 2 (f)

3 11

4 4

3.14 3

(1)25

f , 4

(2) ,25

f 5

(3)25

f , 6

(4)25

f , 7

(5)25

f

( )F x

0

3 / 25

7 / 25

12 / 25

18 / 25

1

1

1 2

2 3

3 4

4 5

5

x

x

x

x

x

x

6 3

(1)50 25

F , 14 7

(2)50 25

F , 24 12

(3)50 25

F , 36 18

(4)50 25

F , 50

(5) 150

F , checks

3.15 (a) 1 1 1( ) 1 ( ) 1 ( )P x x P x x F x for i = 1, 2, …, n

(b) 1 1( ) 1 ( ) 1 ( )i iP x x P x x F x for i = 2, …, n and

1( ) 1P x x

Chapter 3 25


3.16

0

1( ) ( 2)

51

F x x

2

2 7

7

x

x

x

3.17 (a) 7

2

71 1 1( ) (7 2) 1

25 5 5f x dx dx x

(b) 5

3

1 1 2(5 3)

5 5 5dx

3.18 (a) ( ) 0,f x 0 ,x and 0

0 0

( ) 1xf x dx e dx e

(c) 1

1

( 1) xP x e dx e

3.19 (a) ( ) 0,f x 0 1x and 1

0

( ) 1f x dx

(c) 0.5

2

0.1

(0.1 0.5) 3 0.124P x x dx

3.20 (a) 3.2

2

3.21 1 1( 1) (8.32 4) 0.54

28 8 2 8

zyy dy y

(b) 3.2

2.9

3.21 1 1( 1) (8.32 7.105) 0.1519

2.98 8 2 8

xyy dy y

3.21 2 2 2

2

1 1 1 1 1( 1) 4 4

28 8 2 8 2 8 8 2

y yt y yt dt y y y

2

0 2

1( ) 4 2 4

8 2

1 4

y

yF y y y

y

(a) 21 3.2

(3.2) 3.2 4 0.548 2

F



(b) 21 2.9

(3.2) (2.9) 0.54 2.9 4 0.54 0.3881 0.15198 2

F F

3.22 (a) 4 4 1/2

1/2

0 0

41 4

01/ 2

c xdx c x dx c c

x

1

4c

(b) 1/4

1/2

0

1 / 41 1 1 1 1 1 104 4 4 1 / 2 2 2 44

xP x dx x dx

x

1

0

11 1 1( 1) 1 1

02 24P x dx x

x

3.23 1

( )2

F x x

0

1( )

21

F x x

0

0 4

4

x

x

x

1 1 1 1

4 2 2 4F

and 1 = 1 1

(1) 12 2

F

3.24 0 0

1( ) [1 ] (1 )

2 2 2

z zz z

z u u zk kF z k ze dz k e du e e k = 2

3.25 0 0

( )1 0

zz

zF z

e z

3.26 2 3 1 / 41 3 1 5(3 2 )

04 16 32 32P x x x

1

2 3

1/2

11 3 1 16 (1 ) (3 2 ) 1

1/ 22 4 4 2P x x x dx x x

3.27 2 3

0

( ) 6 (1 ) 3 2x

F x x x dx x x 2 3

0

( ) 3 2

1

F x x x

0

0 1

1

x

x

x

1 3 2 5

4 16 64 32P x

and 1 3 2 1

12 4 8 2

P x

Chapter 3 27


3.28 2

0

( ) 2

xx

F x x dx 0 to 1

2 2

1

2

1 1 1 3( ) (2 ) 2 2

12 2 2 2 2 2

2 12

x xx xF x x dx x x

xx

2

2

0 0

0 12( )

2 1 1 22

1 2

x

xx

F xx

x x

x

3.29 0

1 1( )

3 3

x

F x dx x 0 to 1 1

( )3

F x 1 to 2

1

( ) ( 2) 2 to 431

( 13

F x x

x

0 0

1 0 1

31

( ) 1 23

1( 1) 2 4

3 1 4

x

x x

F x x

x x

x

3.30 (a) 1 1.2 2 2

0.8 1

1 1.2 1 1 (2 ) 2 0.32) 2.4 0.72 2 0.36

0.8 12 2 2 2

x xx dx x dx x

(b) 2 2(1.2) (0.8)

(1.2) (0.8) 2(1.2) 12 2

2.4 0.72 1 0.32 0.36

F F

3.31 0x ( ) 0F x

0 1x 2

( )4

xF x

1(1)

4F

1 2x 1 1

( )2 4

F x x 3

(2)4

F

2 3x 23 5

( )2 4 4

xF x x (3)F = 1

3 x ( ) 1F x



3.32 (a) 1 1 3 1 1

;2 2 4 4 2

F F

(3) (2) 1 1 0F F

3.33 dF 1

dx 2 ,

1( )

2f x for 1 1x ; 0 elsewhere

1 1 1 1

12 2 2 2

P x

; (2 3)P x = 0

3.34 (a) 9 16

(5) 125 25

F

(b) 9 9

1 (8) 1 164 64

F

3.35 2

dF 18

dy y for 0;y elsewhere

(a) 5

2 23

518 9 9 161

3 25 25dy

y y ; (b) 2 2

8

18 9 9 90

8 64 64dy

y y

3.37 2( 2) (2) 1 3 1 3(0.1353) 1 0.4074 0.5926P x F e

2 1 2(1 3) (3) (1) 1 4 1 2 4

=2(0.3679) 4(0.0498) 0.7358 0.1992 0.5366

P x F F e e e

4( 4) 1 (4) 5 5(0.0183) 0.0915P x F e

3.38 xdFxe

dx for > 0; 0 elsewhere

3.39 (a) for 0x ( ) 0F x

(b) for 0 0.5x 1

( )2

F x x

(c) for 0.5 1x 1 1 3 1( ) 1

2 2 4 2F x x x

(d) for 1x ( ) 0f x

3.40 (a) ( ) 0;f x (b) 1

( ) ;2

f x (c) 1

( ) ;2

f x (d) ( ) 0f x

3.41 2 4 1 1 5 1 3

( 2) , ( 2) , ( 2 1)8 4 4 8 4 8

P Z P Z P Z

and 1 1

(0 2) 12 2

P z

Chapter 3 29


3.42 (a) 1

;20

(b) 1 1 3

;4 8 8 (c)

1 1 1 1;

6 4 12 2 (d)

1 1 1 28 7

6 24 40 120 30

3.43 (a) 1 1 1

6 12 4 ; (b) 0 ; (c)

1 1 1 7

12 6 24 24 ; (d)

1 1191

120 120

3.44 (2 5 10 1 4 9 2 5 10 10 13 18) 1c

1

89c

3.45 (a) 1 29

(10 9 10)89 89

; (b) 1 5

(1 4)89 89

(c) 1 55

(9 5 10 13 18)89 89

3.46 (a) (0 2 8 0 1 2) 1k f(3, 1) differs in sign from all other terms 3.47

x y 0 1 2 3 0 1 2 3

0 0 1

30

1

15

1

10 0 0

1

30

1

10

1

5

y 1 1

30

1

15

1

10

2

15 1

1

30

2

15

3

10

8

15

2 1

15

1

10

2

15

1

6 2

1

10

3

10

3

5 1

density joint distribution function 3.48 (a) ( , ) 0P x y (b) ( , ) 1P x y (c) ( , ) ( , ) three probabilitiesF b c F a c ( , ) ( , )F b c F a c

3.49 1 1 2

2

0 0

( ) 2

x

x

xxyk x x y dy dx k x y dx

x

1 13 3

3 3 3

0 0

2 12 2 2

x x kk x x dx k x dx

k = 2



3.50 1/21/2 1/2 1/2 22

0 0 0 0

1/2 2 2 42 3

0

1 / 2 124 24 12

02 2

1/ 2 1 1 112 12 12

04 8 3 4 32 24 64

12 12 1(6 8 3)

64 3 3 64 16

x xxyxy dy dx dx x x dx

x x x xx x dx

3.51 (a) 1

2

(b) 1 2 2 4 5

1 2 12 3 3 9 9

(c) 1 2 1 1 1 1 3 1

22 3 3 2 3 3 9 3

( , ) 2F x y xy for 0, 0, 1x y x y

3.52 (a) 1 1 1

22 2 2

3.53 1/2 1

1/4 1/2 1/2 0

1 1y y

y

dx dy dx dyy y

1

1 ln 2=1-0.3466=0.65342

3.54 2 2 2 2 2 2( )2 2 4 4x y x y x yF

xe ye xy e xyey x

0, 0x y

and ( , ) 0f x y elsewhere

3.55 2 2

2 22 2 41 4 2

1 1 1

42 2 ( )

1x y u uxe dx ye dy e du e e e

3.56 2

x x y x yF Fe e e

x x y

0, 0x y

= 0 elsewhere

3.57

23 32 3 2

2 2

3( )

2X y xe dx e dy e e e

3.58 ( , ) ( , ) ( , ) ( , )F b d F a d F b c F a c

Chapter 3 31


3.59 a = 1, b = 3, c = 1, d = 2

3 2 1 2 3 1 1 1

2 3 1 1 2 1

2 1 3 1

1 2 1 3

(3,2) (1,2) (3,1) (1,1)

(1 )(1 ) (1 )(1 ) (1 )(1 ) (1 )(1 )

(1 ) (1 ) (1 ) (1 ) (1 ) (1 )

(1 )(1 ) (1 ) (1 )

( )( ) 0

F F F F

e e e e e e e e

e e e e e e

e e e e

e e e e

.074

3.60

4 4 1 4 1 4 1 1

4 4 1 1 4 1

4 1 4 1 1 4

1 4 1 4 1

(2,2) (1,2) (2,1) (1,1)

(1 )(1 ) (1 )(1 ) (1 )(1 ) (1 )(1 )

(1 ) (1 ) (1 ) (1 ) (1 ) (1 )

(1 )( ) (1 )( )

( )( ) (

F F F F

e e e e e e e e

e e e e e e

e e e e e e

e e e e e

4 2)e

3.61

3 3 6

2 3 5 2 2 5 2 2 4

4 5 6 2 3 2

(3,3) (2,3) (3,2) (2,2)

(1 )

(1 ) (1 (1 )

2 ( ) QED

F F F F

e e e

e e e e e e e e e

e e e e e

3.62 x = 1, 2 y = 1, 2, 3 z = 1, 2

(1 2 2 4 3 6 2 4 4 8 6 12) 1

1

54

k

k

3.63 (a) 1 1

(1 2)54 18

(b) 1 14 7

(8 6)54 54 27

3.64 (a) 1 9 1

(1 2 2 4)54 54 6

; (b) 0; (c) 1

3.65 11 1

0 0 0

(1 )y zz

xy z dx dy dz

1 1

2

0 0

1(1 ) (1 )

2

z

y z y z dy dz

11 1

0 0 0

(1 ) 1y zz

k xy z dx dy dz

k = 144



3.66 11/21/2

0 0 0

144 (1 ) 0.15625x yx

xy z dz dy dx

3.68 (a) 1/2 1/2 1/2

0 0 0

1(2 3 )

3x y z dz dy dx

1/2 1/2 2

0 0

1/2 1/2

0 0

1/2 1/22

0 0

1 / 21(2 3 )

03 2

1 3 1

3 2 8

1/ 21 3 1 1 1 3 103 4 8 3 2 16 16

1 1 3 1 1 6 1

3 16 32 32 3 32 16

zx y z dy dx

x y dy dx

xy y y dx x dx

3.69 (a) 1 3

( 1) , (1)4 4

g g

(b) 5 1 1

( 1) , (0) , (1)8 4 8

h h h

(c) 1/ 8 1

( 1 1)1 / 8 1/ 2 5

f

; 1/ 2 4

(1 1)1 / 8 1/ 2 5

f

3.70 (a) 1 1 1 1 7

(0)12 4 8 120 15

g ; 1 1 1 7

(1)6 4 20 15

g

1 1 1

(2)24 40 15

g

(b) 1 1 1 7

(0)12 6 24 24

h ; 1 1 1 21

(1)4 4 40 40

h

1 1 7

(2)8 20 40

h ; 1

(3)120

h

(c) 1 / 4 10

(0 1)21/ 40 21

f ; 10

(1 1)21

f ; 1/ 40 1

(2 1)21 / 20 21

f

(d) 1/12 5

(0 0)56 /120 28

w ; 1 / 4 15

(1 0)56 /120 28

w ; 1 / 8 15

(2 0)56 /120 56

w

1 /120 1

(3 0)56 /120 56

w

Chapter 3 33


3.71 (a) ( , ) (1 2) for 1,2,3108 36

xy xym x y x ; 1,2,3y

(b) ( , ) (1 2 3) for 1,2,3108 18

xz xzn x z x ; 1,2z

(c) ( ) (1 2 3) for 1,2,336 6

x xg x x

(d) / 64

( 1,2)2 / 36 3

z zzφ for 1,2z

(e) / 36

( , 3)1 / 2 18

yz yzy zψ for 1,2,3y ; 1,2z

3.72 (a) 5

(0)12

g , 1

(1)2

g ; 1

(2)12

g

0

5 /12( )

11/12

1

G x

0

0 1

1 2

2

x

x

x

x

(b)

2 / 9 4(0 1)

7 /18 71 / 6 3

(1 1)7 /18 7

f

f

0

( 1) 4 / 7

1

F x

0

0 1

1

x

x

x

3.73 (a) 1

( )2

f x for 1, 1x ; 1

( )2

g y for 1, 1y ; 1 1 1

2 2 4 , independent

(b) 2

(0)3

f , 1

(1)3

f , 1

(0) ,3

g 2

(1)3

g

1 2 1 2

(0,0)3 3 3 9

f not independent

3.74 (a) 2 2

0

21 1 1 1(2 ) 2 (4 2) (2 1)

04 4 2 4 2

yx y dy xy x x

for 0 1x

= 0 elsewhere

(b)

1 11 14 2

(2 1)1 34 62 2

yf y y

for 0 2y

= 0 elsewhere



3.75 (a) 1

2

0

11 1 1(2 ) ( ) (1 )

04 4 4x y dx x xy y for 0 2y

= 0 elsewhere

(b)

1(2 1) 14( 1) (2 1)1 2(2)4

xf x x

for 0 1x

= 0 elsewhere

3.76 (a) 1 2 3

2

0

2 2 3

3 3 3

1( ) 24 ( ) 24

02 2 3

12(1 ) 12 (1 ) 8(1 )

12(1 ) 8(1 ) 4(1 )

x xy xy yf x y xy y dy

x x x x

x x x

34(1 ) 0 1

( )0 elsewhere

x xf x

(b) 1

2 2 2

0

1( ) 24 ( ) 24 (1 ) (1 ) (1 )

2

1 124(1 ) 1 (1 ) 24 (1 )

2 2 2

y

g y y xy y dy y y y y y y

yy y y y y

212 (1 ) 0 1

0 elsewhere

y y y

( , ) ( ) ( )f x y f x g y not independent 3.77

(a) 1 1 ln 0 11

( ) ln ln1 ln 0 elsewhere

x

x xg x dy y x

xy

(b) 0

1 0 11 1( ) ( 0)

0 elsewhere

y xh y dx y

y y

1

1 ( ln ) independentx noty

Chapter 3 35


3.78 (a) 3

2

1 2 1 22 1 3

12

( )( , )

1122

x

x

x x e x xf x x x

xx e

22

22

1 2 61 0 13,2 51 13

0 elsewhere3 2

xx xf x

(b) 33

2 2 31 22 3 1

2

10 1, 0( )

( , ) 21

0 elsewhere2

xx x e x xx x eg x x x

x

3.79 -

( ) ( , ) g x f x y dy

0

( ) ( , ) ( , )x

G x f x y dy F x

2

1 0( ) ( , )

0 elsewhere

xe xG x F x

3.80 1 3 1 2 3 2 1 3( , ) ( , , ) ( , , )M x x f x x x dx F x x

1 1 2 3 2 3 1( ) ( , , ) ( , , )G x f x x x dx dx F x

(a) 3

3

1 3

2 3 1 1 1 3

1 3

0 0 or 0

1( , ) ( 1)( 1 ) 0 1, 0

2

1 x 1, 0

x

x

x x

M x x x x e x x

e x

(b)

1

1 1 1 1

1

0 0

1( ) ( 1) 0 1

2 1 1

x

G x x x x

x



3.81 1 11

1 0 1

( ) 2 0 elsewhere

x xg x

2 22

1 0 1

( ) 2 0 elsewhere

x xh x

3

33

1( )

0 elsewhere

xe xxφ

1, 2 3 1 2 3( , ) ( ) ( ) ( )f x x x g x h x xφ not independent

1 3 1 3( , ) ( ) ( )m x x g x xφ independent

2 3 2 3( , ) ( ) ( )n x x h x xφ independent

3.82 (a)

10 2, 0 3

( , ) 60 elsewhere

x yg x y

(b) / 4

1 16 24

π π

(a) 5

(0)14

g , 5

(1)28

g , 3

(2)28

g

(b) 3.28 3

(0 0)10 / 28 10

φ , 6 / 28 6

(1 0)10 / 28 10

φ , 1 / 28 1

(2 0)10 / 28 10

φ

3.83 Heads Tails Probability H-T 0 4 1/16 4 1 3 4/16 2 2 2 6/16 0 3 1 4/16 2 4 0 1/16 4 3.84 1 2 3 1 3 4 1 4 5 2 3 5 2 4 6 3 4 7

(a) x 3 4 5 6 7 ( )f x 1/6 1/6 2/6 1/6 1/6

0 3

1 / 6 3 4

2 / 6 4 5( )

4 / 6 5 6

5 / 6 6 7

1 7

x

x

xF x

x

x

x

Chapter 3 37


3.85 2

( )3

P H

(a) 1

(0)27

P , 6

(1)27

P , 1 2 2 12

(2) 3, , ,3 3 3 27

P , 8

(3)27

P

(b) 1 6 12 19

27 27 27 27

3.86

0

1 / 27

( ) 7 / 27

19 / 27

1

F x

0

0 1

1 2

2 3

3

x

x

x

x

x

7 20 1

27 2719 8

127 27

(a)

(b)

3.87

0

0.40

( ) 0.70

0.90

1

F V

0

0 1

1 2

2 3

3

V

V

V

V

V

3.88 (a) 0.20 0.10 0.30 (b) 1 0.70 0.30

3.89 Yes; ( ) 0f x for 2,3, 12x and 12

2

( ) 1x

f x

3.90 S 2 3 4 5 6 7 8 9 10 11 12 1/36 3/36 6/36 10/36 15/36 21/36 26/36 30/36 33/36 35/36 1

3.91 (a) 1

(228.65 227.5) 0.235

; (b) 1

(231.66 229.34) 0.4645

;

(c) 1

(232.5 229.85) 0.535

3.92 3

21 1 1( ) (36 ) 36

288 288 3 2

xF x x dx c x

so that F(6) = 0 and F(6) = 1.

(a) 1 8 1 1 208 1 7

( 2) ( 72 )288 3 2 288 3 2 27

F

(b) 1 1 1 1 107 1 757 1 325

(6) (1) 1 (36 ) 1288 3 2 288 3 2 864 2 854

F F

(c) 1 1 1 99 1 107 190 95

(3) (1) (108 9) 36288 288 3 288 288 3 288 3 432

F F

(d) 0



3.93 /30

/30 /30 /301 1( ) 1

30 30 1/ 30

xx x xe

F x e dx c c c e e

(a) 18/30 0.6(18) 1 1 1 0.5488 0.4512F e e

(b) 27/30 36/30 0.9 1.2(36) (27) 0.4066 0.3012 0.1054F F e e e e

(c) 48/30 1.61 (48) 0.2019F e e

3.94 3 2 2

20,000 20,000 10,000( ) 1 1

( 100) 2( 100) ( 100)F x dx c

x x x

(a) 2

10,000 11 (200)

9300F

(b) 10,000 3

(100) 140,000 4

f

3.95 (a) 2

25 11 (10) 0.25

410F

(b) 2

25 39(8) 1

648F

(c) 2 2 2

25 25 25(25 16) 1(15) (12)

1612 15 15 16F F

3.96 /3

/3 /3

0

1 1 1 1( ) 1 1

9 9 1/ 9 3 3

x xx xe

F x x e dx c x c c e x

c =1

(a) 2 2(6) 1 3 1 3 1 3(0.1353) 0.5491F e e

(b) 31 (9) 4 4(0.0498) 0.1992F e

3.97 3 2 3

(0,0,2) 30 0 2

3

(0,0)28

f , 6

(0,1)28

f , 1

(0,2)28

f

3 2 3

(1,0,1) 91 0 1

9

(1,0)28

f , 3

(2,0)28

f , 6

(1,1)28

f

3 2 3

(0,1,1) 60 1 1

3 2 3

(2,0,0) 32 0 0

3 2 3

(1,1,0) 61 1 0

3 2 3

(0,2,0) 10 2 0

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

3.100 (a) Probability = 1/8 (b) 1 1 1

4 4π

π

3.101 (a)

0.3 0.3

0.2 2 0.2

0.3 22 0.4 0.6

0.2

5 52

0.35 55 0.3038

0.22 2

ps ps

pp

pe ds dp e dp

ee dp e e

(b) 0.30 1 0.30 0.30

0.25 0 0.25 0.25

0.30 0.30 0.25

15 5 5(1 )

0

5[ ] 5(0.30 0.25 ] 0.01202

ps ps p

p

pe ds dp e dp e dp

p e e e

3.102 (a) 0.4 0.4 0.4

2

0 0 0

0.42 2(2 3 ) ( 3 )

05 5x y dx dy x xy dy

0.4

0

2(0.16 1.2

5

2 1.2(0.16)(0.16)(0.4) 0.064

5 2

y dy

x 0 1 2 0 1

9

1

9

1

36

y 1 1

3

1

6

2 1

4



(b) 0.5 1 0.5

2

0 0.8 0

12 2(2 3 ) ( 3 )

0.85 5x y dx dy x xy dy

0.5 0.5

0 0

2

2 2(1 3 ) (0.64 2.4 ) (0.6 0.36)

5 5

2 20.5(0.3 0.36 ) (0.075 0.18) 0.10205 5

y y dy y dy

y y

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

3.104 (a) 1 1 1 1

2

0.3 0 0.3 0.3

2

12 2 2( 4 ) ( 2 ) ( 2)

05 5 5

1 22

0.35 2

2 1 0.09 22 0.6 (1.855) 0.742

5 2 2 5

x y dy dx xy y dx x dx

xx

(b) 1

0

2 2( ) ( 4 ) ( 2)

5 5g x x y dy x

(2 / 5)( 4 )

( )(2 / 5)( 2)

x yg y x

x

, 4 0.2

( 0.2)2.2

yg y

0.5

0

1 1 0.6(4 0.2) (0.5 0.1) 0.273

2.2 2.2 2.2y dy

3.105 (a) 48 47 188

(0,0)52 51 221

f , 48 4 16

(0,1)52 51 221

f

4 48 16

(1,0)52 51 221

f , 48 4 16

(1,1)52 51 221

f , 4 3 1

(1,2)52 51 221

f

(b) 188 16 204

(0)221 221

g ,

16 1 17(1)

221 221g

(c) 16 / 221 16

(0 1) ,17 / 221 17

φ 1/ 221 1

(11)17 / 221 17

φ

Chapter 3 41


3.106 ( , ) 5 psf p s pe 0.2 0.4p 0s

(a) 0

5 0.2 0.45 5 5

0 0 elsewhere

psps ps pe

p e ds p ep

(b) for 0( , ) 5

( ) 5 0 elsewhere

psps pe sf p s pe

g s

(c) 3

(1/4) /4 0.75

0

311

04s se ds e e

3.107 (a) /2

20 10 201 20

5025

0 elsewhere

x

x

xxx

dyx

(b)

1 20225

( ) ,20

50

xx

y xx x

φ

1 / 6 6 12

( 12) 0 elsewhere

yyφ

(c) 1 1 2

(12 8) 46 6 3

3.108 2

( , ) (2 3 )5

f x y x y

2 12 3 2 3( ) 2 2

05 2 5 2

4 3 0 1

5 5 0 elsewhere

yg x xy x

x x

2 12( ) ( 3 )

05

(2 / 5)(1 3 ) 0 1

0 elsewhere

h y x xy

y y

( , ) ( ) ( )f x y g x h y



3.109 (a)

3

1 2 33 3 31 2 3 2 2 3

(20,000) 0, 0, 0

( , , ) ( 100) ( 100) ( 100)

0 elsewhere

x x xf x x x x x x

(b) 100 100

1 2 33 3 31 2 30 0 200

20,000 20,000 20,000

( 100) ( 100) ( 100)

3 3 1 1

4 4 9 16

dx dx dxx x x

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

(b) 5f 4 5s 9 5 7 9 9 8 6f 1 3 0 2 0 1 0 4 2 0 2 1 3 1 6s 5 7 8 5 (c) The double-stem display is more informative. 3.111 *=Station 105 o = Station 107 o o o o o

o * * o o o * * o o * * * * * o * * *

54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69

3.112 *=Lathe A o = Lathe B

* * o o o o * * * * * * * o o o o

1.28 1.30 1.32 1.34 1.36 1.38 1.40 1.42 1.44 1.46 1.48 1.50 1.52 1.54 1.56

3.115 Class Limits Frequency 40.0 44.9 5 45.0 49.9 7 50.0 54.9 15 55.0 59.9 23 60.0 64.9 29 65.0 69.9 12 70.0 74.9 8 75.0 79.9 1 100

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; the class interval is 2; the class marks are: 3.95, 5.95, 7.95, 9.95, 9.95, 11.95, 13.95.

3.116 Class Limits Frequency 3.0 4.9 15 5.0 6.9 25 7.0 8.9 17 9.0 10.9 11 11.0 12.9 8 13.0 14.9 4 80

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 4 5 4 3.5 5.5 4.5 6 7 5 5.5 7.5 6.5 8 9 1 7.5 9.5 8.5 10 11 0 9.5 11.5 10.5 12 13 1 11.5 13.5 12.5 30

3.120 Class Limits Frequency Percentage 3.0 4.9 15 18.75% 5.0 6.9 25 31.25 7.0 8.9 17 21.25 9.0 10.9 11 13.75 11.0 12.9 8 10.00 13.0 14.9 4 5.00 80 100.00

3.121 Class Limits Frequency Percentage 40.0 44.9 5 5.0% 45.0 49.9 7 7.0 50.0 54.9 15 15.0 55.0 59.9 23 23.0 60.0 64.9 29 29.0 65.0 69.9 12 12.0 70.0 74.9 8 8.0 75.0 79.9 1 1.0 100 100.0



3.122 Percentage

Class Limits Shipping

Department Security

Department 0 1 43.3% 45.0% 2 3 30.0 27.5 4 5 16.7 17.5 6 7 6.7 7.5 8 9 3.3 2.5 100.0 100.0 The patterns seem comparable for the two departments.

3.125 Cumulative Percentage

Class Limits Shipping

Department Security

Department 1.5 43.3% 45.0% 3.5 73.3 72.5 5.5 90.0 90.0 7.5 96.7 97.5 9.5 100.0 100.0

3.123 Upper Class Boundary

Frequency

Cumulative Frequency

44.95 5 5 49.95 7 12 54.95 15 27 59.95 23 50 64.95 29 79 69.95 12 91 74.95 8 99 79.95 1 100 100

3.124 Upper Class Boundary

Frequency

Cumulative Frequency

4.95 15 15 6.95 25 40 8.95 17 57 10.95 11 68 12.95 8 76 14.95 4 80 100

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 4 5 4 6 7 5 8 13 2 30 3.127 (a) Class Limits Frequency Class Marks 0 99 4 49.5

(b) Yes, [see part (a)..

100 199 3 149.5 200 299 4 249.5 300 324 7 312.0 325 349 14 337.0 350 399 6 374.5 38 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 ***** 7.0 4 **** 7.5 5 ***** 8.0 5 ***** 8.5 3 *** 9.0 2 ** 9.5 2 **

10.02 2 **

MIDDLE OF INTERVAL

NUMBER OF OBSERVATIONS

40 1 * 45 7 ******* 50 11 *********** 55 21 ********************* 60 21 ********************* 65 23 *********************** 70 10 ********** 75 6 ******

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