36
79 5.2. ( Siméon Denis Poisson). n p - , - , - , - , - , - , . . . . , k : n k k e k X P k , , 2 , 1 , 0 , ! : n p p n , 0 , . , . . X M 2 ) ( X D

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Page 1: N H QSS Q 3ww qq 3ww i [ w q q 1 q 1 wq q q w 3www q q 3w q w ; q s qs st t

79

5.2.

( Siméon Denis Poisson).

n p

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

.

. .

, k

:

nkk

ekXP

k

,,2,1,0,!

:

nppn ,0, .

, .

.

XM 2)(XD

Page 2: N H QSS Q 3ww qq 3ww i [ w q q 1 q 1 wq q q w 3www q q 3w q w ; q s qs st t

80

5.2. 12 . 10 :

) , ) .

:

X

.2

8647,0101111 2eXPXPXP

10 86,47%.

4060,02!1

2

!0

2101 22

2120

eeee

XPXPXP

10

40,60%.

Page 3: N H QSS Q 3ww qq 3ww i [ w q q 1 q 1 wq q q w 3www q q 3w q w ; q s qs st t

81

5.3.

( . Johann Carl Friedrich Gauß) ,

. ,

.

5,0p N N

.

, . ,

. . X

xexfx

,2

1 2

2

x

,xF

x : xXPxF

5.1.

.

.

Page 4: N H QSS Q 3ww qq 3ww i [ w q q 1 q 1 wq q q w 3www q q 3w q w ; q s qs st t

82

,

44 XPF

40,0XP .xf .

5.2.

40,0140,0140,04040,0 FXPXPFXP

3446,06554,0140,0XP

xFxF 1 aXPaXP 1 .

X , :a b

aFbFaXPbXPbXaP

: 121 xFxFxFxFxFxXxP

:

. ,

. 0,95 95%. :

975,021295,0 xFxF

3

96,1;96,1 .

:

95,0xFzZPxXP

1121111111 FFFFFXP

x96,1x

x

Page 5: N H QSS Q 3ww qq 3ww i [ w q q 1 q 1 wq q q w 3www q q 3w q w ; q s qs st t

83

.64,1x

X

,N

2

221

, , , 02

x

f x e x

f x

e -

x x . .

N

X .

x

xfx

e xf .

xf 0xe

N .

1 12

0 0

Page 6: N H QSS Q 3ww qq 3ww i [ w q q 1 q 1 wq q q w 3www q q 3w q w ; q s qs st t

84

1

Z

Z

X X npZ

npq

Z X

1)

2) 2

3) 3

5.3.

Page 7: N H QSS Q 3ww qq 3ww i [ w q q 1 q 1 wq q q w 3www q q 3w q w ; q s qs st t

85

5.3. 3% . 500 :

) 10 20 , ) 15 .

:

X 500n .03,0p

500210

500

...

...210:

ppppX

2011102010 pppXP

iii i

iXPp 50097,003,0500

. .

,

, 2010 X 9,5 20,5 :X

npnpXnpPXPXP 5,205,95,205,92010

npq

np

npq

npX

npq

npPXP

5,205,92010

1503,0500pn .55,1497,003,0500qpn

44,181,3

5,5

55,14

155,95,9

npq

np

44,181,3

5,5

55,14

155,205,20

npq

np

npq

npXZ

44,144,144,144,12010 FFZPXP

44,1144,1 FF 85,019251,02144,12 F

500 10 20 85%.

Page 8: N H QSS Q 3ww qq 3ww i [ w q q 1 q 1 wq q q w 3www q q 3w q w ; q s qs st t

86

) 5,155,015015 XPXPXP

55,14

155,15

55,14

15

55,14

155,0 XP

06,413,013,006,4 FFZP 06,4113,0 FF

5507,0115517,0106,413,0 FF

500 15 55,07%.

5.4.

4.500 kg/ 000.2 kg/ .

: ) 3.000 kg/ 4.000 kg/ , ) 4.000 kg/ , ) 4.000 kg/ .

:

000.2

500.4000.4

000.2

500.4000.3000.4000.3

XPXP

25,075,0000.2

500.4000.4

000.2

500.4000.3ZPZP

75,0125,0175,025,0 FFFF

3721,02266,05987,07734,015987,01

3.000 4.000 kg 37,21%. 000.4

140001000.4X

PXPXP

5987,025,01125,01000.2

500.4000.41 FZPZP

4.000 kg 59,87%.

25,0125,0000.2

500.4000.4000.4 FZPZPXP

.4013,05987,01

4.000 kg 40,13%.

Page 9: N H QSS Q 3ww qq 3ww i [ w q q 1 q 1 wq q q w 3www q q 3w q w ; q s qs st t

87

5.5. - 120 20 .

, 95% ?

:

95,020

120xZP

xXPxXP

64,195,095,020

120zzF

xF

8,1521208,3264,120

120x

x

- 153

.

Page 10: N H QSS Q 3ww qq 3ww i [ w q q 1 q 1 wq q q w 3www q q 3w q w ; q s qs st t

88

5.4. 2

nXXX ,...,, 21

1,0N22

221

2nXXX .n

2

n

.2n

2

2

30,,2,1n

, 10 .

01,0 05,02;n 2

n

, .2;

2nnP

5.4.

30n 2n

2n 2

m 2n + 2

m = 2mn .

2

12;

2nnP .

95,01 %505,0 215;0,05 25, 95,0252

152

05,0;15215 PP

Page 11: N H QSS Q 3ww qq 3ww i [ w q q 1 q 1 wq q q w 3www q q 3w q w ; q s qs st t

89

5.5.

William Sealy Gosset t

Z 1,0N 2 n

n

ZT

n2

t n

,;nt 10

nt ,;nt

;nn ttP 1;nn ttP .

5.5. 5.5.

)(xf - t

n

120n 1,0N 130n

1,0N .

Page 12: N H QSS Q 3ww qq 3ww i [ w q q 1 q 1 wq q q w 3www q q 3w q w ; q s qs st t

90

nt

5n

5n 025,02

05,095,01

t

95,0025,0;5025,0;5 tttP n 95,0571,2571,2 ntP

571,2571,2 nt .

5.6.

5.6.

5.8. X 30n .3,0p

:

.93,030pn

7,03,011 qqp

3,67,03,0302 qpn

5.9. 39% . 80 ?

39,0p 61,01 pq ).

80 0,39 31M X n p

Page 13: N H QSS Q 3ww qq 3ww i [ w q q 1 q 1 wq q q w 3www q q 3w q w ; q s qs st t

91

5.10. 800 420 380 .

: ) , ) , ) , .

:

, , ,,

;525,0p .475,0q

, 4 ,

.

, X 4 ( )

.

373,02256,02756,012

34475,0525,0

2

42 22XP

37,3%.

2XP 04 475,0525,00

4+ 13 475,0525,0

1

4+ 22 475,0525,0

2

4

726,0226,0276,06475,0145,041076,012XP

72,6%.

31 XP 13 475,0525,01

4+ 22 475,0525,0

2

4+ 31 475,0525,0

3

4

925,0475,0145,04226,0276,06475,0145,0431 XP

, 92,5%.

Page 14: N H QSS Q 3ww qq 3ww i [ w q q 1 q 1 wq q q w 3www q q 3w q w ; q s qs st t

92

5.11. . : )

, ) 1 , ) 2 6.

.,,2,1,0,!

nkk

ekXPk

0498,01

11

!0

30 3

03

eeXP

4,98%.

101111 XPXPXPXP

!1

3

!0

31

13

03 ee 8008,01494,00498,01

1 80,08%.

62 XP 2XP + 3XP + 4XP + 5XP + 6XP

=!6

3

!5

3

!4

3

!3

3

!2

3 63

53

43

33

23 eeeee

767,0050,0101,0168,0224,0224,0

2 6 76,7%.

5.12. 2,4. ,

: ) 1 , ) 3 , ) 7 .

:

.,,2,1,0,!

nkk

ekXPk

218,0718,2

4,2

!1

4,21

4,2

14,2eXP

1 21,8%.

3

Page 15: N H QSS Q 3ww qq 3ww i [ w q q 1 q 1 wq q q w 3www q q 3w q w ; q s qs st t

93

) 209,00204,11

3040,2

718,2

3040,2

!3

4,23

4,2

34,2eXP

3 20,9%.

) 7 1 6 0,0116P X P X

7 1,16%.

5.13. 2,3. , 365

, ?

:

:

1002,09718,9

1

718,2

1

!0

3,20

3,2

03,2eXP

365 : 3757,361002,0365

5.14. X .

, : 2XP ,

32 XP 96,196,1 XP 58,258,2 XP .

:

9772,00202202 FFXPXPXPXP

0215,09772,09987,0232332 FFXPXPXP

96,1196,196,196,196,196,1 FFXPXPXP

95,019500,119750,02196,12 F

58,2158,258,258,258,258,2 FFXPXPXP

9902,019902,119951,02158,22 F

Page 16: N H QSS Q 3ww qq 3ww i [ w q q 1 q 1 wq q q w 3www q q 3w q w ; q s qs st t

94

5.15. X 4 .2

5,2XP ,

4XP

96,196,1 XP .

75,0175,02

45,25,2 FFFXPxXP

2266,07734,01

5000,05000,0101414 FXPXP

96,196,196,196,1 PX

PXP

95,019500,1196,1296,196,1 FPZP

Page 17: N H QSS Q 3ww qq 3ww i [ w q q 1 q 1 wq q q w 3www q q 3w q w ; q s qs st t

95

6.

-

-

-

nN

Page 18: N H QSS Q 3ww qq 3ww i [ w q q 1 q 1 wq q q w 3www q q 3w q w ; q s qs st t

96

N N . N n

N

nN n n

N

nXXX ,,, 21

n X

nxxx ,,, 21

nXXX ,,, 21

n nXXX ,,, 21 .

K (K -

)N

nXXX ,,, 21 nxxx ,,, 21

Page 19: N H QSS Q 3ww qq 3ww i [ w q q 1 q 1 wq q q w 3www q q 3w q w ; q s qs st t

97

6.1.

nXXX ,,, 21

n

XXXX n21

iX

X XMXM i

nn XXXM

nn

XXXMXM 21

21 1

nXMXMXMn

XM 21

1

XMnn

XMXMXMn

XM11

XMXM

,

.

X K

K nxxx ,,, 21

nXXX ,,, 21

nXXX ,,, 21 XM

n

xxxXM n

__

2

_

1

Page 20: N H QSS Q 3ww qq 3ww i [ w q q 1 q 1 wq q q w 3www q q 3w q w ; q s qs st t

98

X 2 2

2 2 2( )D X M X M X M X M X M X

nSXD X

22)( 2

nSX

, :

22( )

1X

N nD X S

n N

1X

N nS

Nn

1N

nN

n

S X .

( ). ( , .) ( )

. ( )

.

( , .) (

, .).

( ) ( . ).

V , W , : M(W)=V, W

V.

X 2S

2

Page 21: N H QSS Q 3ww qq 3ww i [ w q q 1 q 1 wq q q w 3www q q 3w q w ; q s qs st t

99

X

XM .

VWM W V .

VWM WM V n

W V .

V

1W 2W

6.2.

X

Page 22: N H QSS Q 3ww qq 3ww i [ w q q 1 q 1 wq q q w 3www q q 3w q w ; q s qs st t

100

:

2 2

1

1

n

i ii

f x n XS

n, .

X

.

( X - ).

.

2 .

,N

nXXX ,,, 21 X

nN ,

nXXX ,,, 21 X

nN , .

n

Page 23: N H QSS Q 3ww qq 3ww i [ w q q 1 q 1 wq q q w 3www q q 3w q w ; q s qs st t

101

6.3.

n

Xn

nnn

XnXnXnX

r

iii

r

rr 1

21

2211

rnnnn 21

n

SnS

r

iii

1

2

Page 24: N H QSS Q 3ww qq 3ww i [ w q q 1 q 1 wq q q w 3www q q 3w q w ; q s qs st t

102

6.4.

6.4.1. µ (

X n

%1100

2 2

X z X zn n

2

z

/ 2. :

=90% =10% 2

1,64z .

=95% =5% 2

1,96z .

=99% =1% 2

2,58z .

6.4.2. µ

(n

n XS

n

%1100

2 2

S SX z X z

n n

/ 2.

2 2

1P X z X zn n

2

z

Page 25: N H QSS Q 3ww qq 3ww i [ w q q 1 q 1 wq q q w 3www q q 3w q w ; q s qs st t

103

:

2 2

1S S

P X z X zn n

=90%, =10% 2

1,64z .

=95%, =5% 2

1,96z .

=99%, =1% 2

2,58z .

6.4.3. µ

nXXX ,,, 21

µ 2.

X

TS

n

t

1n

X S

2, %1100

; 1 ; 12 2

n nX t X t

n

S

n

S

; 12

nt - t 1n / 2.

1, 1,2 2

1n n

S SP X t X t

n n

Page 26: N H QSS Q 3ww qq 3ww i [ w q q 1 q 1 wq q q w 3www q q 3w q w ; q s qs st t

104

6.4.4. ( )

nXXX ,,, 21

2. 2S

: 2

22

1n S 2)

1n

2S n 2 ,

%1100 2 2 2

22 2

, 1 1 , 12 2

1 1

n n

n S n S

2

, 12

n 2

1 , 12

n 2 1n

2 1

2

6.4.5.

n X ( n

P X n p

n p

P p (1 ) /p p n p

n 5np (1 ) 5n p .

n

(1 ) (1 )

X np P pZ

np p p pn

Page 27: N H QSS Q 3ww qq 3ww i [ w q q 1 q 1 wq q q w 3www q q 3w q w ; q s qs st t

105

P n %1100

p

2 2

(1 ) (1 )z p z

P P P PP P

n n

2

z

/ 2.

2 2

(1 ) (1 )1

P P P PP P

nz p z

nP

Page 28: N H QSS Q 3ww qq 3ww i [ w q q 1 q 1 wq q q w 3www q q 3w q w ; q s qs st t

106

6.5.

6.1. 800 .

6.1.

(kg) 2 100 3 200 4 400 5 70 6 30

95% : ) 800 ; )

1.132.000.

:

6.2.

ix if ii fx ii fx 2

2 100 200 400 3 200 600 1.800 4 400 1.600 6.400 5 70 350 1.750 6 30 180 1.080

800 2.930 11.430

) 7,3800

930.2X

7735,0799

478

1800

7,3800430.11

1

222

n

XnxfS ii

n

2 2

S SX z X z

n n

2 2

1S S

P X z X zn n

Page 29: N H QSS Q 3ww qq 3ww i [ w q q 1 q 1 wq q q w 3www q q 3w q w ; q s qs st t

107

95% 5%2

1,96z .

2

z

.975,01295,0 zFzF

z

1,96 1,96S S

X Xn n

0,7735 0,77353,7 1,96 3,7 1,96

800 800

75,365,3

0,95 3,65 3,75 kg.

2 2

S SN X z N N X z

n n

3,751.132.000N3,651.132.000

000.245.4N800.131.4 , 95%

1.132.000 4.131.800 kg 4.245.000 kg.

6.2. 400 50 .

5,7 , 0,95 .

:

µ

( :

2 2

X z X zn n

2 2

1P X z X zn n

Page 30: N H QSS Q 3ww qq 3ww i [ w q q 1 q 1 wq q q w 3www q q 3w q w ; q s qs st t

108

95% 5%2

1,96z .

7,5,50,400 Xn

2 2

X z X zn n

5,7 5,750 1,96 50 1,96

400 400

6,504,49

6.3. 80 .

6.3. (mc/ )

22 10 22,1-30 25 30,1-38 30 38,1-46 15

80

0,01 : ) ; ) ,

450.000 .

: 6.4.

six if isi fx isi fx 2

18 10 180 3.240 26 25 650 16.900 34 30 1.020 34.680 42 15 630 26.460

80 2.480 81.280

3180

480.2X

46,779

400.4

180

3180280.81

1

222

n

XnxfS sii

Page 31: N H QSS Q 3ww qq 3ww i [ w q q 1 q 1 wq q q w 3www q q 3w q w ; q s qs st t

109

99% 1%2

2,58z .

2 2

S SX z X z

n n

2,58 2,58S S

X Xn n

80

46,758,231

80

46,758,231

15,3385,28

0,01

28,85 33,15 .

2 2

S SN X z N N X z

n n

2,58 2,58S S

N X N N Xn n

33,15.000504N85,82450.000

500.917.14N500.982.12

6.4. 700 :

6.5.

1 200 2 250 3 150 4 70 5 30

700 ) 0,05

2 .) 2

, 200.000.

Page 32: N H QSS Q 3ww qq 3ww i [ w q q 1 q 1 wq q q w 3www q q 3w q w ; q s qs st t

110

:

150 70 30 250X

250 / 700 0,36P X n

2 2

(1 ) (1 )z p z

P P P PP P

n n

2 2

(1 ) (1 )1

P P P PP P

nz p z

nP

2 2

(1 ) (1 )z p z

P P P PP P

n n

0,36(1 0,36) 0,36(1 0,36)0,36 1,96 0,36 1,96

700 700p

0,36 1,96 0,018 0,36 1,96 0,018p

396,0324,0 p

2 2

(1 ) (1 )z Np

P P P PN P P

nN

nz

396,0000.200324,0000.200 Np

200.79800.64 Np

Page 33: N H QSS Q 3ww qq 3ww i [ w q q 1 q 1 wq q q w 3www q q 3w q w ; q s qs st t

111

6.5. 80 :

6.6.

(mc/ ) 22 10

22,1-30 25 30,1-38 30 38,1-46 15

80

, 0,05 30 mc/ .

:

30 mc/ : 30 15 45X

: 45 / 80 0,56P X n

:

2 2

(1 ) (1 )z p z

P P P PP P

n n

0,56(1 0,56) 0,56(1 0,56)0,56 1,96 0,56 1,96

80 80p

0,56 1,96 0,055 0,56 1,96 0,055p

67,045,0 p

30 mc/ , 0,95 45% 67%.

6.6. 500 , 60 . 95%

.

:

60X

60 / 500 0,12P X n

Page 34: N H QSS Q 3ww qq 3ww i [ w q q 1 q 1 wq q q w 3www q q 3w q w ; q s qs st t

112

:

2 2

(1 ) (1 )z p z

P P P PP P

n n

0,12(1 0,12) 0,12(1 0,12)0,12 1,96 0,12 1,96

500 500p

0,12 1,96 0,015 0,12 1,96 0,015p

15,009,0 p

%15%9 p

, 500.

15,050009,0500 Np

7545 Np

500 45 75.

6.7. ( ) 7.000 , : 4.000 ; 1.000 2.000 . :

) ;

) 7.000 , ( 0,95) 70 .

:

6.7.

iN

in

iN

iS

4.000 40 500 3 1.000 10 200 4 2.000 20 400 2

7.000 70 1100 9

Page 35: N H QSS Q 3ww qq 3ww i [ w q q 1 q 1 wq q q w 3www q q 3w q w ; q s qs st t

113

:

57,42870

000.30

70

4002020010500403

1

3

1

ii

iii

n

XnX

n

SnS i

ii

3

1

2

93,270

600

70

220410340 222

µ

( n :

2 2

S SX z X z

n n

n

SX

n

SX 96,196,1

70

93,296,157,428

70

93,296,157,428

26,42988,427

95% , , 427,88

429,26 .

2 2

S SN X z N N X z

n n

1,96 1,96S

nN

S

nN X N X

26,429000.788,427000.7 Np

802.004.388,427160.995.2 Np

( 95%) 7.000 2.995.160 3.004.820

.

6.8. 1.000 , 100

km.

Page 36: N H QSS Q 3ww qq 3ww i [ w q q 1 q 1 wq q q w 3www q q 3w q w ; q s qs st t

114

6.8. -

(cm3)

1.000 500 7,1 0,2 1.000 - 1.300 300 8,4 0,4

1.300 200 10,1 0,6

0,95 : ) 100 km; ) ,

1.930.000.

: 6.9.

- cm3) in X S ii Xn

2iS

2iiSn

1.000 500 7,1 0,2 3.550 0,04 20 1.000 - 1.300 300 8,4 0,4 2.520 0,16 4&

1.300 200 10,1 0,6 2.020 0,36 72 1.000 8.090 140

09,8000.1

090.8

i

ii

n

XnX

n

SnS i

ii

3

1

2

37,01000

140

µ ( n :

2 2

S SX z X z

n n

n

SX

n

SX 96,196,1

1000

37,096,109,8

1000

37,096,109,8

11,807,8 0,95

100 km 8,07 8,11 .

n

SXNN

n

SXN 96,196,1

1.930.000 8,07 1.930.000 8,11N

300.652.15100.575.15 N

0,95 1.930.000 15.575.100 15.652.300

.