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.

[111]

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

78

brs, ɬɚɤ, ɱɬɨɛɵ ars = brs = 0 ɩɪɢ ɫɨɜɩɚɞɟɧɢɢ ɪɚɧɝɨɜ (ɷɬɨ ɟɫɬɟɫɬɜɟɧɧɨ). ȼɫɟɝɨ ɫɨɩɨɫɬɚɜɥɟɧɢɣ

ȼ ɨɬɥɢɱɢɟ ɨɬ ȡ ɤɨɷɮɮɢɰɢɟɧɬ IJ ɛɟɡ ɩɨɩɪɚɜɤɢ ɦɟɧɶɲɟ, ɱɟɦ ɤɨɷɮɮɢɰɢɟɧɬ IJ ɫ ɩɨɩɪɚɜɤɨɣ, ɬ.ɟ. ɢɫɩɨɥɶɡɨɜɚɧɢɟ IJ ɛɟɡ ɩɨɩɪɚɜɨɤ ɩɨɜɵɲɚɟɬ ɨɲɢɛɤɭ II ɪɨɞɚ ɢ ɦɟɧɟɟ ɨɩɚɫɧɨ, ɱɟɦ ɢɫɩɨɥɶɡɨɜɚɧɢɟ ȡ ɛɟɡ

ȼX – ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢɬɪɢɨɛɴɟɞɢɧɟɧɢɹ: t1 = 2; t2 = 3; t3 = 2; Uɯ = 5; ɜɨɜɬɨɪɨɣ– ɱɟɬɵɪɟ: u1

=u2 = u3 = u4 = 2; Uy = 4. Ɍɟɩɟɪɶɩɨɮɨɪɦɭɥɟ(II,6,5): IJ= 0,55.

ɍɩɪɚɠɧɟɧɢɟ 53. ȼ ɭɩɨɦɢɧɚɜɲɟɣɫɹ ɤɧɢɝɟ «Ɇɟɬɨɞɢɤɚ ɢ ɬɟɯɧɢɤɚ ɫɬɚɬɢɫɬɢɱɟɫɤɨɣ ɨɛɪɚɛɨɬɤɢ ɩɟɪɜɢɱɧɨɣɫɨɰɢɨɥɨɝɢɱɟɫɤɨɣɢɧɮɨɪɦɚɰɢɢ» ɩɪɢɜɨɞɢɬɫɹɬɚɛɥɢɰɚ«ȼɵɱɢɫɥɟɧɢɟ

[115]

ɤɨɷɮɮɢɰɢɟɧɬɚ ɤɨɪɪɟɥɹɰɢɢ ɪɚɧɝɨɜ Ʉɟɧɞɷɥɚ ɦɟɠɞɭ ɨɬɜɟɬɚɦɢ ɪɚɛɨɱɢɯ: «ɢɧɬɟɪɟɫɧɚɹ ɪɚɛɨɬɚ» ɢ «ɨɛɪɚɡɨɜɚɧɢɟɫɨɨɬɜɟɬɫɬɜɭɟɬɪɚɛɨɬɟ» (ɫ. 17). ȼɨɫɩɪɨɢɡɜɟɞɟɦɱɚɫɬɶɟɟ.

Ɋɚɫɫɱɢɬɚɬɶ IJ. ȼ ɫɥɭɱɚɟ ɧɟɨɛɯɨɞɢɦɨɫɬɢ ɩɨɦɨɱɶ ɜ ɷɬɨɦ ɦɨɠɟɬ ɰɢɬɢɪɭɟɦɚɹ ɤɧɢɝɚ. Ɍɚɦ, ɜ ɱɚɫɬɧɨɫɬɢ, ɩɨɤɚɡɵɜɚɟɬɫɹ, ɱɬɨ

Ɍɚɛɥɢɰɚ28

ɉɪɢɦɟɪɜɵɱɢɫɥɟɧɢɹɤɨɷɮɮɢɰɢɟɧɬɚɪɚɧɝɨɜɨɣɤɨɪɪɟɥɹɰɢɢɄɟɧɞɷɥɚ

79

Ɋ = 61, Q = 28, ɨɞɧɚɤɨ ɩɪɢ ɜɵɱɢɫɥɟɧɢɢ IJ ɧɟ ɭɱɬɟɧɨ, ɱɬɨ ɢɦɟɸɬɫɹ ɨɛɴɟɞɢɧɟɧɢɹ ɪɚɧɝɨɜ. Ⱦɚɠɟ ɟɫɥɢ ȼɵ ɢɫɩɨɥɶɡɭɟɬɟ ɤɧɢɝɭ, ɪɚɫɫɱɢɬɚɣɬɟ IJ ɫɚɦɨɫɬɨɹɬɟɥɶɧɨ, ɫ ɭɱɟɬɨɦ ɨɛɴɟɞɢɧɟɧɢɣ. Ⱦɥɹ ɤɨɧɬɪɨɥɹ: Ux = 1, Uy = 2. Ɉɬɜɟɬ: IJ= + 0,39.

ɈɛɨɰɟɧɤɟɫɭɳɟɫɬɜɟɧɧɨɫɬɢIJɜɫɥɭɱɚɟɨɛɴɟɞɢɧɟɧɧɵɯɪɚɧɝɨɜɫɦ. § 8 ɝɥɚɜɵV. Ⱦɨɫɢɯɩɨɪɢɫɩɨɥɶɡɨɜɚɥɢɫɶɮɨɪɦɭɥɵ, ɫɩɪɚɜɟɞɥɢɜɵɟɞɥɹɥɸɛɵɯN, ɨɞɧɚɤɨɭɞɨɛɧɵɟɥɢɲɶɞɥɹ

ɦɚɥɵɯ(ɧɟɛɨɥɟɟ20–30); ɜɩɪɨɬɢɜɧɨɦɫɥɭɱɚɟɜɵɱɢɫɥɟɧɢɹɫɭɳɟɫɬɜɟɧɧɨɡɚɬɪɭɞɧɹɸɬɫɹ.

ɋɟɣɱɚɫ ɦɵ ɪɚɫɫɦɨɬɪɢɦ ɛɨɥɶɲɢɟ N. ȼ ɬɚɤɢɯ ɫɥɭɱɚɹɯ ɩɪɢɡɧɚɤɢ ɲɤɚɥɢɪɭɸɬɫɹ. Ʉɚɤ ɢ ɪɚɧɟɟ,

ɛɭɞɟɦ ɫɱɢɬɚɬɶ, ɱɬɨ ɩɪɢɡɧɚɤ X ɩɪɢɧɢɦɚɟɬ ɡɧɚɱɟɧɢɹ ɯi ɝɞɟ i 1,k , ɚ ɩɪɢɡɧɚɤ Y – ɡɧɚɱɟɧɢɹ yj, ɝɞɟ j 1,l (ɨɛɵɱɧɨ k, l § 5–10). ɗɦɩɢɪɢɱɟɫɤɢɣ ɦɚɬɟɪɢɚɥ ɫɜɨɞɢɬɫɹ ɜ ɤɨɪɪɟɥɹɰɢɨɧɧɭɸ ɬɚɛɥɢɰɭ ^Nij `,

ɉɪɢ ɛɨɥɶɲɢɯ N ɜɵɩɨɥɧɢɬɶ ɫɭɦɦɢɪɨɜɚɧɢɟ ɩɨ r ɢ s ɨɬ 1 ɞɨ N ɱɪɟɡɜɵɱɚɣɧɨ ɡɚɬɪɭɞɧɢɬɟɥɶɧɨ, ɩɨɷɬɨɦɭɩɟɪɟɣɞɟɦɤɫɭɦɦɢɪɨɜɚɧɢɸɩɨi ɢj ɨɬ1 ɞɨk ɢl ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ.

Ɋɚɫɫɦɨɬɪɢɦ A. ɇɚɦ ɧɭɠɧɨ ɫɪɚɜɧɢɬɶ ɪɚɧɝɢ ɩɨ X ɤɚɠɞɨɣ ɩɚɪɵ ɨɛɴɟɤɬɨɜ, ɚ ɪɟɡɭɥɶɬɚɬɵ ɩɪɨɫɭɦɦɢɪɨɜɚɬɶ22. Ɉɱɟɜɢɞɧɨ, ɦɨɠɧɨ ɧɟ ɫɪɚɜɧɢɜɚɬɶ ɦɟɠɞɭ ɫɨɛɨɣ ɷɥɟɦɟɧɬɵ ɫɬɪɨɤɢ, ɬɚɤ ɤɚɤ ɭ ɧɢɯ ɨɞɢɧɚɤɨɜɵɟ ɪɚɧɝɢ ɩɨ X. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɜɫɟ ɷɥɟɦɟɧɬɵ, ɭ ɤɨɬɨɪɵɯ X = ɯ1 (ɜɫɟɝɨ ɢɯ N (x1)), ɦɨɠɧɨ ɧɟ ɫɪɚɜɧɢɜɚɬɶ ɞɪɭɝ ɫ ɞɪɭɝɨɦ, ɧɨ ɫɥɟɞɭɟɬ ɫɪɚɜɧɢɬɶ ɫ ɷɥɟɦɟɧɬɚɦɢ, ɭ ɤɨɬɨɪɵɯ X = ɯ2. Ɍɚɤɨɟ

22 ȼ ɞɚɥɶɧɟɣɲɟɦ ɢɡɥɨɠɟɧɢɢ ɩɪɟɞɩɨɥɚɝɚɟɬɫɹ, ɱɬɨ ɡɧɚɱɟɧɢɹ X ɢ Y ɜɵɩɢɫɚɧɵ ɜ ɬɚɛɥɢɰɟ ɜ ɩɨɪɹɞɤɟ ɜɨɡɪɚɫɬɚɧɢɹ (ɫɜɟɪɯɭ ɜɧɢɡɢɫɥɟɜɚɧɚɩɪɚɜɨ).

80

ɫɪɚɜɧɟɧɢɟ ɩɨɪɨɞɢɬ N (ɯ1) • N (x2) ɟɞɢɧɢɰ, ɚɫɪɚɜɧɟɧɢɟɷɥɟɦɟɧɬɨɜɫ X = ɯ1 ɫɷɥɟɦɟɧɬɚɦɢ, ɭɤɨɬɨɪɵɯ X = x3, ɞɚɟɬN (ɯ1) N (ɯ3) ɟɞɢɧɢɰɢɬ.ɞ. ɉɨɷɬɨɦɭ

A = N(x1) [N(x2) + N(x3) + … +N(xk)] + N(x2) [N(x3) + N(x4) + … +N(xk)] + … + N(xk-1)N(xk) =

AN(x1)>N(x2 ) N(x3 ) ... N(xk )@ N(x2 )>N(x3 ) N(x4 ) ... N(xk )@ ...

ɉɟɪɟɣɞɟɦ ɤ ɪɚɫɫɦɨɬɪɟɧɢɸ S. Ɍɟɩɟɪɶ ɞɥɹ ɤɚɠɞɨɣ ɩɚɪɵ ɷɥɟɦɟɧɬɨɜ ɧɭɠɧɨ ɫɪɚɜɧɢɜɚɬɶ ɢ ɪɚɧɝɢ ɩɨX (ars), ɢɪɚɧɝɢɩɨY (brs).

Ɋɚɫɫɦɨɬɪɢɦ ɷɥɟɦɟɧɬɵ ɤɥɟɬɤɢ (i, j). əɫɧɨ, ɱɬɨ ɢɯ ɧɟ ɧɭɠɧɨ ɫɪɚɜɧɢɜɚɬɶ ɧɢ ɫ ɷɥɟɦɟɧɬɚɦɢ i-ɨɣ ɫɬɪɨɤɢ(ɨɛɷɬɨɦɦɵɭɠɟɝɨɜɨɪɢɥɢ), ɧɢɫɷɥɟɦɟɧɬɚɦɢ j-ɝɨɫɬɨɥɛɰɚ(ɭɷɥɟɦɟɧɬɨɜɫɬɨɥɛɰɚɨɞɢɧɚɤɨɜɵɟ ɪɚɧɝɢ ɩɨ Y, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɡɚ ɫɱɟɬ brs ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɟ ɫɥɚɝɚɟɦɨɟ ɨɛɪɚɬɢɬɫɹ ɜ ɧɭɥɶ). ɋɬɚɧɟɦ ɫɪɚɜɧɢɜɚɬɶ ɧɟɤɨɬɨɪɵɣ ɷɥɟɦɟɧɬ ɢɡ ɤɥɟɬɤɢ (i, j) ɫ ɷɥɟɦɟɧɬɨɦ ɤɥɟɬɤɢ (i', j'), ɟɫɥɢ i'>i, j'> j. Ɍɚɤɨɟ ɫɪɚɜɧɟɧɢɟ ɞɥɹɤɚɠɞɨɣ ɩɚɪɵɨɛɴɟɤɬɨɜɩɨɪɨɞɢɬ +1 ɜɫɢɥɭɭɩɨɪɹɞɨɱɟɧɧɨɫɬɢ ɩɭɧɤɬɨɜɲɤɚɥɵ (ars = 1, brs = 1). ȿɫɥɢi'>i, ɚj'> j, ɬɨɤɚɠɞɚɹɩɚɪɚɩɨɪɨɞɢɬ–1 (ars = 1, brs = – 1). ɋɭɦɦɢɪɭɹɩɨi',j', ɦɵ

[117]

ɩɟɪɟɛɟɪɟɦ ɜɫɟɜɨɡɦɨɠɧɵɟ ɫɪɚɜɧɟɧɢɹ ɜɵɞɟɥɟɧɧɨɝɨ ɷɥɟɦɟɧɬɚ ɢɡ ɤɥɟɬɤɢ (i, j) ɫɨ ɜɫɟɦɢ ɷɥɟɦɟɧɬɚɦɢ,

k l

ɥɟɠɚɳɢɦɢ ɧɢɠɟ ɢ ɫɩɪɚɜɚ (j'> j, i'>i) ɤɨɬɨɪɵɟ ɞɚɞɭɬ, ɬɚɤɢɦ ɨɛɪɚɡɨɦ, ¦ ¦Nicjc . ɋɨɩɨɫɬɚɜɥɟɧɢɟ

ic i 1 jc j 1

ɷɥɟɦɟɧɬɚ ɢɡ ɤɥɟɬɤɢ (i,j) ɫ ɷɥɟɦɟɧɬɚɦɢ, ɪɚɫɩɨɥɨɠɟɧɧɵɦɢ ɧɢɠɟ ɢ ɫɥɟɜɚ ɨɬ ɷɬɨɣ ɤɥɟɬɤɢ, ɩɨɪɨɠɞɚɟɬ

kj 1

ɫɥɚɝɚɟɦɨɟ ¦¦Nicjc . Ɍɚɤɤɚɤɜɫɟɷɥɟɦɟɧɬɵɤɥɟɬɤɢ(i,j) ɪɚɜɧɨ-

ic i 1 jc 1

Ɍɚɛɥɢɰɚ29

ɋɜɹɡɶɭɞɨɜɥɟɬɜɨɪɟɧɧɨɫɬɢɪɚɛɨɬɨɣɫɭɞɨɜɥɟɬɜɨɪɟɧɧɨɫɬɶɸɫɩɟɰɢɚɥɶɧɨɫɬɶɸ

ɩɪɚɜɧɵ, ɬɨ ɭɦɧɨɠɚɹ ɪɟɡɭɥɶɬɚɬ ɧɚ NiJ ɢ ɫɭɦɦɢɪɭɹ ɡɚɬɟɦ ɩɨ i ɢ j, ɦɵ ɨɫɭɳɟɫɬɜɢɦ ɜɨɨɛɳɟ ɜɫɟ ɜɨɡɦɨɠɧɵɟɫɪɚɜɧɟɧɢɹɩɚɪɷɥɟɦɟɧɬɨɜ.

ɍɩɪɚɠɧɟɧɢɟ55. ɉɨɱɟɦɭɧɟɧɭɠɧɨɪɚɫɫɦɚɬɪɢɜɚɬɶɫɥɭɱɚɣi'<i? ɂɬɚɤ,

k l k l k j 1

S ¦¦Nij ( ¦ ¦ Nicjc ¦¦Nicjc) (II,6,9)

i 1 j 1 ic i 1 jc j 1 ic i 1 jc 1

ɌɟɦɫɚɦɵɦɦɵɡɚɜɟɪɲɢɥɢɩɟɪɟɯɨɞɤɤɨɪɪɟɥɹɰɢɨɧɧɨɣɬɚɛɥɢɰɟɜɨɜɫɟɯɦɧɨɠɢɬɟɥɹɯIJ23.

Ⱦɥɹ ɢɥɥɸɫɬɪɚɰɢɢ ɷɬɨɣ «ɫɬɪɚɲɧɨɣ» ɮɨɪɦɭɥɵ ɩɪɢɜɟɞɟɦ ɩɪɢɦɟɪ, ɤɨɬɨɪɵɣ ɩɨɤɚɠɟɬ ɫɩɪɚɜɟɞɥɢɜɨɫɬɶɩɨɫɥɨɜɢɰɵ«ɧɟɬɚɤɫɬɪɚɲɟɧɱɟɪɬ, ɤɚɤɟɝɨɪɢɫɭɸɬ».

23 ȺɜɬɨɪɵɜɵɪɚɠɚɸɬɛɥɚɝɨɞɚɪɧɨɫɬɶȽ.ɂ. ɋɚɝɚɧɟɧɤɨɡɚɩɨɦɨɳɶɩɪɢɜɵɜɨɞɟɫɨɨɬɧɨɲɟɧɢɹ(II,6,9).

81

ɉɪɢɦɟɪ 23. ɂɡɭɱɚɹ ɫɜɹɡɶ ɭɞɨɜɥɟɬɜɨɪɟɧɧɨɫɬɢ ɪɚɛɨɬɨɣ (Y) ɫ ɭɞɨɜɥɟɬɜɨɪɟɧɧɨɫɬɶɸ ɫɩɟɰɢɚɥɶɧɨɫɬɶɸ(X) ɦɵ, ɜɱɚɫɬɧɨɫɬɢ, ɩɨɥɭɱɢɥɢɤɨɪɪɟɥɹɰɢɨɧɧɭɸɬɚɛɥɢɰɭ29 (ɦɚɫɫɢɜ, ɈɋɊɁ).

[118]

ɌɟɩɟɪɶȺ= 1587 (243 + 333) + 243·333 = 995031; ȼ= 1734 (157 + 272) + 157·272 = 786590;

S = 1472 (65 + 42 + 42 + 165) + 50 (42 + 165 – 136 – 126) – 65 (136 + 65 + 126 + 42) + 136 (42 +

165)+ 65 (165 – 126) – 42 (126 + 42) = 459104;

IJ= +0,52.

Ɍɚɤɢɦɨɛɪɚɡɨɦ, ɦɟɠɞɭɢɡɭɱɚɟɦɵɦɢɭɞɨɜɥɟɬɜɨɪɟɧɧɨɫɬɹɦɢɟɫɬɶɬɟɫɧɚɹɩɨɥɨɠɢɬɟɥɶɧɚɹɫɜɹɡɶ. ɍɩɪɚɠɧɟɧɢɟ 56. Ⱦɥɹ ɩɪɢɡɧɚɤɨɜ ɭɞɨɜɥɟɬɜɨɪɟɧɧɨɫɬɶ ɪɚɛɨɬɨɣ (Y), ɭɞɨɜɥɟɬɜɨɪɟɧɧɨɫɬɶ

ɨɛɳɟɫɬɜɟɧɧɨɣɪɚɛɨɬɨɣ(X) ɤɨɪɪɟɥɹɰɢɨɧɧɚɹɬɚɛɥɢɰɚɢɦɟɟɬɜɢɞ:

Ɍɚɛɥɢɰɚ30

ɋɜɹɡɶɭɞɨɜɥɟɬɜɨɪɟɧɧɨɫɬɢɪɚɛɨɬɨɣ(Y) ɫɭɞɨɜɥɟɬɜɨɪɟɧɧɨɫɬɶɸɨɛɳɟɫɬɜɟɧɧɨɣɪɚɛɨɬɨɣ

(X)

ȼɵɱɢɫɥɢɬɶIJ. Ɉɬɜɟɬ: IJ= + 0,31.

ɋɜɹɡɶ, ɬɚɤɢɦ ɨɛɪɚɡɨɦ, ɬɨɠɟ ɩɨɥɨɠɢɬɟɥɶɧɚɹ, ɧɨ ɦɟɧɟɟ ɬɟɫɧɚɹ. ȿɳɟ ɦɟɧɟɟ ɬɟɫɧɨɣ, ɧɚɩɪɢɦɟɪ, ɨɤɚɡɵɜɚɟɬɫɹ ɫɜɹɡɶ ɦɟɠɞɭ ɭɞɨɜɥɟɬɜɨɪɟɧɧɨɫɬɶɸ ɪɚɛɨɬɨɣ ɢ ɭɞɨɜɥɟɬɜɨɪɟɧɧɨɫɬɶɸ ɞɨɫɭɝɨɦ (ɞɥɹ ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɣ ɤɨɪɪɟɥɹɰɢɨɧɧɨɣ ɬɚɛɥɢɰɵ IJ = +0,14), ɱɬɨ ɞɨɩɭɫɤɚɟɬ ɟɫɬɟɫɬɜɟɧɧɭɸ ɢɧɬɟɪɩɪɟɬɚɰɢɸ.

Ʉɨɷɮɮɢɰɢɟɧɬ IJ, ɨɩɪɟɞɟɥɹɟɦɵɣ ɮɨɪɦɭɥɨɣ (II,6,6), ɦɨɠɟɬ ɨɛɪɚɳɚɬɶɫɹ ɜ ±1 ɬɨɥɶɤɨ ɜ ɬɨɦ ɫɥɭɱɚɟ, ɤɨɝɞɚɬɚɛɥɢɰɚɞɢɚɝɨɧɚɥɶɧɚ.

ȼ ɫɚɦɨɦ ɞɟɥɟ, ɫɨɝɥɚɫɧɨ ɧɟɪɚɜɟɧɫɬɜɭ Ʉɨɲɢ24 |S| ɦɚɤɫɢɦɚɥɟɧ, ɟɫɥɢ ɧɚɛɨɪɵ ars ɢ brs ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɵ: brs = Į · ars. ɗɬɨ ɜɨɡɦɨɠɧɨ ɥɢɲɶ ɬɨɝɞɚ, ɤɨɝɞɚ ɜɫɟ ɧɚɛɥɸɞɟɧɢɹ ɥɢɛɨ ɧɚ ɩɨɥɨɠɢɬɟɥɶɧɨɣ (Į = 1), ɥɢɛɨ ɧɚ ɨɬɪɢɰɚɬɟɥɶɧɨɣ (Į = – 1) ɝɥɚɜɧɨɣ ɞɢɚɝɨɧɚɥɢ ɬɚɛɥɢɰɵ, ɬ.ɟ. ɟɫɥɢ ɬɚɛɥɢɰɚɤɜɚɞɪɚɬɧɚɹ(ɟɫɥɢɟɫɬɶɧɟɞɢɚɝɨɧɚɥɶɧɵɟɷɥɟɦɟɧɬɵ, ɬɨĮɧɟɛɭɞɟɬɡɧɚɤɨ-

[119]

ɩɨɫɬɨɹɧɧɨɣɜɟɥɢɱɢɧɨɣ, ɫɨɨɬɧɨɲɟɧɢɟbrs = Įars ɧɟɛɭɞɟɬɜɵɩɨɥɧɹɬɶɫɹɞɥɹɜɫɟɯɩɚɪɷɥɟɦɟɧɬɨɜ). Ⱦɥɹ ɩɪɹɦɨɭɝɨɥɶɧɨɣ ɬɚɛɥɢɰɵ |S| ɞɨɫɬɢɝɚɟɬ ɦɚɤɫɢɦɭɦɚ, ɟɫɥɢ: 1) ɜɫɟ ɧɚɛɥɸɞɟɧɢɹ ɥɟɠɚɬ ɜ

ɤɥɟɬɤɚɯ ɫɚɦɨɣ ɞɥɢɧɧɨɣ ɞɢɚɝɨɧɚɥɢ ɬɚɛɥɢɰɵ, ɬ.ɟ. ɞɢɚɝɨɧɚɥɢ, ɫɨɞɟɪɠɚɳɟɣ m = min (k, l) ɤɥɟɬɨɤ, ɬɚɤ ɤɚɤ ɜ ɫɥɭɱɚɟ ɩɨɹɜɥɟɧɢɹ ɧɟɞɢɚɝɨɧɚɥɶɧɵɯ ɷɥɟɦɟɧɬɨɜ ɜ S, ɤɪɨɦɟ ɧɭɥɟɣ ɬɢɩɚ 0·0, ɞɨɛɚɜɥɹɸɬɫɹ ɧɭɥɢ ɬɢɩɚars · 0 ɢ0 · brs, ɩɪɢɱɟɦɡɚɫɱɟɬɭɦɟɧɶɲɟɧɢɹɱɢɫɥɚɫɥɚɝɚɟɦɵɯ, ɪɚɜɧɵɯ1;

2) ɜɫɟɧɚɛɥɸɞɟɧɢɹɪɚɜɧɨɦɟɪɧɨɪɚɫɩɪɟɞɟɥɟɧɵɦɟɠɞɭɞɢɚɝɨɧɚɥɶɧɵɦɢɤɥɟɬɤɚɦɢ, ɬ.ɟ. Nii = N/m (ɬɚɤɤɚɤɨɛɵɱɧɨN >> m, ɬɨɦɨɠɧɨɫɱɢɬɚɬɶ, ɱɬɨɨɧɨɤɪɚɬɧɨm ɛɟɡɫɭɳɟɫɬɜɟɧɧɨɣɩɨɬɟɪɢɬɨɱɧɨɫɬɢ).

ɉɪɨɢɥɥɸɫɬɪɢɪɭɟɦɩɟɪɜɨɟɭɬɜɟɪɠɞɟɧɢɟ, ɧɚɩɪɢɦɟɪ, ɞɥɹɫɥɟɞɭɸɳɟɣɬɚɛɥɢɰɵ:

24 Ⱦɥɹɱɢɬɚɬɟɥɹ, ɧɟɡɧɚɤɨɦɨɝɨɫɷɬɢɦɧɟɪɚɜɟɧɫɬɜɨɦ, ɦɵɩɪɢɜɨɞɢɦɟɝɨɜɵɜɨɞɜɤɨɧɰɟɩɚɪɚɝɪɚɮɚ.

82

SN11 (N22 1) % N11N22

ɉɪɨɢɥɥɸɫɬɪɢɪɭɟɦ ɜɬɨɪɨɟ ɭɬɜɟɪɠɞɟɧɢɟ. Ɋɚɫɫɦɨɬɪɢɦ, ɧɚɩɪɢɦɟɪ, ɞɢɚɝɨɧɚɥɶɧɭɸ ɬɚɛɥɢɰɭ

3 × 3:

ɬ.ɟ. [120]

ɟɫɥɢɜɫɟɧɚɛɥɸɞɟɧɢɹɪɚɫɩɪɟɞɟɥɟɧɵɪɚɜɧɨɦɟɪɧɨ. Ɂɞɟɫɶɦɵɢɫɩɨɥɶɡɨɜɚɥɢɢɡɜɟɫɬɧɨɟɧɟɪɚɜɟɧɫɬɜɨ ab bc ac d a2 b2 c2 ,

ɤɨɬɨɪɨɟɥɟɝɤɨɩɨɥɭɱɢɬɶ, ɫɤɥɚɞɵɜɚɹɩɨɱɥɟɧɧɨɬɪɢɨɱɟɜɢɞɧɵɯɧɟɪɚɜɟɧɫɬɜɚ

(a b)2 t 0,(a c)2 t 0,(b c)2 t 0.

ȼ ɨɛɳɟɦ ɫɥɭɱɚɟ ɜ ɤɚɠɞɨɣ ɤɥɟɬɤɟ ɫɚɦɨɣ ɞɥɢɧɧɨɣ ɞɢɚɝɨɧɚɥɢ ɞɨɥɠɧɨ ɛɵɬɶ N/m ɷɥɟɦɟɧɬɨɜ.

ɋɨɩɨɫɬɚɜɥɹɹ ɷɥɟɦɟɧɬɵ ɩɟɪɜɨɣ ɤɥɟɬɤɢ ɫ ɨɫɬɚɥɶɧɵɦɢ, ɦɵ ɩɨɥɭɱɢɦ N N (m 1) ɟɞɢɧɢɰ, ɚ m m

ɷɥɟɦɟɧɬɵ ɜɬɨɪɨɣ ɫ ɩɪɨɱɢɦɢ N N (m 2) , ɬɚɤ ɤɚɤ ɢɯ ɭɠɟ ɧɟ ɧɭɠɧɨ ɫɪɚɜɧɢɜɚɬɶ ɫ ɷɥɟɦɟɧɬɚɦɢ m m

ɩɟɪɜɨɣ ɢ ɬ.ɞ. ȼɢɬɨɝɟ

Ɉɱɟɜɢɞɧɨ, ɨɧ ɩɪɢɧɢɦɚɟɬ ɡɧɚɱɟɧɢɹ, ɤɨɬɨɪɵɟ ɦɨɝɭɬ ɞɨɫɬɢɱɶ ± 1 (ɟɫɥɢ ɧɟ ɫɱɢɬɚɬɶ ɧɟɡɧɚɱɢɬɟɥɶɧɨɝɨ ɷɮɮɟɤɬɚ, ɜɨɡɧɢɤɚɸɳɟɝɨ ɜ ɫɥɭɱɚɟ, ɤɨɝɞɚ N ɧɟ ɤɪɚɬɧɨ ɬ) ɞɚɠɟ ɞɥɹ ɩɪɹɦɨɭɝɨɥɶɧɵɯ ɬɚɛɥɢɰ.

83