Материал: chap2

ɂɡ ɮɨɪɦɭɥɵ (II, 6, 1) ɜɢɞɧɨ, ɱɬɨ ɤɨɷɮɮɢɰɢɟɧɬ IJ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɪɚɡɧɨɫɬɶ ɞɨɥɢ ɩɚɪ ɨɛɴɟɤɬɨɜ, ɭ ɤɨɬɨɪɵɯ ɫɨɜɩɚɞɚɟɬ ɩɨɪɹɞɨɤ ɩɨ ɨɛɨɢɦ ɩɪɢɡɧɚɤɚɦ (ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɱɢɫɥɭ ɜɫɟɯ ɩɚɪ)

ɡɧɚɱɟɧɢɟ ɤɨɷɮɮɢɰɢɟɧɬɚ 0,60 ɨɡɧɚɱɚɟɬ, ɱɬɨ ɭ 80% ɩɚɪ ɩɨɪɹɞɨɤ ɨɛɴɟɤɬɨɜ ɫɨɜɩɚɞɚɟɬ, ɚ ɭ 20% ɧɟ ɫɨɜɩɚɞɚɟɬ (80% + 20% = 100%; 0,80 – 0,20 = 0,60). Ɍ.ɟ. IJ ɦɨɠɧɨ ɬɪɚɤɬɨɜɚɬɶ ɤɚɤ ɪɚɡɧɨɫɬɶ ɜɟɪɨɹɬɧɨɫɬɟɣ ɫɨɜɩɚɞɟɧɢɹ ɢ ɧɟ ɫɨɜɩɚɞɟɧɢɹ ɩɨɪɹɞɤɨɜ ɩɨ ɨɛɨɢɦ ɩɪɢɡɧɚɤɚɦ ɞɥɹ ɧɚɭɝɚɞ ɜɵɛɪɚɧɧɨɣ ɩɚɪɵɨɛɴɟɤɬɨɜ.

ȼɨɛɳɟɦ ɫɥɭɱɚɟ ɪɚɫɱɟɬIJ (ɬɨɱɧɟɟ Ɋ ɢɥɢ Q) ɞɚɠɟ ɞɥɹ N ɩɨɪɹɞɤɚ 10 ɨɤɚɡɵɜɚɟɬɫɹ ɝɪɨɦɨɡɞɤɢɦ. ɉɨɤɚɠɟɦ, ɤɚɤɭɩɪɨɫɬɢɬɶɜɵɱɢɫɥɟɧɢɹ.

Ɋɚɫɩɨɥɨɠɢɦ ɨɛɴɟɤɬɵ ɬɚɤ, ɱɬɨɛɵ ɢɯ ɪɚɧɝɢ ɩɨ X ɩɪɟɞɫɬɚɜɢɥɢ ɧɚɬɭɪɚɥɶɧɵɣ ɪɹɞ. Ɍɚɤ ɤɚɤ ɨɰɟɧɤɢ, ɩɪɢɩɢɫɵɜɚɟɦɵɟ ɤɚɠɞɨɣ ɩɚɪɟ ɷɬɨɝɨ ɪɹɞɚ, ɩɨɥɨɠɢɬɟɥɶɧɵɟ, ɡɧɚɱɟɧɢɹ «+1», ɜɯɨɞɹɳɢɟ ɜ Ɋ, ɛɭɞɭɬɩɨɪɨɠɞɚɬɶɫɹɬɨɥɶɤɨɬɟɦɢɩɚɪɚɦɢ, ɪɚɧɝɢɤɨɬɨɪɵɯɩɨ Y ɨɛɪɚɡɭɸɬɩɪɹɦɨɣɩɨɪɹɞɨɤ. ɂɯ ɥɟɝɤɨ

ȼɪɹɞɭ Y ɫɩɪɚɜɚ ɨɬ 1 ɪɚɫɩɨɥɨɠɟɧɨ 9 ɪɚɧɝɨɜ, ɩɪɟɜɨɫɯɨɞɹɳɢɯ 1, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, 1 ɩɨɪɨɞɢɬ ɜ

Ɋɫɥɚɝɚɟɦɨɟ 9. ɋɩɪɚɜɚɨɬ

74

Ⱦɥɹɜɵɱɢɫɥɟɧɢɹ IJ ɪɚɧɠɢɪɭɟɦ ɝɪɭɩɩɵ ɩɨ KT ɜ ɩɨɪɹɞɤɟɧɚɬɭɪɚɥɶɧɨɝɨɪɹɞɚ:

Ɂɚɦɟɬɢɦ, ɱɬɨ ɞɥɹ ɧɚɯɨɠɞɟɧɢɹ IJ ɞɨɫɬɚɬɨɱɧɨ ɛɵɥɨ ɧɚɣɬɢ ɥɢɲɶ Ɋ ɢ ɩɪɢɦɟɧɢɬɶ ɮɨɪɦɭɥɭ (II,6,3). Ɂɞɟɫɶ ɜɨɡɧɢɤɚɟɬ ɟɫɬɟɫɬɜɟɧɧɵɣ ɜɨɩɪɨɫ: ɤɚɤ ɨɰɟɧɢɬɶ ɷɬɨ ɡɧɚɱɟɧɢɟ IJ. əɫɧɨ, ɱɬɨ ɫɜɹɡɶ ɨɬɪɢɰɚɬɟɥɶɧɚɹ (ɨɛɪɚɬɧɚɹ), ɧɨ ɧɚɫɤɨɥɶɤɨ ɡɧɚɱɢɦɚ ɨɧɚ?

ɉɪɨɜɟɪɤɚ ɫɭɳɟɫɬɜɟɧɧɨɫɬɢ. Ɂɚɞɚɞɢɦɫɹ ɜɨɩɪɨɫɨɦ: ɤɚɤɨɜɚ ɫɭɳɟɫɬɜɟɧɧɨɫɬɶ ɩɨɥɭɱɟɧɧɨɝɨ ɧɚ ɨɩɵɬɟɡɧɚɱɟɧɢɹɤɨɷɮɮɢɰɢɟɧɬɚɤɨɪɪɟɥɹɰɢɢɪɚɧɝɨɜIJ ɢɥɢ, ɞɪɭɝɢɦɢɫɥɨɜɚɦɢ, ɩɪɢɞɚɧɧɨɦ IJ ɫ ɤɚɤɨɣ ɫɬɟɩɟɧɶɸ ɧɚɞɟɠɧɨɫɬɢ ɦɨɠɧɨ ɭɬɜɟɪɠɞɚɬɶ, ɱɬɨ ɫɜɹɡɶ ɦɟɠɞɭ ɞɜɭɦɹ ɩɪɢɡɧɚɤɚɦɢ ɞɟɣɫɬɜɢɬɟɥɶɧɨ ɫɭɳɟɫɬɜɭɟɬ?

ɉɪɟɞɩɨɥɨɠɢɦ, ɱɬɨ ɫɜɹɡɢ ɧɟɬ. ɗɬɨ ɨɡɧɚɱɚɟɬ, ɱɬɨ, ɧɚɩɪɢɦɟɪ, ɩɪɢ ɮɢɤɫɢɪɨɜɚɧɧɨɣ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ Y-ɪɚɧɝɨɜ ɨɛɴɟɤɬɚ ɩɨɹɜɥɟɧɢɟ ɥɸɛɨɣ ɏ-ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ ɪɚɜɧɨɜɨɡɦɨɠɧɨ. Ɉɛɴɟɤɬɵɜɫɟɝɞɚɦɨɠɧɨɩɟɪɟɫɬɚɜɢɬɶɬɚɤ, ɱɬɨɛɵ Y-ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɨɤɚɡɚɥɚɫɶ ɭɩɨɪɹɞɨɱɟɧɧɨɣ ɜ ɜɢɞɟ ɧɚɬɭɪɚɥɶɧɨɝɨ ɪɹɞɚ: 1, 2, ..., N. ȼɫɟɝɨ ɪɚɡɥɢɱɧɵɯ ɏ-ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɟɣ (N!). Ʉɚɠɞɚɹ,

ɬɚɤɢɦ ɨɛɪɚɡɨɦ, ɢɦɟɟɬ ɜɟɪɨɹɬɧɨɫɬɶ ɩɨɹɜɥɟɧɢɹ 1 . Ʉɚɠɞɨɣ ɏ-ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ ɫɨɨɬɜɟɬɫɬɜɭɟɬ

N!

ɧɟɤɨɬɨɪɨɟ S = Ɋ – Q (ɢ IJ, ɡɚɤɥɸɱɟɧɧɨɟ ɦɟɠɞɭ –1 ɢ +1). ɋɪɟɞɢ ɷɬɢɯ IJ ɧɟ ɜɫɟ ɛɭɞɭɬ ɪɚɡɥɢɱɧɵɦɢ (ɫɦ. ɧɢɠɟ). ɋɨɜɨɤɭɩɧɨɫɬɶ IJ ɜɦɟɫɬɟ ɫ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɦɢ ɱɚɫɬɨɬɚɦɢ ɢɯ ɩɨɹɜɥɟɧɢɹ ɨɛɪɚɡɭɟɬ ɧɟɤɨɬɨɪɨɟ ɪɚɫɩɪɟɞɟɥɟɧɢɟ. ȼ ɞɚɥɶɧɟɣɲɟɦ, ɨɞɧɚɤɨ, ɧɚɦ ɛɭɞɟɬ ɭɞɨɛɧɨ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɱɚɫɬɨɬ S (ɪɚɡɭɦɟɟɬɫɹ, ɢɞɟɧɬɢɱɧɨɟ ɪɚɫɩɪɟɞɟɥɟɧɢɸ IJ, ɬ.ɤ. IJ ɨɬɥɢɱɚɟɬɫɹ ɨɬ S ɥɢɲɶ ɩɨɫɬɨɹɧɧɵɦɦɧɨɠɢɬɟɥɟɦ CN2 , ɧɟɦɟɧɹɸɳɢɦɪɚɫɩɪɟɞɟɥɟɧɢɟ).

ȿɫɥɢ, ɧɚɩɪɢɦɟɪ, N = 4, ɬɨ ɩɪɢ ɡɚɞɚɧɧɨɣ Y-ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ 1,2,3,4 ɜɨɡɦɨɠɧɵ 4! = 1 • 2 • 3 • 4 = 24 ɏ-ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ (ɩɨɥɟɡɧɨɪɚɫɩɢɫɚɬɶɢɯ).

75

ɉɨɤɚɠɟɦ, ɱɬɨ ɧɟ ɜɫɟ ɨɧɢ ɪɚɡɥɢɱɧɵ (ɜ ɫɦɵɫɥɟ S) ɢ ɧɚɣɞɟɦɪɚɫɩɪɟɞɟɥɟɧɢɟɱɚɫɬɨɬ:

SP Q 2P 1 (N 1)N

2

ɋɪɟɞɢ 24-ɯ ɩɟɪɟɫɬɚɧɨɜɨɤ ɧɚɣɞɟɬɫɹ ɥɢɲɶ ɨɞɧɚ (4, 3, 2; 1) ɫ Ɋ = 0 (ɢ S = – 6

ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ), ɬɪɢ (4, 3, 1, 2; 4, 2, 3, 1; 3,4, 2, 1)ɫ Ɋ = 1(S= – 4), ɩɹɬɶ (4, 2, 1,3; 4, 1,3,2; 3, 4, 1, 2; 3,2, 4, 1; 2, 4, 3, 1) ɫ Ɋ = 2 (S = – 2), ɲɟɫɬɶ ɫ P = 3 (S = 0), ɩɹɬɶ ɫ Ɋ = 4 (S = 2), ɬɪɢ ɫ Ɋ = 5 (S = 4), ɨɞɧɚ ɫ Ɋ = 6 (S = 6).

Ɍɚɤɢɦɨɛɪɚɡɨɦ, ɦɵɢɦɟɟɦ7 ɪɚɡɥɢɱɧɵɯS (ɢIJ) ɫɫɢɦɦɟɬɪɢɱɧɵɦɪɚɫɩɪɟɞɟɥɟɧɢɟɦɱɚɫɬɨɬ:

[110]

Ⱥɧɚɥɨɝɢɱɧɨ ɦɨɠɧɨ ɩɨɥɭɱɢɬɶ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɢ ɞɥɹ ɞɪɭɝɢɯ N. ɇɚɩɪɢɦɟɪ, ɞɥɹ N = 8 ɱɢɫɥɨ ɪɚɡɥɢɱɧɵɯ S ɪɚɜɧɨ 15: Ɉ ± 2 ± 4 ± ... ±28. ɉɪɢɜɟɞɟɦ ɱɚɫɬɨɬɵ ɞɥɹ S 0 (ɞɥɹ S < Ɉ ɱɚɫɬɨɬɵ ɬɟ ɠɟ, ɱɬɨ ɞɥɹ S > 0 ɩɪɢ ɨɞɢɧɚɤɨɜɵɯ ɦɨɞɭɥɹɯ):

Ɇɚɤɫɢɦɚɥɶɧɚɹ ɱɚɫɬɨɬɚ ɫɨɨɬɜɟɬɫɬɜɭɟɬ S = 0, ɫ ɪɨɫɬɨɦ S ɱɚɫɬɨɬɵ ɦɨɧɨɬɨɧɧɨ

ɭɦɟɧɶɲɚɸɬɫɹ, ɞɨɫɬɢɝɚɹ 1 ɩɪɢ S ɬɚɯ = CN2 ; (|IJ| = 1). ȿɫɥɢ N ɧɟɱɟɬɧɨ, ɬɨ, ɨɤɚɡɵɜɚɟɬɫɹ, ɢɦɟɸɬɫɹ 2

ɦɚɤɫɢɦɭɦɚ, ɩɪɢɯɨɞɹɳɢɟɫɹ ɧɚ S = ± 1 ɫɭɜɟɥɢɱɟɧɢɟɦ |S| ɱɚɫɬɨɬɵɬɚɤɠɟɭɦɟɧɶɲɚɸɬɫɹ. ɉɭɫɬɶ N = 3, ɢɦɟɟɦ 6 ɩɟɪɟɫɬɚɧɨɜɨɤ:

1)3 2 1 Ɋ = 0 S = – 3 nS = 1

2)3 1 2 Ɋ = 1 S = – 1 nS = 2

3)2 3 1 Ɋ = 1

4)2 1 3 Ɋ = 2 S = +1 nS = 2

5)1 3 2 Ɋ = 2

6)1 2 3 Ɋ = 3 S = 3 nS = 1

ɍɩɪɚɠɧɟɧɢɟ 51. Ⱦɥɹ ɫɥɭɱɚɹ N = 5 ɭɛɟɞɢɬɶɫɹ ɜ ɫɩɪɚɜɟɞɥɢɜɨɫɬɢ ɬɨɝɨ, ɱɬɨ ɢɦɟɸɬɫɹ 2 ɦɚɤɫɢɦɭɦɚ (S = ±1), ɚ ɫ ɭɜɟɥɢɱɟɧɢɟɦ |S| ɱɚɫɬɨɬɚ ɭɦɟɧɶɲɚɟɬɫɹ, ɞɨɫɬɢɝɚɹ 1 ɩɪɢ S CN2

ɍɠɟ ɢɡ ɪɚɫɫɦɨɬɪɟɧɢɹ ɫɥɭɱɚɟɜ N = 4, 5, 8 ɹɫɧɨ, ɱɬɨ ɨɫɧɨɜɧɚɹ ɱɚɫɬɶ ɡɧɚɱɟɧɢɣ S (ɢ IJ) ɤɨɧɰɟɧɬɪɢɪɭɟɬɫɹ ɜɛɥɢɡɢ ɧɭɥɹ. ȿɫɥɢ ɧɟɤɨɬɨɪɨɟ ɡɧɚɱɟɧɢɟ S ɞɨɫɬɚɬɨɱɧɨ ɞɚɥɟɤɨ ɨɬ ɫɪɟɞɧɟɝɨ (ɧɭɥɟɜɨɝɨ), ɬɨ ɢ ɜɟɪɨɹɬɧɨɫɬɶ ɟɝɨ ɩɨɹɜɥɟɧɢɹ ɨɱɟɧɶ ɦɚɥɚ.

76

ɉɪɢɦɟɪ 21. ɉɭɫɬɶ ɩɪɢ N = 8 ɡɧɚɱɟɧɢɟ S = 18 ɢɦɟɟɬ ɱɚɫɬɨɬɭ nS = 343. ȼɵɱɢɫɥɢɦ ɜɟɪɨɹɬɧɨɫɬɶ ɬɨɝɨ, ɱɬɨ ɡɧɚɱɟɧɢɟ S = 18 ɩɨɹɜɢɬɫɹ ɫɥɭɱɚɣɧɨ, ɬ.ɟ. ɫ ɤɚɤɨɣ ɜɟɪɨɹɬɧɨɫɬɶɸ ɦɵ ɨɬɜɟɪɝɚɟɦɝɢɩɨɬɟɡɭ ɧɟɡɚɜɢɫɢɦɨɫɬɢ (ɢɭɬɜɟɪɠɞɚɟɦɧɚɥɢɱɢɟɫɜɹɡɢ).

ɋɨɛɵɬɢɸ «S ɧɟ ɦɟɧɶɲɟ 18» ɛɥɚɝɨɩɪɢɹɬɫɬɜɭɸɬ 343 + 174 + 76 + 27 + 7 + 1 = 628 ɪɚɜɧɨɜɨɡɦɨɠɧɵɯ ɷɥɟɦɟɧɬɚɪɧɵɯ ɫɨɛɵɬɢɣ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɜɟɪɨɹɬɧɨɫɬɶ ɪɚɜɧɚ 628/8! § 0.016, ɨɧɚ ɧɟɜɟɥɢɤɚ.

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Ɉɛɵɱɧɨ ɢɫɩɨɥɶɡɭɸɬɫɥɟɞɭɸɳɢɣɤɪɢɬɟɪɢɣɫɭɳɟɫɬɜɟɧɧɨɫɬɢ: ɟɫɥɢ ɧɚɛɥɸɞɚɟɦɨɟ ɡɧɚɱɟɧɢɟ S ɬɚɤɨɜɨ, ɱɬɨ ɜɟɪɨɹɬɧɨɫɬɶ ɩɨɹɜɥɟɧɢɹ ɷɬɨɝɨ ɢɥɢ ɛɨɥɶɲɟɝɨ ɩɨ ɚɛɫɨɥɸɬɧɨɣ ɜɟɥɢɱɢɧɟ ɡɧɚɱɟɧɢɹ ɞɨɫɬɚɬɨɱɧɨ ɦɚɥɚ (ɜ ɫɨɰɢɚɥɶɧɵɯ ɢɫɫɥɟɞɨɜɚɧɢɹɯ, ɤɚɤ ɭɠɟ ɨɬɦɟɱɚɥɨɫɶ, ɦɚɥɨɣ ɫɱɢɬɚɸɬ ɜɟɪɨɹɬɧɨɫɬɶ 0,05, ɚ ɨɱɟɧɶ ɦɚɥɨɣ 0,01), ɬɨ ɝɢɩɨɬɟɡɚ ɧɟɡɚɜɢɫɢɦɨɫɬɢ ɨɬɜɟɪɝɚɟɬɫɹ. ɗɬɨ ɡɧɚɱɢɬ, ɱɬɨ S – ɜ «ɯɜɨɫɬɚɯ» ɪɚɫɩɪɟɞɟɥɟɧɢɹ. Ʉɨɝɞɚ ɝɨɜɨɪɹɬ, ɱɬɨ .«ɧɚɛɥɸɞɟɧɧɨɟ S ɥɟɠɢɬ ɜɧɟ 5- ɩɪɨɰɟɧɬɧɨɝɨ ɩɪɟɞɟɥɚ ɫɭɳɟɫɬɜɟɧɧɨɫɬɢ», ɬɨ ɢɦɟɸɬ ɜ ɜɢɞɭ, ɱɬɨ ɜɟɪɨɹɬɧɨɫɬɶ ɩɨɹɜɥɟɧɢɹ ɪɚɜɧɨɝɨ ɢɥɢ ɛɨɥɶɲɟɝɨ ɩɨ ɚɛɫɨɥɸɬɧɨɣ ɜɟɥɢɱɢɧɟ ɡɧɚɱɟɧɢɹ ɦɟɧɶɲɟ, ɱɟɦ 0,05. (Ʉ ɷɬɨɦɭ ɜɨɩɪɨɫɭ ɦɵ ɜɟɪɧɟɦɫɹɜɝɥɚɜɟ V).

ȼ ɧɚɲɟɦ ɩɪɢɦɟɪɟ (N = 8, S = 18, IJ = 0,64) ɜɟɪɨɹɬɧɨɫɬɶ ɬɨɝɨ, ɱɬɨ |S| 18, ɪɚɜɧɚ 2·0,016, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɫ ɧɚɞɟɠɧɨɫɬɶɸ, ɧɟ ɦɟɧɶɲɟɣ 0,968, ɦɨɠɧɨ ɫɱɢɬɚɬɶ, ɱɬɨ ɦɟɠɞɭ X ɢ Y ɟɫɬɶ ɩɨɥɨɠɢɬɟɥɶɧɚɹ ɫɜɹɡɶ.

Ⱦɨɩɭɫɬɢɦ, ɱɬɨ ɞɥɹ N = 10 IJ = – 0,16. əɜɥɹɟɬɫɹ ɥɢ ɷɬɨ ɡɧɚɱɟɧɢɟ IJ ɫɭɳɟɫɬɜɟɧɧɵɦ? ȼ ɞɚɧɧɨɦ ɫɥɭɱɚɟ S = – 7. ȼɟɪɨɹɬɧɨɫɬɶ ɬɨɝɨ, ɱɬɨ S – 7, ɤɚɤ ɜɢɞɧɨ ɢɡ ɬɚɛɥɢɰɵ19 Ƚ ɉɪɢɥɨɠɟɧɢɹ 3, ɪɚɜɧɚ 0,30 > 0,0520. Ɇɵ ɧɟ ɦɨɠɟɦ ɨɬɜɟɪɝɧɭɬɶ ɝɢɩɨɬɟɡɭɧɟɡɚɜɢɫɢɦɨɫɬɢɢɫɱɢɬɚɬɶɨɬɪɢɰɚɬɟɥɶɧɭɸ ɫɜɹɡɶɭɫɬɚɧɨɜɥɟɧɧɨɣ.

Ⱦɥɹ N = 10 ɢ IJ = 0,51 (S = + 23) ɜɟɪɨɹɬɧɨɫɬɶ ɬɨɝɨ, ɱɬɨ S > 23, ɪɚɜɧɚ (ɫɦ. ɬɚɛɥɢɰɭ Ƚ) 0,023, ɚ ɜɟɪɨɹɬɧɨɫɬɶ ɬɨɝɨ, ɱɬɨ |S| > 23, ɪɚɜɧɚ 0,046. Ɉɛɟ ɜɟɪɨɹɬɧɨɫɬɢ ɦɟɧɶɲɟ 0,05. Ƚɢɩɨɬɟɡɭ ɨ ɧɟɡɚɜɢɫɢɦɨɫɬɢɦɨɠɧɨɨɬɜɟɪɝɧɭɬɶɫɛɨɥɶɲɨɣɧɚɞɟɠɧɨɫɬɶɸ (ɧɟ ɦɟɧɶɲɟɣ, ɱɟɦ 0,95).

ɍɩɪɚɠɧɟɧɢɟ 52. Ⱦɥɹ N = 9 ɢ IJ = – 0,72 ɪɚɫɫɦɨɬɪɟɬɶ ɜɨɩɪɨɫɨɫɭɳɟɫɬɜɟɧɧɨɫɬɢIJ. Ɉɬɜɟɬ: ɫ ɧɚɞɟɠɧɨɫɬɶɸ, ɛɨɥɶɲɟɣ0,99 ɝɢɩɨɬɟɡɚ ɧɟɡɚɜɢɫɢɦɨɫɬɢ ɨɬɜɟɪɝɚɟɬɫɹ.

ɍɩɨɦɢɧɚɜɲɚɹɫɹ ɬɚɛɥɢɰɚ ɫɭɳɟɫɬɜɟɧɧɨɫɬɢ ɫɨɫɬɚɜɥɟɧɚ ɥɢɲɶ ɞɥɹ N 10. Ɉɤɚɡɵɜɚɟɬɫɹ, ɱɬɨ ɞɥɹ N > 10 ɧɟɬ ɧɭɠɞɵ ɫɨɡɞɚɜɚɬɶ ɫɩɟɰɢɚɥɶɧɵɟ ɬɚɛɥɢɰɵ. Ɇɨɠɧɨ ɩɨɤɚɡɚɬɶ, ɱɬɨ ɫ ɪɨɫɬɨɦ N ɨɱɟɪɬɚɧɢɹ ɩɨɥɢɝɨɧɚ ɱɚɫɬɨɬ ɩɪɢɛɥɢɠɚɸɬɫɹ ɤ ɯɨɪɨɲɨ ɢɡɭɱɟɧɧɨɣ ɜ ɫɬɚɬɢɫɬɢɤɟ ɤɪɢɜɨɣ ɧɨɪɦɚɥɶɧɨɝɨɪɚɫɩɪɟɞɟɥɟɧɢɹ (ɫɦ. (1,3,4)) ɞɥɹ

ı2 = (1/18)N(N—1 )(2N+5)

ɉɨɷɬɨɦɭ ɦɨɠɧɨ ɢɫɩɨɥɶɡɨɜɚɬɶ ɬɚɤ ɧɚɡɵɜɚɟɦɭɸ ɬɚɛɥɢɰɭ ɩɥɨɳɚɞɟɣ ɩɨɞ ɧɨɪɦɚɥɶɧɨɣ ɤɪɢɜɨɣ21 (ɫɦ. § 8 ɝɥɚɜɵ V, ɚɬɚɤɠɟɬɚɛɥɢɰɭ Ⱥɉɪɢɥɨɠɟɧɢɹ 3).

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ɉɨɡɧɚɤɨɦɢɦɫɹ ɫ ɟɳɟ ɨɞɧɨɣ ɮɨɪɦɨɣ ɡɚɩɢɫɢ ɤɨɷɮɮɢɰɢɟɧɬɚ Ʉɟɧɞɷɥɚ. ɉɭɫɬɶ ɤɚɠɞɵɣ ɢɡ N ɢɡɭɱɚɟɦɵɯ ɨɛɴɟɤɬɨɜ ɦɨɠɟɬ ɛɵɬɶ ɨɯɚɪɚɤɬɟɪɢɡɨɜɚɧ ɩɨ ɫɬɟɩɟɧɢ ɢɧɬɟɧɫɢɜɧɨɫɬɢ ɤɚɤ ɩɪɢɡɧɚɤɚ X, ɬɚɤɢɩɪɢɡɧɚɤɚ Y, ɬ.ɟ. ɦɵɡɧɚɟɦɭɤɚɠɞɨɝɨɨɛɴɟɤɬɚɪɚɧɝ ɩɨ X ɢ ɪɚɧɝ ɩɨ Y.

ȼɜɟɞɟɦɜɟɥɢɱɢɧɭ

19ɗɬɚ ɬɚɛɥɢɰɚ ɩɨɫɬɪɨɟɧɚ ɧɚ ɨɫɧɨɜɟ ɪɚɫɱɟɬɨɜ, ɚɧɚɥɨɝɢɱɧɵɯ ɬɟɦ, ɤɨɬɨɪɵɟ ɜɵɩɨɥɧɟɧɵ ɜ ɩɪɟɞɵɞɭɳɟɦ ɩɪɢɦɟɪɟ (ɞɥɹ ɪɚɡɧɵɯ N ɢ S).

20Ʌɟɝɤɨ ɩɨɧɹɬɶ, ɱɬɨ ɜɟɪɨɹɬɧɨɫɬɶ |S| 7 ɪɚɜɧɚ 2·0,300 = 0,600.

21ɉɪɢ ɨɬɫɭɬɫɬɜɢɢ ɨɛɴɟɞɢɧɟɧɧɵɯ ɪɚɧɝɨɜ ɫɭɳɟɫɬɜɟɧɧɨɫɬɶ IJ ɨɩɪɟɞɟɥɹɟɬɫɹ ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɨ ɩɨ ɡɧɚɱɟɧɢɸ IJ ɩɨ ɬɚɛɥɢɰɟ Ⱦ ɉɪɢɥɨɠɟɧɢɹ 3.

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ɝɞɟ Rr(x) – ɪɚɧɝ ɩɨ X r-ɨɝɨ ɨɛɴɟɤɬɚ, ɚ Rs(x) – s-ɨɝɨ. Ⱥɧɚɥɨɝɢɱɧɨ ɜɜɨɞɢɬɫɹ ɜɟɥɢɱɢɧɚ brs ɞɥɹ

ɩɪɢɡɧɚɤɚ Y. ɋɬɚɧɟɦ ɫɨɩɨɫɬɚɜɥɹɬɶ ɩɚɪɵ ɨɛɴɟɤɬɨɜ ɢ ɜɵɱɢɫɥɹɬɶ ɩɪɨɢɡɜɟɞɟɧɢɟ ars · brs. ȿɫɥɢ ɛɨɥɶɲɟɦɭ ɪɚɧɝɭ ɩɨ X ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɛɨɥɶɲɢɣ ɩɨ Y (ɢɥɢ ɦɟɧɶɲɟɦɭ – ɦɟɧɶɲɢɣ), ɬɨ ɷɬɨ ɩɪɨɢɡɜɟɞɟɧɢɟ ɛɭɞɟɬ ɪɚɜɧɨ 1, ɬɚɤ ɤɚɤ ɩɪɢ ɷɬɨɦ ars = brs = 1 (ɥɢɛɨ ars = brs = –1). ȼɩɪɨɬɢɜɧɨɦ ɫɥɭɱɚɟ (ɛɨɥɶɲɟɦɭɪɚɧɝɭɩɨX ɫɨɨɬɜɟɬɫɬɜɭɟɬɦɟɧɶɲɢɣɩɨY ɢɥɢɧɚɨɛɨɪɨɬ) ɩɪɨɢɡɜɟɞɟɧɢɟars brs = –1.

Ɂɚɜɟɪɲɢɜ ɜɫɟɜɨɡɦɨɠɧɵɟ ɫɪɚɜɧɟɧɢɹ ɩɚɪ ɷɥɟɦɟɧɬɨɜ, ɫɨɫɬɚɜɢɦ ɫɭɦɦɭ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɩɪɨɢɡɜɟɞɟɧɢɣ S ¦¦ars ubrs . ɑɬɨɛɵ ɨɞɧɭ ɢ ɬɭ ɠɟ ɩɚɪɭ ɨɛɴɟɤɬɨɜ ɧɟ ɫɨɩɨɫɬɚɜɥɹɬɶ ɞɜɚɠɞɵ,

rs

ɦɵɛɭɞɟɦɨɫɭɳɟɫɬɜɥɹɬɶɫɭɦɦɢɪɨɜɚɧɢɟɩɨr, ɫɤɚɠɟɦ, ɨɬ 1 ɞɨ N, ɧɨ ɬɨɝɞɚ ɩɨ s ɨɬ r + 1 ɞɨ N, ɬ.ɟ. ɩɨ s > r.

ɇɟɬɪɭɞɧɨ ɜɢɞɟɬɶ, ɱɬɨ S > 0, ɟɫɥɢ ɫɜɹɡɶ ɩɪɹɦɚɹ ɢ S < Ɉ, ɟɫɥɢ ɨɛɪɚɬɧɚɹ. S ɛɥɢɡɤɨ ɤ 0, ɟɫɥɢ ɫɜɹɡɢɧɟɬ. ɋɤɨɧɫɬɪɭɢɪɭɟɦɜɟɥɢɱɢɧɭ

NN

ɇɚɣɞɟɦ ɦɚɤɫɢɦɚɥɶɧɨɟ ɡɧɚɱɟɧɢɟ ɱɢɫɥɢɬɟɥɹ. Ɉɧɨ ɞɨɫɬɢɝɚɟɬɫɹ ɬɨɝɞɚ, ɤɨɝɞɚ ɜɫɟ ars · brs.= 1.

ɉɪɢ ɷɬɨɦ IJɬɚɯ = +1 ( ars2 =brs2 =1).

ȺɧɚɥɨɝɢɱɧɨIJmin = –1.

ȼɵɱɢɫɥɢɦ ¦¦ars2 . ɋɨɩɨɫɬɚɜɥɟɧɢɟ ɤɚɠɞɨɝɨ ɢɡ N ɷɥɟɦɟɧɬɨɜ c ɞɪɭɝɢɦɢ ɩɨɪɨɞɢɬ N – 1

NN

¦ ¦ars brs

2

ɑɢɫɥɢɬɟɥɶɦɨɠɧɨɧɟɫɤɨɥɶɤɨɭɩɪɨɫɬɢɬɶ. ɊɚɫɩɨɥɨɠɢɦɨɛɴɟɤɬɵɩɨɪɚɧɝɭX, ɬɨɝɞɚɜɫɟars = 1. ɉɪɢɷɬɨɦ

ɝɞɟɊ, ɨɱɟɜɢɞɧɨ, ɩɨɥɭɱɢɦ, ɫɭɦɦɢɪɭɹɱɢɫɥɚ, ɩɨɤɚɡɵɜɚɸɳɢɟ, ɫɤɨɥɶɤɨɪɚɧɝɨɜɨɛɪɚɡɨɜɚɜɲɟɝɨɫɹ ɪɚɧɝɨɜɨɝɨɪɹɞɚ Y ɩɪɟɜɵɲɚɸɬɪɚɧɝɢ, ɡɚɧɢɦɚɟɦɵɟɩɟɪɜɵɦ, ɜɬɨɪɵɦɢɬ.ɞ. N-ɧɵɦ, ɚ Q – ɚɧɚɥɨɝɢɱɧɚɹ ɫɭɦɦɚ, ɩɨɤɚɡɵɜɚɸɳɚɹ, ɫɤɨɥɶɤɨ ɪɚɧɝɨɜ ɪɹɞɚ Y ɧɢɠɟ ɪɚɧɝɨɜ, ɡɚɩɢɫɚɧɧɵɯ ɩɟɪɜɵɦ, ɜɬɨɪɵɦ ɢ ɬ.ɞ. N- ɧɵɦ. Ɍɚɤɢɦɨɛɪɚɡɨɦ, ɩɪɢɯɨɞɢɦɤɭɠɟɢɡɜɟɫɬɧɨɦɭɤɨɷɮɮɢɰɢɟɧɬɭ: ɫɦ. (II,6,1).

ɂɬɚɤ, ɦɵɩɨɡɧɚɤɨɦɢɥɢɫɶɫɧɨɜɨɣɮɨɪɦɨɣɡɚɩɢɫɢɤɨɷɮɮɢɰɢɟɧɬɚɄɟɧɞɷɥɚ(II,6,4).

Ⱦɚɥɟɟ, ɞɨɩɭɫɬɢɦ, ɱɬɨt ɪɚɧɝɨɜɩɨX ɫl+ 1 ɩɨl + t ɨɛɴɟɞɢɧɟɧɵ, ɬ.ɟ. ɪɚɧɝɨɜɵɣɪɹɞɢɦɟɟɬɜɢɞ:

1,2,...,l,l 1 t ,l 1 t ,...l 1 t ,l t 1,..., N 2 2 2

ɋɨɩɨɫɬɚɜɥɟɧɢɟ ɜɫɟɯ ɧɟ ɨɛɴɟɞɢɧɟɧɧɵɯ ɪɚɧɝɨɜ ɫ ɞɪɭɝɢɦɢ, ɨɛɴɟɞɢɧɟɧɧɵɦɢ ɢ ɧɟ ɨɛɴɟɞɢɧɟɧɧɵɦɢ, ɞɚɞɭɬ ɬɟ ɠɟ ɪɟɡɭɥɶɬɚɬɵ, ɱɬɨ ɢ ɪɚɧɟɟ: ɜ ɧɚɲɟɦ ɩɪɢɦɟɪɟ ɪɚɧɝ ɨɛɴɟɞɢɧɟɧɧɵɯ ɜɫɟ ɪɚɜɧɨ ɜɵɲɟ ɪɚɧɝɨɜ 1, 2, ..., l ɢ ɧɢɠɟ ɪɚɧɝɨɜ l+ t+ 1, ..., N. ɇɨ ɫɨɩɨɫɬɚɜɥɟɧɢɟ ɨɛɴɟɞɢɧɟɧɧɵɯ ɪɚɧɝɨɜ ɦɟɠɞɭ ɫɨɛɨɣ ɧɟ ɛɭɞɟɬ ɩɨɪɨɠɞɚɬɶ ɧɢ +1, ɧɢ –1, ɬɚɤ ɤɚɤ ɷɬɢ ɪɚɧɝɢ ɪɚɜɧɵ. Ⱦɨɨɩɪɟɞɟɥɢɦ ɬɟɩɟɪɶ ars ɢ

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