. 1.10. .
Y X y = ax + b ,
j(a, b) = M[(aX + b) Y)2] (1.41)
. , y = ax + b , ((ax + b) y)2 .
j(a, b) = M[y2] + a2M[x2] + b2 2×a×M[x y] 2×b×my + 2×a×b×mx
, :
; , (1.42)
mx my X Y; X Y,
r = ; Kxy = M[xy] mxmy = M[(x mx)(y mx)]. (1.43)
, Y X :
y = r (x mx) + my. (1.44)
, :
M[(a×Y + b X)2] (1.45)
X Y:
x = r (y my) + mx. (1.46)
, (1.46) (1.44) x. (1.44) (1.46) , , . , (1.41) Y aX + b Y, , (1.45) aY + b , .
xy (1.43) X Y, r = . , 1 £ r £ 1 Y.
r = 0, Y . , . r = 0 (1.44) (1.46) y = my x = mx, , ( ) X Y . , r= 1, (1.44) (1.46) ( ) . Y . . |r| , X Y. r > 0, . , . , X Y . , (r < 0) . , , , .
. 1.11 .
|
|
. 1.11. r > 0, r < 0, r= 1, r = 0.
(1.44) (1.46) mx, my, sx, sy r, , Fxy(x, y) fxy(x, y). , . , , , 1.4. [3]
= , (1.47)
:
. (1.48)
, n (1.47) .
, Y X:
y = r* (x ) + (1.49)
X Y:
x = r* (y ) + . (1.50)
, (1.49) (1.50) , , , (1.44) (1.45). , , , , r* n ¥ (1.49) (1.50) (1.44) (1.46).
(1.49) (1.50) , , :
,
(1.51)
.
(1.44) (1.46) (1.41) (1.45).
. (1.48), - r* r. , r=0. , (1.48). , , , ?
. n r* H0 = (r = 0). b
(1.52)
H0 ,
(1.53)
H0 ,
( 2). b r = n 2.
, X Y. . y = g(x, a1, a2, , ak), 1, 2, , :
.
2. ר.
1. Y, n = 100. .
2. Y.
3. Y.
4. X Y k = 8 10.
5. . X Y. 1.
|
|
6. , , , , , , , ,.
7. 95% 99% mx my.
8. Y, ( ).
9. Y , . !
10. b = 0,01 , c2. 1 , - .
11. Y . y = ax + b ( (xi, yi) ).
12. b = 0,01.
.
1. .
2. , .
3. . .
4. .
5. .
6. , .
7. , .
8. .
9. . .
10. .
11. . .
12. . .
13. c2 .
14. .
15. .
16. .
4. ר.
( Y) ( ). 100 . 4.1.
, 4.1, . ( 4.2) Y ( 4.3).
( 4.4).
( 4.5) , ( 4.5).
4.1. 100 .
N | X | y | N | x | y | N | x | y |
1. | 44.0 | 118.0 | 35. | 69.2 | 150.0 | 69. | 22.7 | 100.0 |
2. | 33.1 | 67.0 | 36. | 45.1 | 120.0 | 70. | 48.9 | 128.0 |
3. | 53.6 | 147.0 | 37. | 40.2 | 113.0 | 71. | 55.0 | 164.0 |
4. | 56.4 | 129.0 | 38. | 34.3 | 107.0 | 72. | 69.2 | 162.0 |
5. | 73.1 | 191.0 | 39. | 21.3 | 104.0 | 73. | 48.4 | 130.0 |
6. | 57.8 | 137.0 | 40. | 26.8 | 105.0 | 74. | 37.6 | 114.0 |
7. | 37.2 | 119.0 | 41. | 72.1 | 169.0 | 75. | 19.6 | 96.0 |
8. | 18.2 | 100.0 | 42. | 20.7 | 102.0 | 76. | 32.0 | 107.0 |
9. | 45.7 | 115.0 | 43. | 26.0 | 103.0 | 77. | 22.0 | 100.0 |
10. | 40.2 | 114.0 | 44. | 33.8 | 112.0 | 78. | 53.2 | 126.0 |
11. | 32.2 | 109.0 | 45. | 45.6 | 114.0 | 79. | 47.5 | 126.0 |
12. | 76.4 | 169.0 | 46. | 37.8 | 102.0 | 80. | 43.2 | 111.0 |
13. | 61.3 | 130.0 | 47. | 18.2 | 97.0 | 81. | 75.0 | 148.0 |
14. | 46.7 | 119.0 | 48. | 57.9 | 142.0 | 82. | 63.2 | 152.0 |
15. | 35.7 | 109.0 | 49. | 74.0 | 149.0 | 83. | 51.8 | 129.0 |
16. | 68.1 | 156.0 | 50. | 60.4 | 133.0 | 84. | 43.0 | 106.0 |
17. | 79.6 | 156.0 | 51. | 55.8 | 125.0 | 85. | 76.4 | 165.0 |
18. | 49.6 | 125.0 | 52. | 33.4 | 112.0 | 86. | 49.5 | 118.0 |
19. | 50.1 | 130.0 | 53. | 58.0 | 127.0 | 87. | 42.1 | 116.0 |
20. | 45.5 | 125.0 | 54. | 67.5 | 143.0 | 88. | 30.7 | 105.0 |
21. | 43.9 | 125.0 | 55. | 53.4 | 121.0 | 89. | 46.1 | 119.0 |
22. | 43.2 | 116.0 | 56. | 38.1 | 115.0 | 90. | 63.5 | 144.0 |
23. | 17.0 | 75.0 | 57. | 46.8 | 128.0 | 91. | 72.8 | 161.0 |
24. | 58.6 | 128.0 | 58. | 59.9 | 130.0 | 92. | 56.6 | 141.0 |
25. | 60.4 | 131.0 | 59. | 19.5 | 91.0 | 93. | 65.7 | 140.0 |
26. | 76.1 | 155.0 | 60. | 71.7 | 141.0 | 94. | 49.8 | 123.0 |
27. | 57.2 | 134.0 | 61. | 50.6 | 134.0 | 95. | 31.8 | 89.0 |
28. | 39.4 | 113.0 | 62. | 35.9 | 109.0 | 96. | 24.0 | 111.0 |
29. | 22.3 | 100.0 | 63. | 18.8 | 108.0 | 97. | 72.4 | 156.0 |
30. | 74.4 | 141.0 | 64. | 64.1 | 135.0 | 98. | 35.7 | 105.0 |
31. | 52.0 | 127.0 | 65. | 51.1 | 137.0 | 99. | 21.8 | 96.0 |
32. | 31.5 | 90.0 | 66. | 31.8 | 94.0 | 100. | 22.0 | 79.0 |
33. | 18.2 | 99.0 | 67. | 25.3 | 75.0 | * | * | * |
34. | 19.5 | 63.0 | 68. | 25.0 | 51.0 | * | * | * |
4.2. .
|
|
N | x | N | x | N | x | N | x | N | x |
1. | 44.0 | 21. | 69.2 | 41. | 25.0 | 61. | 33.1 | 81. | 45.1 |
2. | 33.1 | 22. | 53.6 | 42. | 40.2 | 62. | 48.9 | 82. | 56.4 |
3. | 53.6 | 23. | 55.0 | 43. | 73.1 | 63. | 21.3 | 83. | 69.2 |
4. | 57.8 | 24. | 26.8 | 44. | 48.4 | 64. | 37.2 | 84. | 72.1 |
5. | 37.6 | 25. | 18.2 | 45. | 20.7 | 65. | 19.6 | 85. | 45.7 |
6. | 26.0 | 26. | 32.0 | 46. | 40.2 | 66. | 33.8 | 86. | 22.0 |
7. | 32.2 | 27. | 45.6 | 47. | 53.2 | 67. | 76.4 | 87. | 37.8 |
8. | 47.5 | 28. | 61.3 | 48. | 18.2 | 68. | 43.2 | 88. | 46.7 |
9. | 57.9 | 29. | 75.0 | 49. | 35.7 | 69. | 74.0 | 89. | 63.2 |
10. | 68.1 | 30. | 60.4 | 50. | 51.8 | 70. | 79.6 | 90. | 55.8 |
11. | 43.0 | 31. | 49.6 | 51. | 33.4 | 71. | 76.4 | 91. | 50.1 |
12. | 58.0 | 32. | 49.5 | 52. | 45.5 | 72. | 67.5 | 92. | 42.1 |
13. | 43.9 | 33. | 53.4 | 53. | 30.7 | 73. | 43.2 | 93. | 38.1 |
14. | 46.1 | 34. | 17.0 | 54. | 46.8 | 74. | 63.5 | 94. | 58.6 |
15. | 59.9 | 35. | 72.8 | 55. | 60.4 | 75. | 19.5 | 95. | 56.6 |
16. | 76.1 | 36. | 71.7 | 56. | 65.7 | 76. | 57.2 | 96. | 50.6 |
17. | 49.8 | 37. | 39.4 | 57. | 35.9 | 77. | 31.8 | 97. | 22.3 |
18. | 18.8 | 38. | 24.0 | 58. | 74.4 | 78. | 64.1 | 98. | 72.4 |
19. | 52.0 | 39. | 51.1 | 59. | 35.7 | 79. | 31.5 | 99. | 31.8 |
20. | 21.8 | 40. | 18.2 | 60. | 25.3 | 80. | 22.0 | 100. | 19.5 |
4.3. Y.
N | Y | N | y | N | y | N | y | N | y |
1. | 118.0 | 21. | 123.0 | 41. | 121.0 | 61. | 109.0 | 81. | 96.0 |
2. | 67.0 | 22. | 100.0 | 42. | 119.0 | 62. | 133.0 | 82. | 114.0 |
3. | 147.0 | 23. | 135.0 | 43. | 128.0 | 63. | 106.0 | 83. | 114.0 |
4. | 191.0 | 24. | 105.0 | 44. | 91.0 | 64. | 130.0 | 84. | 126.0 |
5. | 105.0 | 25. | 99.0 | 45. | 140.0 | 65. | 143.0 | 85. | 119.0 |
6. | 114.0 | 26. | 150.0 | 46. | 113.0 | 66. | 105.0 | 86. | 149.0 |
7. | 115.0 | 27. | 100.0 | 47. | 108.0 | 67. | 75.0 | 87. | 129.0 |
8. | 112.0 | 28. | 129.0 | 48. | 156.0 | 68. | 130.0 | 88. | 125.0 |
9. | 126.0 | 29. | 104.0 | 49. | 90.0 | 69. | 141.0 | 89. | 127.0 |
10. | 130.0 | 30. | 130.0 | 50. | 75.0 | 70. | 134.0 | 90. | 116.0 |
11. | 142.0 | 31. | 100.0 | 51. | 51.0 | 71. | 109.0 | 91. | 116.0 |
12. | 152.0 | 32. | 103.0 | 52. | 147.0 | 72. | 111.0 | 92. | 128.0 |
13. | 156.0 | 33. | 100.0 | 53. | 107.0 | 73. | 127.0 | 93. | 161.0 |
14. | 112.0 | 34. | 169.0 | 54. | 162.0 | 74. | 94.0 | 94. | 155.0 |
15. | 118.0 | 35. | 97.0 | 55. | 119.0 | 75. | 79.0 | 95. | 134.0 |
16. | 125.0 | 36. | 148.0 | 56. | 102.0 | 76. | 67.0 | 96. | 89.0 |
17. | 115.0 | 37. | 156.0 | 57. | 107.0 | 77. | 113.0 | 97. | 141.0 |
18. | 144.0 | 38. | 125.0 | 58. | 109.0 | 78. | 164.0 | 98. | 137.0 |
19. | 131.0 | 39. | 165.0 | 59. | 102.0 | 79. | 137.0 | 99. | 96.0 |
20. | 141.0 | 40. | 125.0 | 60. | 111.0 | 80. | 169.0 | 100. | 63.0 |
|
|
4.4.
17.0 | 18.2 | 18.2 | 18.2 | 18.8 | 19.5 | 19.5 | 19.6 | 20.7 | 21.3 |
21.8 | 22.0 | 22.2 | 22.3 | 22.7 | 24.0 | 25.0 | 25.3 | 26.0 | 26.8 |
30.7 | 31.5 | 31.8 | 31.8 | 32.0 | 32.2 | 33.1 | 33.4 | 33.8 | 34.3 |
35.7 | 35.7 | 35.9 | 37.2 | 37.6 | 37.8 | 38.1 | 39.4 | 40.2 | 40.2 |
42.1 | 43.0 | 43.2 | 43.2 | 43.9 | 44.0 | 45.1 | 45.5 | 45.6 | 45.7 |
46.1 | 46.7 | 46.8 | 47.5 | 48.4 | 48.9 | 49.5 | 49.5 | 49.8 | 50.1 |
50.6 | 51.1 | 51.8 | 52.0 | 53.2 | 53.4 | 53.6 | 55.0 | 55.8 | 56.4 |
56.6 | 57.2 | 57.8 | 57.9 | 58.0 | 58.6 | 59.9 | 60.4 | 60.4 | 61.3 |
63.2 | 63.5 | 64.1 | 65.7 | 67.5 | 68.1 | 69.2 | 69.2 | 71.7 | 72.1 |
72.4 | 72.8 | 73.1 | 74.4 | 74.8 | 75.0 | 76.1 | 76.4 | 76.4 | 79.6 |
4.5. .
(a i ; a i+1) | m i | z i | Pi* | f i* | |
1. | 17 20 | 18.5 | 0.08 | 0.0267 | |
2. | 20 30 | 25.0 | 0.12 | 0.0120 | |
3. | 30 35 | 32.5 | 0.10 | 0.0200 | |
4. | 35 40 | 37.5 | 0.08 | 0.0160 | |
5. | 40 45 | 42.5 | 0.08 | 0.0160 | |
6. | 45 50 | 47.5 | 0.13 | 0.0260 | |
7. | 50 55 | 52.5 | 0.08 | 0.0160 | |
8. | 55 60 | 57.5 | 0.10 | 0.0200 | |
9. | 60 70 | 65.0 | 0.11 | 0.0110 | |
10. | 70 80 | 75.5 | 0.12 | 0.0120 |
( (1.12), (1.14), (1.15), (1.16) ):
= = 0.01 × 4650 = 46.50,
= = 0.01 × 4640.5 = 46.41,
= 260.18,
= 16.13,
= = 298.29,
= 17.27.
, b = 0.95 b = 0.99 ( (1.23)):
I0.95 = (46.41 1.96 × ; 46.41 + 1.96 × (43.03; 49.80),
I0.99 = (46.41 2.58 × ; 46,41 + 2.58 × (41.95; 50.87).
, ( 4.5), (. 4.1, 1 2, ).
.4.1. , .
, ,
1/(b a) £ £ b,
fx(x) =
0 < a b < x.
a b (1.25):
a = min xi = 17; b = max xi = 79.6.
i i
,
0.016 17 £ £ 79.6,
f(x) =
0 <17 >79.6,
0 <17,
Fx(x) = ( 17)/62.6 17 £ £ 79.6,
1 >79,6.
f(x) .4.1. ( 3).
c2. r = ks1=1021=7, s=2 : a b. b = 0.01 r = 7 = 18.5. c2 4.6.
4.6. c2 X.
N | mi | F(ai) | F(ai+1) | Pi | nPi | mi nPi | (mi -nPi)2 | ||
1. | 17-20 | 0.000 | 0.048 | 0.048 | 4.8 | 3.2 | 10.24 | 2.133 | |
2. | 20-30 | 0.048 | 0.208 | 0.160 | 16.0 | - 4.0 | 16.00 | 1.000 | |
3. | 30-35 | 0.208 | 0.288 | 0.080 | 8.0 | 2.0 | 4.00 | 0.500 | |
4. | 35-40 | 0.288 | 0.367 | 0.079 | 7.9 | 0.1 | 0.01 | 0.013 | |
5. | 40-45 | 0.367 | 0.447 | 0.080 | 8.0 | 0.0 | 0.00 | 0.000 | |
6. | 45-50 | 0.447 | 0.527 | 0.080 | 8.0 | 5.0 | 25.00 | 3.125 | |
7. | 50-55 | 0.527 | 0.607 | 0.080 | 8.0 | 0.0 | 0.00 | 0.000 | |
8. | 55-60 | 0.607 | 0.687 | 0.080 | 8.0 | 2.0 | 4.00 | 0.500 | |
9. | 60-70 | 0.687 | 0.847 | 0.160 | 16.0 | - 5.0 | 25.00 | 1.563 | |
70-80 | 0.847 | 1.000 | 0.153 | 15.3 | - 3.3 | 10.89 | 0.712 | ||
Σ | 1.000 | 9.546 |
|
|
, c2 = 9.546. 9.546 < 18.5, .
Y.
4.7. Y.
( 4.8) 9 - , ( 4.8).
4.8. Y.
N | mi | zi | Pi* | fi* | |
1. | 50-100 | 75.0 | 0.14 | 0.0028 | |
2. | 100-105 | 102.0 | 0.08 | 0.0160 | |
3. | 105-110 | 107.0 | 0.10 | 0.0200 | |
4. | 110-115 | 112.0 | 0.09 | 0.0180 | |
5. | 115-120 | 117.0 | 0.09 | 0.0180 | |
6. | 120-130 | 125.0 | 0.16 | 0.0160 | |
7. | 130-140 | 135.0 | 0.11 | 0.0110 | |
8. | 140-150 | 145.0 | 0.10 | 0.0100 | |
9. | 150-200 | 175.0 | 0.13 | 0.0026 |
Y ( (1.12), (1.14) (1.15), (1.16) ):
= = 0.01 × 12127 = 121.27,
= = 0.01 × 12225 = 122.25,
= = 592.44,
= 24.34,
= = 816.86,
= 28.58.
Y, b = 0.95 b = 0.99 ( (1.23)):
I0.95 = (122.25 1.96 × 2.858; 122.25 + 1.96 × 2.858) (116.65; 127.85),
I0.95 = (122.25 2.58 × 2.858; 122.25 + 2.58 × 2.058) (114.08; 129.62).
, ( 4.8), (. 4.2).
.4.2. , Y.
, , . , Y ( 1)
f(y) = ,
Fy(y) = 0.5 + F0((y m)/s)
F0() [5].
m# = 122.25, - = 816.86 ( = 28.58). m = m# s = s#. ,
f(y) = ,
Fy(y) = F0 ((y 122.25)/28.58) + 0.5
f(y) .4.2 ( 3).
c2, : r = 9 2 1 = 6. b = 0.01, =16.8. c2 4.9.
4.9. c2 Y.
N | mi | F(ai) | F(ai+1) | Pi | nPi | mi nPi | (mi nPi)2 | ||
1. | -¥-100 | 0.000 | 0.218 | 0.218 | 21.8 | -7.8 | 60.84 | 2.790 | |
2. | 100-105 | 0.218 | 0.274 | 0.056 | 5.6 | 2.4 | 5.76 | 1.030 | |
3. | 105-110 | 0.274 | 0.334 | 0.060 | 6.0 | 4.1 | 16.81 | 2.850 | |
4. | 110-115 | 0.334 | 0.401 | 0.067 | 6.7 | 2.3 | 4.84 | 0.710 | |
5. | 115-120 | 0.401 | 0.468 | 0.067 | 6.7 | 2.3 | 5.29 | 0.790 | |
6. | 120-130 | 0.468 | 0.607 | 0.149 | 14.9 | 1.1 | 1.21 | 0.081 | |
7. | 130-140 | 0.607 | 0.732 | 0.125 | 12.5 | -1.5 | 2.25 | 0.180 | |
8. | 140-150 | 0.732 | 0.834 | 0.101 | 10.1 | -0.1 | 0.01 | 0.001 | |
9. | 150-¥ | 0.834 | 1.000 | 0.167 | 16.7 | -3.7 | 13.69 | 0.820 | |
Σ | 1.000 | 9.252 |
, c2 = 9.252. 9.252< 16.8, .
X Y . Oxy 100 (.4.3). (1.47):
= ,
= 46.50, = 121.27, = 16.13 = 24.34. ( (1.48)):
r* = .
Y X (1.49):
y = .
Y X (.4.3):
.4.3. Y X.
, b = 0,01 r* = 0,82. (1.51)
( 2) b = 0.01 r = 100 2 = 98 . , , H0 = (r = 0) , r* = 0,82 . , , Y.
1.
.