Μαθηματικά Και Στοιχεία Στατιστικής Γ' Ενιαίου...

2

status (, ) , - . - (, -, ). , , , , ., , . 2238 .. Yao. , , . ( ) (). . , - . , - . , . - , 1348 400 , . 1620 Graunt - 88 3 . , 13.200 1620, 387.200 .

22-0088-02.indd 55 6/9/2013 11:11:59 pi

56

, 11 , , .. 16 19 , , , . - , - : , . - -, , . R.A. Fisher (1890-1962), :

:

.

- (experimental design) , - (descriptive statistics), . , - (inferential statistics) , . , , 100 - 20 100 -, . 20% , () . . . - . , (, , , ,

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57

), (, , , ), (, , , ), (-, , , ) - (, , , Marketing, ), . - . ( ) . , , - - , , - , . - , - . . - . - . - , -, . , , . . , . ( ) , , , . . . - .

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58

2.1

-

, - . , , :

) )

) , ,

) .

- . - (population). . , .. . - : , , . - (variables) , , , , . , - . . , , : , , , , , , , , , . : , , . :1. ,

. , , ( , , , ), ( , ), ( , ), ( , ,

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59

), , , , .

2. , -: i) , . -

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

ii) , (, ). , - , .

() . (census). , - () 10 , , , . 1991. - . , , , . , , . , - , . - - . , - ,

22-0088-02.indd 59 6/9/2013 11:11:59 pi

60

, . . , , , , , . , , , . . - . , , - - . , . . , - .

. - 1936. Literary Digest 2.400.000 Landon 57%. , G. Gallup 50.000 Roosvelt 62%! . - (Sampling), . , - , , .

22-0088-02.indd 60 6/9/2013 11:11:59 pi

61

1. ; ;

) ) ) ) ) ) ) ) ) ) .

2. ; :

) . ) . ) . ) .

3. 500 . , , -, ; :

) , ) , ) ) , ) .

4. ;) -

, , - .

) - . .

) - .

) .

) - . - .

22-0088-02.indd 61 6/9/2013 11:11:59 pi

62

2.2

- , - . , .

:) , ( ) -- ,) , . . :) , ,) , ,) (), ,) , , , , .

, - .

1 ( )

( )

1971

1981

1991

1993

1994

0-14 2,22 2,31 1,97 1,85 1,8115-64 5,58 6,19 6,88 6,99 7,04

65 0,96 1,24 1,40 1,54 1,58

: , 1996

22-0088-02.indd 62 6/9/2013 11:11:59 pi

63

2 , 1991, 10.000 .

.. 1971 1981 1991

8.261,183 3.661,637 1.635,998 1.401,459

842,796 734,014 585,312 477,942 476,288 406,612 389,434 383,672 301,106 287,611 110,581 91,503 84,069 77,014

456.471 162.986

97.008 66.606 52.487 31.787 89.578 32.664 17.367 30.180 14.201 13.31622.917 16.650 13.097 23.065 18.642

9.553

502.082 185.626

88.601 87.831 48.700 27.649 96.533 31.629 15.721 30.011 14.037 13.111 19.947 20.350 14.295 28.574 19.668 11.127

539.938 205.502

87.151 98.181 51.060 29.392

104.781 33.032 17.645 32.556 14.838 13.527 19.350 26.379 15.706 34.272 19.870 11.639

: , 1991

3

1990-94

: ,

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64

4 40 .

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65

x x x1 2, ,..., , v, . xi () (frequency) i, xi -. , :

1 2+ + + =... (1)

, : 4 - x

1 = 0, x2 0= , x3 2= , x4 3= , , 1 8= , 2 22= ,

3 7= , 4 3= 1 2 3 4 40+ + + = . , 5. - | .

5 :

4.

i , (relative frequency) fi xi,

fi i=

, i =1 2, , ..., . (2)

:

i) 0 1 fi i =1 2, , ,... 0 i .

ii) f f f1 2 1+ + + =... ,

f f f1 2 1 2 1 2 1+ + + = + + + =+ + +

= =... ... ...

.

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66

, fi , - fi %, f fi i% =100 . , - x1 0= , x2 1= , x3 2= , x4 3= : - :

f1840

0 20= = , , f22240

0 55= = , , f3740

0 175= = , f4340

0 075= = ,

f f f f1 2 3 4 0 20 0 55 0 175 0 075 1+ + + = + + + =, , , , .

f1 20% = , f2 55% = , f3% = 17,5 f4 7 5% ,= f f f f1 2 3 4 100% % % %+ + + = .

xi , i , fi -, -. , ( , )xi i ( , )x fi i , ( , %)x fi i , - . 5 : 4.

i fi - (cumulative frequencies) Ni (cumulative relative frequencies) Fi , - x

i. Fi 100 ,

F Fi i% =100 , 6. x x x1 2, , ..., , - xi Ni i= + + + 1 2 .... . , , i =1 2, , ...,. , - : 4 1 1 8= = , 2 1 2 30= + = , 3 1 2 3 37= + + = 4 1 2 3 4 40= + + + = = , F f1 1 0 20= = , , F f f2 1 2 0 75= + = , , F f f f3 1 2 3 0 925= + + = , F f f f f4 1 2 3 4 1= + + + = , F1 20% = , F2 75% = , F3 92 5% ,= F4 100% = . :

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67

1 1= , 2 2 1= , , = 1

f F1 1= , f F F2 2 1= , , f F F = 1.

6

4.

. , - , . . , - . ) , ) , ) ) .

)

(barchart) - . . . .

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68

. 7 - : 1(), () -.

7

4.

, 1(), . - , - 1(), 4.

() () 7.

1

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69

() 4

.

)

(line diagram). xi ( x x x1 2< <

70

)

(piechart) , . - , , ,

i - fi xi . i ,

i i i

f= =360 360

i =1 2, , ..., .

3 - 4.

)

, (dot diagram), . 4 ( ) 4, 2, 3, 1, 5, 6, 4, 2, 3, 4, 7, 4, 8, 6, 3 , .

4.

3

4

22-0088-02.indd 70 6/9/2013 11:12:13 pi

71

)

- , -. . 5 1990 1995. ( ). - - , 8 . 1993-95 6,5% 15% .

- , - . - , , , . () , (class intervals), . (class boundaries). ( ) ( ), [ , ). , , .

. . :

5

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72

. - . . , , , , . , (range) R , . c R , , , .

. - , , c -. ( ) .

, . i i - xi , i =1 2, , ..., .

, , 4 . , , 8.

8 ( cm) ,

4. .

R = =191 156 35. = 40 , = 6 .

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73

c R= = = / / , 35 6 5 83 6. 156, 9.

:

. ,

6. -

. , - 180 [ , )180 186 .

9 (, , )

8.

(histogram) . , - , . , (), , .

)

c - , - , . , . -

22-0088-02.indd 73 6/9/2013 11:12:14 pi

74

, - . , , , (frequency polygon). - , . - 1, ( 6).

() () - 9.

- . ( ) - - (ogive) . 7 - 9.

)

, .

6

7

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75

, , - . , , ( sec) v = 80 - , . , . , ci i i , -

ii

ic= , i =1 2, , ..., . , -

-. .

8(). , -.

() ()

() () .

8

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76

, ( 8())

, ii

i

fc

* %= .

( )

( ), - , (frequency curve), 9. , - .

. - 10. (), (normal distribution) . - [, ], (), . - , () ().

4

10

9

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77

1. -

) x*, 25% ) p 170 cm.

) , ,

( , , )0 0 25 -

0x 0x - ( *, )x 0 . x* =168 .

) , ( , )170 0 , , ( , )0 p . p = =0 35 35, % .

2. [2-5). - $1000, - .

( %)fi , 100. E1 1 0 10 10= =( ) , E2 2 1 20 20= =( ) , E4 10 5 5 25= =( ) . , E3 100 10 20 25 45= + + =( ) . 5 2 3 = , - 45 3 15/ = , .

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78

1. 50 :

3 4 5 8 9 7 6 8 7 108 7 6 5 9 3 8 5 6 66 3 5 6 4 2 9 8 7 71 6 3 1 5 8 1 2 3 45 6 7 9 10 9 8 7 6 5

) ( ).

) i) ( 5) ii) (9 10) iii) 7 9.

2. () ():

-.

3. 2 :

)

) ()

) ()

.

>55

22-0088-02.indd 78 6/9/2013 11:12:18 pi

79

4. , - 50 , :

) 1

) 5

) 3 5

) 5

) 5 .

5. .

6. - () 4.

7. 18 - 4 :

Metallica, Iron Maiden, , Scorpions, Oasis, , , Rolling Stones, Metallica, Metallica, Rolling Stones, Metallica, Iron Maiden, Iron Maiden, Scorpions, Scorpions, Scorpions, Metallica.

) ) .

8. 450 , , - . 30% -. 144. -. . ;

22-0088-02.indd 79 6/9/2013 11:12:18 pi

80

9. 1960 1998 ( ) 15 , 12, 9, 2 1. - .

10. 17 - , 1998. .

11. 1987 1997 . (: .)

- .

12. 50 ( 0-10).

6 7 8 5 1 4 7 3 9 9 2 5 3 8 6 7 7 6 8 1 3 0 1 4 9 0 9 7 8 6 1 2 3 5 4 6 6 4 3 2 8 8 7 7 6 5 5 9 2 4

) . ) -

.

22-0088-02.indd 80 6/9/2013 11:12:18 pi

81

) 8, ;

) 36% , ;

13. . 15.000 20.000 ;

14. - . -;

1. ) , ) , ) 2. ;

2. 1988-1998 ( 1998 8 ) - () 62, 72, 66, 79, 79, 78, 146, 176, 222, 222 65 . 7, 4, 2, 2, 1, 4, 8, 7, 14, 22 6 20 , 43, 51, 34, 44, 47, 49, 71, 90, 98, 99 33 21-30 30 .

.

3. 2 - , 1988-1998 8, 10, 7, 5, 9, 8, 11, 14, 20, 20 9 .

4. - 1930 1995 - . ) ; )

.

.

15 20

22-0088-02.indd 81 6/9/2013 11:12:18 pi

82

1995 789. ; ) - 1974 173. ; ) 1985;

5. 1 .

6. 150 17-24 200 , , - . )

130 mm Hg ) ,

.

22-0088-02.indd 82 6/9/2013 11:12:19 pi

83

7. ( ) 55 - :

3,4 13,2 6,7 1,4 1,3 3,8 3,9 2,9 13,8 3,9 2,7

4,4 3,6 1,4 2,4 3,6 3,1 7,5 6,9 7,8 12,7 3,9

3,3 9,7 2,0 4,4 3,3 8,7 3,9 11,6 5,6 9,0 3,4

1,4 3,5 2,8 10,4 11,9 12,3 2,9 2,8 1,5 4,1 5,9

3,1 8,7 2,8 3,8 13,0 3,0 6,4 3,2 5,9 7,0 8,2

) . ) i, fi%, Ni, Fi%. ) -

.

2.3

- . - . , , , , 10 . :

: 13 13 14 15 15 15 15 16 16 18 : 10 13 14 14 15 15 15 16 18 20.

11(), ().

11

22-0088-02.indd 83 6/9/2013 11:12:19 pi

84

15, . , - . . ( ), ) ) -, . (location measures), (measures of variability). , . x x= 0, x0 0x. , , (measures of skewness). - , - . 12 - - x

0,

. - , . x

0, x

0.

.

0x, , (arithmetic mean or average), (median) (mode).

12

22-0088-02.indd 84 6/9/2013 11:12:20 pi

85

)

- . t1, t2, , t x :

x t t tt

ti

ii

i=

+ + += ==

=

1 2 1

1

1...

(1)

tii=

1

t t t1 2+ + +... ti i =1 . , , it t . , x1, x2, , x 1, 2, , , :

x x x xx

xi i

i

ii

i ii

=+ + ++ + +

= ==

=

=

1 1 2 2

1 2

1

1

1

...... 1

1

(2)

:

x x x fi ii

i ii

= == =

1 1

fi .

, (1)

xA =+ + + +

= =13 13 14 18

1015010

15...

(2).

22-0088-02.indd 85 6/9/2013 11:12:23 pi

86

xx

Ai i= = =

15010

15.

, ,

x =15.

, 40 8,

(1) x = =691840

172 95, cm.

, , = 6 . xi i i , (2) :

xx

cmi i= = =

693040

173 25, cm.

. , , ; - - - - - xi. . !

)

() x1, x2, , x , - (weighted mean).

22-0088-02.indd 86 6/9/2013 11:12:24 pi

87

x1, x2, , x , () w1, w2, , w, - :

x x w x w x ww w w

x w

w

i ii

ii

=+ + ++ + +

= =

=

1 1 2 2

1 2

1

1

......

.

, , - x1 () w1 7 5= , , x2 - w2 1= , x3 1 w3 1= x4 2 w4 0 5= , . x1 16 5= , , x2 18= , x3 17= x4 16 6= , , :

x = + + + + + +

= =16 5 7 5 18 1 17 1 16 6 0 5

7 5 1 1 0 516710

16 7, , , ,, ,

, .

) ()

( ) 9 , : 3, 5, 5, 36, 6, 7, 4, 7, 8 x = 9. 9 ( ), , 9. () . , (median), :

() , , () .

, : ) 3, 4, 0, 6, 5, 8, 1, 1, 6, 1, 2, 8, 9 ) 3, 4, 0, 6, 5, 8, 1, 1, 6, 1, 2, 8, 9, 9 :

22-0088-02.indd 87 6/9/2013 11:12:27 pi

88

) =13 , :0 1 1 1 2 3 4 5 6 6 8 8 9.

, ( ), = 4.

) =14 : 0 1 1 1 2 3 4 5 6 6 8 8 9 9.

, (

), =+

=4 52

4 5, .

, - . , 50% 50% .

9 - , 13. , , x = - ( ), 50% . , - Fi = 50%. , (50% ) AB//0x 0x. , -. , 173.

13

22-0088-02.indd 88 6/9/2013 11:12:29 pi

89

) ( )P

, 50% - 50% , (percentiles) P , =1 2 99, , ..., . P1, P2, , P99 100 . , , - P % P ( )%100 .

P25, P50, P75, - (quartiles) Q1, Q2 Q3, . Q1 25% 75%. Q3 75% - 25% . Q P2 50= , Q2 = . . Q1 Q3 , . -, 3, 4, 0, 6, 5, 8, 1, 1, 6, 1, 2, 8, 9, :

:

=13 , : 0 1 1 1 2 3 4 5 6 6 8 8 9.

, :

H , = 4.

-, . Q1 : ,

0 1 1 1 2 3, Q11 12

1= + = .

-, . Q3.

22-0088-02.indd 89 6/9/2013 11:12:32 pi

90

, 5 6 6 8 8 9,

Q36 82

7= + = . ( , Q3 6= ).

( ) - . 13 Q1, Q2 = , Q3 P10, P90 9, :

P10 162 5= , , Q1 168= , =173, Q3 178= P90 184= .

) ( )0M

(mode) M0 . - , -. :

) ( ) - 7 M 0 = .

) 6 M 0 1= , (55%) .

) - 0 1 1 2 2 2 3 4 4 4 5 5 7 8, . 2 4 , 3. . , - (bimodal), (multimodal).

) , - . , 0, 1, 2, 7, 8, 9 - .

22-0088-02.indd 90 6/9/2013 11:12:35 pi

91

() , i, - . , - , - , -, 14, . - 9. (- ) M 0 173 cm.

. , , 2.3 11(), ().

x xA B= =15 = =15, . - , , , , . - , .

(measures of variation, dispersion measures). , - , .

14

22-0088-02.indd 91 6/9/2013 11:12:36 pi

92

) (R)

(range) (R), -, :

R = -

, RA = =18 13 5, RB = =20 10 10, .

, . 9 R = =192 156 36. , . , 8, , - R = =191 156 35. , , -.

) (Q)

(interquartile range) Q1 Q3, :

Q Q Q= 3 1

50% . , -.

13 Q1 168= , Q3 178= Q =10. 50% - 168 178 cm.

22-0088-02.indd 92 6/9/2013 11:12:37 pi

93

) ( )2s

t1, t2, , t x - , :

( ) ( ) ... ( )( )

t x t x t xt xi

i1 2 1 + + + =

=

.

,

( ) ( ) ... ( ) ...t x t x t x t t t x x x1 2 1 2 0 + + + = + + + = =

.

, ti x . (variance)

s t xii

2 2

1

1=

=

( ) (1)

:

s tt

ii

i2 2

1

1

2

1=

=

=

(2)

x . , - :

s x xi ii

2 2

1

1=

=

( ) (3)

:

22-0088-02.indd 93 6/9/2013 11:12:39 pi

94

s xx

i ii

i ii2 2

1

1

2

1=

=

=

.

x1, x2, , x ( ) - 1, 2, , .

, (1)

sA2

2 2 2 213 15 13 15 14 15 18 1510

2010

2= + + + + = =( ) ( ) ( ) ... ( ) ,

sB2 6 6= , ,

.

, 9, (3), - :

:

s xx

i ii

i ii2 2

1

1

2

1 140

1202776=

==

=

=

693040

53 42

,

(4)

22-0088-02.indd 94 6/9/2013 11:12:41 pi

95

- 8, s2 50 9= , . - .

) (s)

, -. -. , cm, cm2. -, - , , . (standard deviation), s :

s s= 2

4 - s = =53 4 7 3, , cm, - 9, s = =50 9 7 13, , cm, 8.

, s :

i) 68%

( , )x s x s +

ii) 95%

( , )x s x s +2 2

iii) 99,7% -

( , )x s x s +3 3

iv) - , R 6 s.

15

22-0088-02.indd 95 6/9/2013 11:12:42 pi

96

M (CV)

- xA = 40 kgr sA = 6 kgr, xB = 75 kgr sB = 6 kgr . , - , 6 kgr 40 kgr 75 kgr. , xA = 2 500. sA = 420 , - xB =1 400. $ sB = 350$. , . , - , , (coefficient of variation), x 0 - :

CV sx

= =pi pi

x < 0, x x . , - , . :

CV sxAA

A

= = = =640

0 15 15, %

CV sxBB

B

= = = =675

0 08 8, %

, - ( - ).

22-0088-02.indd 96 6/9/2013 11:12:45 pi

97

, - CVA =16 8, % CVB = 25% . - , . . , 10%.

1. ( sec) 60 , - . : ) ,

, ) , )

25% .

) :

, :

xxi i= =

400060

66 67, sec.

22-0088-02.indd 97 6/9/2013 11:12:45 pi

98

= 60 , 30 31 , -

65 70, =+

=65 70

267 5, sec.

, M 0 70= sec.) - (4), . :

s xx

i ii i2 2

221 1

60270900 4000

6070=

( )

=

=

,, sec .56 2

s = =70 56 8 4, , sec.) , Q1. - = 67 5, 30 . 30 - 15 16 , Q1 60 60 2 60= + =( ) / sec. , - 25% .

2. =1i

, x= .

f ( ) , = f x x x( ) ( ) ( ) ... ( ) 2 2 21 2 .

: =f ( ) 0

x x x1 2 0 + + + = ...x x x1 2 0+ + + =...

=+ + +

=x x x x1 2 ... .

f ( ) :

= + + + =f ( ) ... 2 2 2 2

= >f x( ) 2 0 , = x f ( ) .

22-0088-02.indd 98 6/9/2013 11:12:49 pi

99

3. x1, x2, , x sx. ) y1, y1, , y

x1, x2, , x c, :

i) y = x + c, ii) s = sy x) y1, y1, , y -

x1, x2, , x c, :

i) y = cx, ii) s = c sy x

) y x ci i= + , i =1 2, , ..., :

i) y y y y x c x c x c= + + + = + + + + + +1 2 1 2... ...

=+ + +

+ = +x x x c x c1 2 ...

ii) sy y y y y y

y2 1

22

2 2

= + + + ( ) ( ) ... ( )

=+ + + + + + ( ) ( ) ... ( )x c x c x c x c x c x c1

22

2 2

= + + +

=( ) ( ) ... ( )x x x x x x sx1

22

2 22

.

s sy x= .

) y cxi i= , i =1 2, , ..., , :

i) y y y y cx cx cx c x x x cx= + + + = + + + = + + + =1 2 1 2 1 2... ... ...

ii) s y y y y y yy2 1

22

2 2

= + + + ( ) ( ) ... ( )

= + + + ( ) ( ) ... ( )cx cx cx cx cx cx1

22

2 2

= + + + =

c x x x x x xc sx

21

22

2 22 2

( ) ( ) ... ( )

.

s c sy x= .

22-0088-02.indd 99 6/9/2013 11:12:52 pi

100

1. 15. - .

2. =10 , - 1, 2 3. ) 1 ) 4 ) 1,8;

3. 8 - . :

5, 16, 10, 0, 27, 14, 20, 34. .

4. 9 205 cm. )

216 cm. ; ) 208 cm,

;

5. 18 12 15,4 -. 15,8 , .

6. , , 10%, 20%, 30% 40% . 10 gr, 11 gr, 12 gr 13 gr. N , , ) 10 , ) 20 , ) .

7. 12, 10, 16, 18, 14. ) . ) 2, 3, 1, 1 3,

; ;

8. 5 () 1249 , 6 1280 ,

22-0088-02.indd 100 6/9/2013 11:12:53 pi

101

4 1360 . ;

9. 6. 5, 8, 9. .

10. . -! ,

) 4,4, ) 4,5, ) ;

11. 40 4 :) ,) ,) ,) -

.

12. 40 .

: ) , ) , ) , ) .

13. 30 170 cm. :) 180 cm,) 170 cm,) 180 cm

170 cm;

14. 50.) (i) (ii) -

; ( ).0, 20, 40, 50, 60, 80, 1000, 48, 49, 50, 51, 52, 1000, 1, 2, 50, 98, 99, 100.

22-0088-02.indd 101 6/9/2013 11:12:53 pi

102

) ;

15. : 7, 11, 10, 13, 15, 3, 12, 11, 4, 14. :

) , ,) Q1 Q3 ,) , .

16. 12.

17. 10 - 2 . , :

) 8 ) 10 ) 14 ) 6 12

.

18. :

) 1 2 6 ) 2 4 12 ) 11 12 16 ) 12 14 22.

19. . ;

) 1, 3, 4, 5, 7 ) 3, 9, 12, 15, 21 ) 6, 8, 9, 10, 12 ) 1, 3, 4, 5, 7.

20. 2 625 - . 27,2%, . xi

2 11 746 700 = . . , 2;

1. 50 10 20 (- ). 12, 14,

22-0088-02.indd 102 6/9/2013 11:12:53 pi

103

18 16.

) . ) : i) , ii) . ) 5% ,

;

2. 5 x = 4

s2 10= , . , ( )x xii

==

1

42 14,

.

3. 10 ... 15, 9, 6, 18, 21, 6, 18, 27, 9, 12 .

) , () -;

) (), - ..., 18%;

) 0,3 ( ...) , ();

4. 0, 2, 4, 6, 8, 10 12 - , 0 1.

5. .

) ;

) 5 ;

) .

) .

)

22-0088-02.indd 103 6/9/2013 11:12:54 pi

104

Q1, Q2, Q3.

6. . :

) .

) .

7. - 1995 - ():

- , , 1995, .

2.4

- (), .. , , . - , , , . , . :) (-

) - .

22-0088-02.indd 104 6/9/2013 11:12:54 pi

105

) - , .

) ( ) , - .

) , , , - .

) .

- . () , . (regression analysis). , regression Galton (1822-1911) 1885. - , , , , , .

- (simple linear regression), X (independent or input variable), Y (dependent or response variable), X. . , , (- ) ( ). - . - , (-

22-0088-02.indd 105 6/9/2013 11:12:54 pi

106

) () . 10 , , . . - , X - Y . , (Y) (X). , , . () ().

10 X ( cm) Y ( kg) 18 4. .

10

( cm) ( kg) 18 4.

() - . X () Y (). ( , )x y - , . (scatter diagram), 16.

22-0088-02.indd 106 6/9/2013 11:12:55 pi

107

10.

- . . , , . , , , , .

- 18 . (x, y) , X Y . , - . X Y, - Y X. , :

y x= + (1)

, , , -

16

22-0088-02.indd 107 6/9/2013 11:12:55 pi

108

() X Y. , ( , )0 yy, - . . 16. , :

, ( , )173 67 (191 86), .

( , )x y (1), :

y xy x1 1

2 2

67 17386 191

= += +

= += +

= 115 6, =1 06, - (1) :

y x= +115 6 1 06, , . (2)

, - ( , , )0 115 6 1,06.

. . , . . - . , , . 1805 Legendre, (1752-1833) Gauss, (1777-1855) Theoria Motus -

22-0088-02.indd 108 6/9/2013 11:12:57 pi

109

. Gauss - 1794 ( 17 ), Legendre .

17 - 18 10. y x= + , ( , )x yi i =18 .

10.

, , , ( , )172 60 , x2 172= cm , 10, y2 60= kg, , , () 64 kg, 2 60 64 4= = , 4 kg . , (177 81), , y kg6 81= , - 71 kg, 6 81 71 10= = , - 10 kg . . - y x= + , . - , ,

17

22-0088-02.indd 109 6/9/2013 11:12:59 pi

110

( , )x yi i y x= + , -

ii

i ii

y x21

2

1= = = ( ) (4)

. , (4), - (least square estimators), ( ) ( ), , ( ) :

=

= = =

= =

x y x y

x x

i ii

ii

ii

ii

i

1 1 1

2

1 1

2

= y x

y yii

==1

1

, x xii

==1

1

.

y x= + (6)

() . = y x (6)

y y x x = ( ) ,

y a x= + ( , )x y . xi yi 10 (5) -:

=1 28, a = 156 1,

- (6),

y x= +156 1 1 28, , .

y x= +115 6 1 06, , 16.

(5)

22-0088-02.indd 110 6/9/2013 11:13:01 pi

111

y x= + a yy, x = 0. a = 0 . x1 x x2 1 1= + . -

y y x x x x2 1 2 1 1 11 = + + = + + + =( ) ( ) ( ) ( )

y y2 1= + . y a x= +

. , x y > 0 < 0 .

1. , , - 10 . - () .

) ) -

.

) , () 0, 0,5, 1 mgr , ; .

22-0088-02.indd 111 6/9/2013 11:13:03 pi

112

) , - . - -. , (X) (Y)., :

) ( , , , )0 3 12 3 ( , , , )0 8 1 9 . y x= + :

12 2 0 31 9 0 8

, ,, , ,

= += +

=18,5 = 20 8, ,

y x= 18 5 20 8, , .

, , , . - ( ) .) x = 0, 0,5 1 mgr, y =18 5, , 8,1 2,3 sec, .

:

18,5 sec.

22-0088-02.indd 112 20/3/2014 1:49:10

113

! 1 mgr 2,3 sec. 2,3 sec !

, , - .

2. () (.. ) ( ) -

) y a x= + ( ) .

) ) (), 8 /

;) -

, 10 ;

) - - ., :

= 8

xx

= = =

728

9

yy

= = =

768

9 5,

( ),

22-0088-02.indd 113 6/9/2013 11:13:05 pi

114

=

( )( ) ( )

=

=

xy x y

x x22 2

8 603 72 768 754 72

648( ) ( )( )( ) ( ) 8848

0 76= ,

= = + =y x 9 5 0 76 9 16 34, ( , )( ) , .

, ( -) y x= 16 34 0 76, , . ( , )0 ( , )x y , - , .) =16 34, - , 0 /. . , . 1 / ( ), 0,76 . ) 8/, x = 8. , x = 8 y

y = = =16 34 0 76 8 16 34 6 08 10 26, , , , , .

) , - , , . , .

1. ( ) .

22-0088-02.indd 114 6/9/2013 11:13:07 pi

115

)

)

2. , . y x= + .

3. - y x = + - . y, x = 6.

4. 1, () (). .

5. ) 5 .

22-0088-02.indd 115 6/9/2013 11:13:07 pi

116

) 15 , , , ;

1. - 10 :

) ;

) . )

. ) 75 ;

2. 4 , : ) 180 cm ) 170 cm.

3. 4 , : ) 167 cm ) 170 cm.

4. 8 , :

) .

22-0088-02.indd 116 6/9/2013 11:13:08 pi

117

) 25 .

) - .

5. (4) : ) , ) 28

, ) -

;

2.5

- , , , . , . , , ( )x yi i, , ( , ) x yi i X , Y :

:

22-0088-02.indd 117 6/9/2013 11:13:08 pi

118

x = 3, y = 5, sx =1 5, , sy = 0 94,

=x 3, =y 5, sx =1 46, , sy =1 66,

18() ( , )x yi i - y x= +2 58 0 47, ,

18() ( , )x yi i - = + y x2 58 0 47, , .

. () - () -. - . (linear correlation coefficient).

( , )x yi i , i =1 2, , ..., r X Y( , ) r :

18

22-0088-02.indd 118 6/9/2013 11:13:11 pi

119

rx x y y

x x y y

i ii

ii

ii

=

=

= =

( )( )

( ) ( )

1

2

1

2

1

(1)

Pearson. r xi yi ( ) ( )x xi ( )y yi -, . xi yi , ( )x xi ( )y yi , . , , , -. r , .

(1)

rx x y y

x x y yi i

i i

=

=

( )( )

( ) ( ), ,

2 2

8 518 7

0 76 .

22-0088-02.indd 119 6/9/2013 11:13:12 pi

120

r 0 41, .

r r> . - , - X , Y .

, , - . :

+1 1r .

:

0 1< < +r , , ( 19(), ())

< 0

r = 1, ( 19()), y x= + , < 0

r = 0, . , ( 19()).

22-0088-02.indd 120 6/9/2013 11:13:14 pi

121

r , - x, y :

rx y x y

x x

i ii

ii

ii

ii

ii

=

= = =

= =

1 1 1

2

1 1

= =

22

1 1

2

y yii

ii

(2)

- ( , )x yi i .

19

22-0088-02.indd 121 6/9/2013 11:13:15 pi

122

, , - . , - . , . , , . . r +1 > 0 . r 1 < 0 . r 0, 0 . - , .

r :

:

22-0088-02.indd 122 6/9/2013 11:13:16 pi

123

:

rxy x y

x x y y=

( )( ) ( ) ( )

22 2 2

=

7 4564 144 2087 3308 144 7 6468 208

0 92 2

( ) ( )( )( ) ( ) ( ) ( )

, .

r , - .

1. r : 0,6, 0,9, 0,7, 0,2, 0, 1.

2. ( , )x yi i - ( ) - X Y , :

)

)

)

22-0088-02.indd 123 6/9/2013 11:13:16 pi

124

)

)

3. ( , )x yi i .

4. .

) )

5. ( , )x yi i :

x = 7, y = 4 5, , xi2 210 = , yi2 92 = , x yi i =138.

.

6. - ( , )x y > 0 :

22-0088-02.indd 124 6/9/2013 11:13:18 pi

125

1. ( ) () () 1982-87 :

) . ) ,

; (1 3,8 )

2. (I.Q.) 10 - () ():

) ) -

)

3. ) ( )( )x x y y x y xyi ii

i ii

= = =

1 1

.

) ( )x yi i, ( )x xi = 2 28, ( )y yi = 2 112, x yi i = 308, x = 4, y = 9 .

4. - , :

.

22-0088-02.indd 125 6/9/2013 11:13:19 pi

126

1. 80 :

) , , .

) - .

) .

2. 60 :

3 4 5 2 6 7 2 2 3 4 5 6 7 4 0 6 4 0 0 2 3 1 1 4 5 4 3 3 7 6 4 3 2 1 0 1 2 3 3 3 4 4 5 6 8 3 9 3 1 0 4 4 5 5 6 6 7 8 9 5 )

.)

.) , ,

.

3. 200 . , - .) -

. )

. ) -

(- 35-) i) 5 , ii) 10 ;

) .

22-0088-02.indd 126 6/9/2013 11:13:19 pi

127

4. - 1990 1994 - ( ).)

.)

1990 1994;

) 1,4% 1990 2,1% 1994 -. ; ;

5. . 230 .

) , :

i) 180 ii) 190 iii) 200 . .

) - ;

6. 15 (..) , (..) 1998, () :

) .) N 15

.. .., i)

22-0088-02.indd 127 6/9/2013 11:13:20 pi

128

ii) 15 .. -;

7. ( -) 400 (freestyle) . ( ) . - ;

8. km2 1960 1974 :

) , .

) 1976 69,5 /km2. ; (: 1960 x =1).

9. (...) 1931-1964 .

) y x= +

... 1965.)

... 1965. -.

22-0088-02.indd 128 6/9/2013 11:13:20 pi

129

) -.

(: 1931 1, 1936 x = 6 1965 35).

10. , .

) () .

) 50 . ;) 30 -

, , ;

11. s x x y yxy i ii

= =1

1

( )( ) -

. sx2 , sy

2 , :

) =ssxy

x2

) r ssx

y

= .

12. (F) (C) , F C= + . , , F C, , :

F C= + -.

( )

X

pi ( )

Y

,

,

( )

X

pi ( )

Y

,

,

22-0088-02.indd 129 6/9/2013 11:13:22 pi

130

13. ( , )x y1 1 , ( , )x y2 2 , ,( , )x y r X Y( , ) . Z Y= , :

r X Z r X Y( , ) ( , )= . , < 0;

14. 1974 1994 (: -).

1995, 2000.

22-0088-02.indd 130 6/9/2013 11:13:23 pi

131

1-10 () ().

1. .

2. , -.

3. , - .

4. - .

5. .

6. , .

7. , .

8. .

9. -, , .

10. .

11. .

12. , - .

13. r = +0 6, r = 0 9, .

22-0088-02.indd 131 6/9/2013 11:13:24 pi

132

14. r X Y( , ) > 0, , .

15-24 .

15. :

. . . . .

16. x1 , x2 ,, x :

. s x xi2 1= ( ) . s x xi i

2 2 21= ( ){ } . s x xi i

2 2 2= ( ) . sx xi i22 2

2= ( )

17. 15% 20% , :

: : : :

18. y x= +10 3 25, y x =10 :

. 3,25 . 22,5 . 10 . .

19. y x= 2 3 , :

. 3 . . . 3/2.

20. , r = +1, :

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

22-0088-02.indd 132 6/9/2013 11:13:27 pi

133

21. r y x= + :

. . . . .

22. r X Y( , ) = 0, , : . . . . .

23. y : . . . , -

. .

24. , r1 0 9= + , , Z, W r2 0 3= + , .

. , Z, W . , ( )

, W . . 25-35 (), () (i), (ii), ,

.

25.

i) x = ii)

22-0088-02.indd 133 6/9/2013 11:13:28 pi

134

26. ) 1 2 10 18 19 i) x =10, s 7 5, ii) x = 20, s 7 5, ) 18 19 20 21 22 iii) x =10, s1,4 iv) x = 20, s1 4, ) 8 9 10 11 12 v) x =15, s 2.

27. ) ) i) ) ) ii) ) )

28. ) 5 7 8 10 13 24 i) x = 9 ) 1 2 8 9 9 25 ii) = 9 ) 1 2 9 12 12 18 iii) 0 9=

29. ) 10 11 12 13 14 i) x < ) 10 11 12 13 24 ) 1 11 12 13 14 ii) x = ) 20 21 22 23 24 ) 30 33 36 39 42 iii) x >

30. () () - (i) (iv) .

i) 24 .

ii) . iii) . iv) .

31. ((i) (iii)) ,

22-0088-02.indd 134 6/9/2013 11:13:30 pi

135

) 50% ) 50% ) 50%.

32. () () (i) (iv) :

33.

34.r = 0 = 0

r > 0 0

22-0088-02.indd 135 6/9/2013 11:13:31 pi

136

35. 2y x= : 0= r > 0 2= r = .

22-0088-02.indd 136 6/9/2013 11:13:32 pi

3

- . - Pascal Fermat 17 .

- , , , , 16 17 .. , -, . .

18 Bernoulli, De Moivre, Laplace Gauss. Laplace . Laplace , . , - Gauss. , . , 19 -, , .

22-0088-02.indd 137 6/9/2013 11:13:32 pi

138

, Chebyshev, Markov, Von Mises, Kolmogorov .., . - -. , , -, , , , .

- . - . , .

3.1 -

, 100 , 760 mm g, . , , - t. - (deterministic) . , ( -) . (random experiment). , - , - . :1. .2. .

22-0088-02.indd 138 6/9/2013 11:13:32 pi

139

3. .

4. .

5. .

6. .7. , 5 .., -

-.8. -

1.9. -

.

- . - (sample space) .

1,

2, ...,

, :

= { , , ..., } 1 2 .

, , - , = { , }. , 1, 2, 3, 4, 5, 6. , = { , , , , , }1 2 3 4 5 6 .

- (event) . , = { , , }2 4 6 , = { , , }1 3 5 = { }6 . , 6.

22-0088-02.indd 139 6/9/2013 11:13:32 pi

140

. . , , . , , . - . , , = { , , }2 4 6 , 2 4 6. , , - . . - . . ( ). -, = { , , , , , }1 2 3 4 5 6 A = { , , }2 4 6 ( ) = 3, ( ) = 6 ( ) = 0.

, . -, , . , , :

, B , .

, , , .

22-0088-02.indd 140 6/9/2013 11:13:34 pi

141

, , . .

, - , . - = .

. -, .

, = { , , , }1 2 3 4 = { , , }2 4 6 . 1, , , , , , , ( ) , ( ) , .

22-0088-02.indd 141 6/9/2013 11:13:35 pi

142

, = { , , }2 4 6 = { , , }1 3 5 . , . . : , =. .

1. .i) . ii)

:1: 2: 2 3: 2 4: 5: K

iii) 3 , 5 2 , 5 4 .

i) , -:

22-0088-02.indd 142 6/9/2013 11:13:36 pi

143

, -

= { , , , , , , , }.

ii) : 1 = { , , , }

2 = { , , }

3 = { , , , } ( 3 = 1)

4 = { , }

5 = { , , , }.

iii) 3 3, 2. , 3 = { , , , }. 5 2 5 2, , . , 5 2 = { , }. 5 4 . , 5 4 = { , , , , }.

2. . Venn :i) . ii) .

i) , , - . - . , ( ) ( ) ( ) ( ) .

22-0088-02.indd 143 6/9/2013 11:13:38 pi

144

ii) - , - . - , ( ) .

1. , , . : , - . . ( ).i) ; ii) ; iii)

;

2. , - .

( ).

3. - . . , . - , :i) ii) :

.

4. . , . :

22-0088-02.indd 144 6/9/2013 11:13:38 pi

145

.i) ii) : iii) : iv) v) : , -

( ) .

5. - , , , . 0 1 , , , , , , . . :i) .ii) : -

.iii) : . iv) : .

6. - :i) . 3

. ii) .

.iii) .

30 30 . iv) .

.

7. - . .

22-0088-02.indd 145 6/9/2013 11:13:39 pi

146

1. . - - , .

2. . :: 1

2 .: : 5 , , , ( ) .

3. , : A , .

4. . - A -.

3.2

, , - . , , . , . (). ; ; - .

22-0088-02.indd 146 6/9/2013 11:13:39 pi

147

,

fA. = { , , ..., } 1 2 -

{ }1 , { }2 , , { } 1 2, , ..., ,

f1 1=

, f2 2=

, , f

= -

: 1. 0 1fi , i =1 2, , ...,, ( 0 i )

2. f f f1 2 1 2 1+ + + =+ + +

= =.... ...

.

: - - . 10, 20, 30, ..., 200 1 - .

f

-

0,5 , 0,5. - , , ,

1

22-0088-02.indd 147 6/9/2013 11:13:41 pi

148

{ } , { } . - ( ), . , -, . - .

. . -

{ } , { } 12

.

{ }1 , { }2 , { }3 , { }4 , { }5 { }6 16

.

, , . . , . - 2 4 6

16

, -

16

16

16

36

+ + = .

,

.

- :

P A A( ) (( )

.=

)=

22-0088-02.indd 148 6/9/2013 11:13:43 pi

149

, , Laplace 1812.

:

1. P( ) ( )( )

= =1

2. P( )( )

= =0 0

3. 0 1 P A( ) ,

.

, . , -. , . -, .

= { , , ..., } 1 2 . { }i , P i( ) , :

0 1 P i( )

P P P( ) ( ) ... ( ) 1 2 1+ + + = .

P i( ) { }i . - () = { , , ..., } 1 2 P P P( ) ( ) ... ( ) 1 2+ + + , P( ) . = 0

P i( ) =

1, i =1 2, , ..., ,

. , - , .

22-0088-02.indd 149 6/9/2013 11:13:46 pi

150

= { , , ..., } 1 2 , -

P i( ) =

1, i =1 2, , ..., .

- , . . .

1. :

P A B P A P B( ) ( ) ( ) = +

( ) = ( ) = , A B + , . , N A B N A N B( ) ( ) ( ). = + = + :

P A BN A BN

( ) ( )( )

=

=+N A B

N( ) ( )

( )

= +N AN

BN

( )( )

( )( )

= +P P( ) ( ) . (simply additive law) . , , P A B P P P( ) ( ) ( ) ( ) = + + .

2. :

P A P A( ) ( ) = 1

22-0088-02.indd 150 6/9/2013 11:13:48 pi

151

A A = , , , :P A A P A P A( ) ( ) ( ) = +

P P A P A( ) ( ) ( ) = + 1= + P A P A( ) ( )

P A P A( ) ( ). = 1

3. :

P A B P A P B P A B( ) ( ) ( ) ( ) = +

N A B N A N B N A B( ) ( ) ( ) ( ), = + (1)

( ) ( )+ A B -. (1) () :

N A BN

N AN

N BN

N A BN

( )( )

( )( )

( )( )

( )( )

= +

P A B P A P B P A B( ) ( ) ( ) ( ) = + .

(additive law).

4. A B , P A P B( ) ( )

A B :

N A N B( ) ( )

N A N B( )

( )( )( )

P A P B( ) ( ) .

22-0088-02.indd 151 6/9/2013 11:13:51 pi

152

5. P A B P A P A B( ) ( ) ( ) = .

A B ( ) ( )A B A B A = , :

P A P A B P A B( ) ( ) ( )= + .

P A B P A P A B( ) ( ) ( ) = .

1. . .

, - , .

36 - , ( ) = 36.

: , = {( , ),( , ),( , ),( , ),( , ),( , ),( , ),( , ),( , ),(1 2 2 1 2 3 3 2 3 4 4 3 4 5 5 4 5 6 66 5, )}

( ) =10

, P A N AN

( ) ( )( )

= = =

1036

518

.

22-0088-02.indd 152 6/9/2013 11:13:53 pi

153

, 518

0 28 , , , 28%.

2. P A ,( ) ,= 0 5 P B( ) ,= 0 4 P A B( ) , = 0 2 . :i) . ii) .

i) ( )A B .

P A B P A B(( ) ) ( ) = 1

= + = + = =

11 0 5 0 4 0 21 0 70 3

( ( ) ( ) ( ))( , , , ),

, .

P A P B P A B

ii) ( ) ( )A B B A . A B B A , :

P A B B AP A B P B A

(( ) ( ))( ) ( )

== +

= + = + = + =

P A P A B P B P A BP A P B P A B( ) ( ) ( ) ( )( ) ( ) ( ), , ,

20 5 0 4 2 0 20,, .5

3. P A( ) ,= 0 6 P B( ) ,= 0 5 . i) . ii) 0 1 0 5, ( ) , P A B .

22-0088-02.indd 153 6/9/2013 11:13:55 pi

154

i) , - :

P A B P A P B( ) ( ) ( ) , , , = + = + =0 6 0 5 11

, , P A B( ) >1, . , -.

ii) A B B A B A , P A B P B( ) ( ) P A B P A( ) ( ) ,

P A B( ) , 0 5 (1)

:

P A B P A P B P A B( ) ( ) ( ) ( ) = +

P A B P A B( ) , , ( ). = + 0 6 0 5

P A B( ) 1 .

: 0 6 0 5 1, , ( )+ P A B

0 6 0 5 1, , ( )+ P A B

0 1, ( ) P A B . (2)

(1) (2) :

0 1 0 5, ( ) , P A B .

1. 52 . i) ii) .

2. .

22-0088-02.indd 154 20/3/2014 1:50:45

155

3. : 10 , 15 , 5 10 . . :

i) ii) iii) .

4. 30 , . :

, - .

5. = { } 10 20 , = { } pipi 3

= { } pipi 4 . - , i) ii) .

6. 30%, - 20% 40%. i) ii) .

7. P A( ) ,= 1730

P B( ) = 715

P A B( ) = 23

. P A B( ) .

8. P A( ) =12

,

P A B( ) = 56

P A B( ) = 13

. P B( ).

9. P A P B( ) ( )= , P A B( ) , = 0 6 P A B( ) , = 0 2. P A( ).

10.

P A( ) = 12

, P B( ) = 23

P A B( ) = 112

. P A B( ) .

11. P A B P A P B( ) ( ) ( ) + .

22-0088-02.indd 155 6/9/2013 11:14:01 pi

156

12. D V. 25% D, 55% V 15% . ;

13. 10% , 6% 2% . -

) ; ) ;

14. 80% , 30% - 20% . . .

1. P A( ) ,= P( ) = P A B( ) = , :

i) ii) iii) .

2. 15% , 40% 10% . . .

3. P AP A( )( )

=34

, P A( ) P A( ) .

4. 0 1<

157

3.3

, :

P A N AN

( ) ( )( )

=

.

, , P A( ) . , - . - .

, , . () () () () . - . . . 4 2 . . - 4 2 8 = . :

22-0088-02.indd 157 6/9/2013 11:14:04 pi

158

:

1, 2, , . 1 1 2 2 - , , , - 1 2 ... .

, -, - - 1 , 2 , , - - , 1 2 ... - -.

5 -, , . : : 1 , 2 3 . 1 5 , . 2 4 -, . 3 3 , . , , - 5 4 3 60 = . 5 3.

: , , .

, :

22-0088-02.indd 158 6/9/2013 11:14:06 pi

159

= +( )( )...( )1 2 1 (1)

. , (1, 2, 3), (1, 4, 3) (3, 2, 1) . , .

- (1)

= ( )( )...1 2 3 2 1 .

1 2 3 2 1 ...( )( ) ! -.

= ! (2)

, 5 , 5 1 2 3 4 5 120= = - < :

= +( )( )...( )1 2 1

= +

( )( )...( )( )( )...

( )( )...1 2 1 1 3 2 1

1 3

=

2 1

!

( )!.

=

!

( )! (3)

(3) = ,

= = !, -

!!

!0

= . 0 1!= .

22-0088-02.indd 159 6/9/2013 11:14:08 pi

160

5 , , , - 3 , . x , 3! . -, 3!x. 3

5 3= !x. , 35 3= !x,

x = =

=

=3

5

35

3 5 31206 2

10!

!!( )!

.

:

{ , , } , { , , } , { , , } , { , , } , { , , } , { , , } , { , , } , { , , } , { , , } { , , } . 5 3.

:

.

,

= =

=

!

!( )!

!!

!( )!.

,

=

!!( )!

.

- .

22-0088-02.indd 160 6/9/2013 11:14:10 pi

161

1. 13 1, 2, : (1), (2), ().i)

. ii) , -

: 12 . : 11 .

i) 13-, - . , , 3 3 3 3 3 1 594 323

13

13 = =... . .pi

a

- .

ii) 12 -

. 1312

12 , 2 .

, ( ) .=

1312

2,

P A( ) ( )( )

,= =

=

=

1312

2

313 23

263

0 00001613 13 13 .

11

2 . 1311

11 2 -

22-0088-02.indd 161 6/9/2013 11:14:11 pi

162

2 . ,

( ) =

1311

2 2 . ,

P( ) ( )( )

,

= =

=3

1311

2 2

3123

0 00019613 13 .

2. 6 49, , : 4 ;

, 49 6,

( ) =

496

.

:

64

4 6 -

. 49 66 4

432

=

2

. ,

( ) =

64

432

.

P( ) , =

=

64

432

496

1354513983816

0 000969 1 .

3. ( 365) ; ( 365 ).

, -

22-0088-02.indd 162 6/9/2013 11:14:12 pi

163

P P( ) ( ) = 1 . P( ) P( ) . ( ) ...= =365 365 365 365 365 , 365 . - 365 365 1 365 2 365 1 ( ) ( )...[( ( )] , .

,

P( ) ( ) ( )...( ) ...( = + = 365 365 1 365 2 365 1365

365 364 365

++1365

)

.

P( ) ...( ) = +1 365 364 365 1365

.

P( ) :

23 - 50%, 70 - .

1. 3 , 4 , 5 , 10 2 , , ; - ;

2. - ; -;

3. 4 6 ; ;

22-0088-02.indd 163 6/9/2013 11:14:14 pi

164

4. 4 3 ; ;

5.

=

.

6. 8 1, 2, ,8. - ; -

1;

1. 1, 2, 3, 4, 5 . , ;

2. 1, 2, 3, , . 1.

3. , , . ;

4. - .

5. 7 6 4 . :

i) ii) iii) .

6. 20 , 5 . 4 . , . .

22-0088-02.indd 164 6/9/2013 11:14:15 pi

165

7. 1 2 . 1 10 - 2 20 . ; , 1 ;

8. . - .

9. - . :

: : : .

10. 10 12 . .

11. 4 24 .

3.4 -

. 7 8 . 3 6 , 4 2 . - :

22-0088-02.indd 165 6/9/2013 11:14:15 pi

166

:

: : .

15 ,

P( ) = 915

P( ) = 815

.

. 8 -

68

. : -

. - P ( )

, P P ( ) = 68

( ) .

P P ( ) = 69

( ).

, - P( ) > 0. . , . - ., , :

P PP

( ) = =

=( )

( )

( )( )( )( )

( )( )

.

.

22-0088-02.indd 166 6/9/2013 11:14:17 pi

167

:

P( ) > 0, PP( )

( ) -

P ( ) . :

P PP

( ) = ( )( )

.

, P( ) > 0, P PP

( ) = ( )( )

.

- :

P PP

( ) = = =( )( )

615815

68

PP

P

( ) = = =( )( )

615915

69

.

P P P P P( ) ( ) ( ) = ( ) = ( ) .

.

, ,

P ( ) = 68

, P( ) = 915

. , P P ( ) ( ). . - . ,

22-0088-02.indd 167 6/9/2013 11:14:19 pi

168

: 1 : , :

P( ) = =636

16

, P( ) = =1836

12

P( ) = 336

.

, P ( ) = =318

16

P ( ) = =36

12

.

, , P P ( ) = ( ) P P ( ) = ( ). . :

P( ) > 0 P( ) > 0 , P P ( ) = ( ) P P ( ) = ( ) .

- P P P( ) ( ) ( ) = . , P( ) > 0 P( ) > 0 . :

,

P P P( ) ( ) ( ) =

.

P P P( ) ( ) ( ) = . , , , - -. , : : ., : : .

22-0088-02.indd 168 6/9/2013 11:14:22 pi

169

1. , . - . 0,003, .

, -. , 0 003 0 003, , . ,

1 0 003 0 003 0 999991 =, , , 99,9991%.

2. () 20%, () 30% () 50% . , 5% , 4% 2% . :i)

; ii) ,

;

1, 2 3 , , P( ) ,1 0 2= , P( ) ,2 0 3= P( ) ,3 0 5= . 1, 2 3 1 2 3 = . ,

= ( ) ( ) ( )1 2 3

P P P P( ) ( ) ( ) ( ) = + + 1 2 3 .

P 1 0 05( ) = , , P 2 0 04( ) = , P 3 0 02( ) = , . :

22-0088-02.indd 169 6/9/2013 11:14:24 pi

170

i) (1) :

P P P P P P P( ) ( ) ( ) ( ) = ( ) + ( ) + ( )1 1 2 2 3 3= + + =0 2 0 05 0 3 0 04 0 5 0 02 0 032, , , , , , , .

ii) P 1( ) .

P PP

P PP

11 1 1 0 2 0 05

0 0320 010 03

( ) = = ( ) = =( )( )

( )( )

, ,,

,, 22

0 3125= , ,

, 31,25%.

, , :

1, 2 3 , , - . , . (i) . 1 , 2 3 - 1 , 2 3 . - . P( ) , , 1 0 2 0 05 = , P( ) , , 2 0 3 0 04 = P( ) , , 3 0 5 0 02 = ., , - .

P( ) , , , , , , , = + + =0 2 0 05 0 3 0 04 0 5 0 02 0 032.

22-0088-02.indd 170 6/9/2013 11:14:27 pi

171

- . P 1( ) (a posteriori) -, P( )1 (a priori) .

1. . 6;

2. 52 . -;

3.

P( ) = 12

, P( ) = 14

P ( ) = 45

. -

P( ) , P ( ) P( ) .

4. P( ) = 56

, P( ) = 14

P( ) =23 , P( ) .

5. . 6 .

6. 6 8 . , - . .

- .

7. P( ) = 14

P( ) = 15

. P( ) P( ) .

22-0088-02.indd 171 6/9/2013 11:14:30 pi

172

8.

P( ) = 13

, P ( ) = 23

P ( ) = 12

.

i) ; ii) ; iii) P(B).

9.

P P ( ) + ( ) =1.

10. 25% - , 15% 10% . .i) , -

; ii)

;

1. ( ) 0,8. . .

2. 10%. - , - , .

3. , 60% 40% . 10% 5% - . , :

i) ii) , .

22-0088-02.indd 172 6/9/2013 11:14:31 pi

173

4. 60% 40% . 50% 30%. , ;

5. . - . . - .

6. 9 12 , -:

.

i) ; ii)

; iii) -

;

7. 1 . - ( , ) 1% ( , ) 5%. . .

22-0088-02.indd 173 6/9/2013 11:14:31 pi

174

1. = { , , , , ..., }0 1 2 3 100 . -

P( ) =12

, =1 2 100, , ..., .

P( )0 .

2. , . P( ) , 0 28 P( ) , . 0 71 i) P P( ) , ( ) 1 01 ii) .

3. :

=

+

1 11

4. , . - . 2 . 1 , 2 ; ;

5. , - : i) ii) , iii) .

22-0088-02.indd 174 6/9/2013 11:14:32 pi

175

1. -, , , -

, 13

.

; ;

2. 5 . -,

55

1= . .

3. , , , - . , .) ;)

i) 3 6 ii) 36 iii) 6! iv) 63 v) 3 63 .( )

( 4-7 - . ).

4. 0,4, ;

) 0,2 ) 0,8 ) 0,6 ) 1,4.

5. P A( ) =12

, P( ) =12

,

P A B( ) = 14

, P A B( ) ;

) 1 ) 34

) 14

) 116

) .

6. Venn ;

) ) ) ) .

22-0088-02.indd 175 6/9/2013 11:14:34 pi

176

7. - 2 3 10 8 ;

() 210

38 () 2

1038+ ()

102

83

+

()

102

83

( 8-13 . , , , ).

8. .

9. .

10. , - .

11. .

12. .

13. , .

14. , P( ) , = 0 6 P( ) , = 0 2. .

P( ) 0,8

P( ) 0,6

P ( ) 0 P ( ) 0,2 0,12

22-0088-02.indd 176 6/9/2013 11:14:36 pi

177

15. , P( ) , = 0 6 P( ) , = 0 2. .

P( ) 0,6

P( ) 0,68

P ( ) 0,2 P ( ) 0,12

16. , P P( ) ( ) , + =1 3;

- .

17. - Venn:

22-0088-02.indd 177 6/9/2013 11:14:37 pi

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