Планирование эксперимента. Учебное пособие-1
.pdfɨɩɪɟɞɟɥɹɸɳɢɦ ɤɨɧɬɪɚɫɬɨɦ x x x x x ɉɚɪɧɵɟ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɞɥɹ ɷɬɨɣ ɪɟɩɥɢɤɢ ɪɚɜɧɵ ɬɪɨɣɧɵɦ ɢ ɫɥɟɞɨɜɚɬɟɥɶɧɨ ɜɵɩɨɥɧɹɸɬɫɹ ɭɫɥɨɜɢɹ ɩɥɚɧɚ Ȼɨɤɫɚ ȿɳɟ ɨɞɢɧ ɩɥɚɧ Ȼɨɤɫɚ ɦɨɠɧɨ ɩɨɫɬɪɨɢɬɶ ɢɫɩɨɥɶɡɭɹ ɜ ɤɚɱɟɫɬɜɟ ɹɞɪɚ ɩɨɥɭɪɟɩɥɢ ɤɭ ɫ ɨɩɪɟɞɟɥɹɸɳɢɦ ɤɨɧɬɪɚɫɬɨɦ x x x x x əɞɪɨ ɩɨɥɭɪɟɩɥɢɤɢ ɛɭɞɟɬ ɫɨɞɟɪɠɚɬɶ ɬɨɱɟɤ ɜɦɟɫɬɨ ɩɪɢ ɢɫɩɨɥɶɡɨɜɚɧɢɢ ɜ ɤɚɱɟɫɬɜɟ ɹɞɪɚ ɉɎɗ ɂɫɩɨɥɶɡɨɜɚɬɶ ɜ ɤɚɱɟɫɬɜɟ ɹɞɪɚ ɩɥɚɧɚ Ȼɨɤɫɚ ɱɟɬɜɟɪɬɶɪɟɩɥɢɤɭ ɧɟɥɶɡɹ ɬɚɤ ɤɚɤ ɪɚɡɪɟɲɚɸɳɚɹ ɫɩɨɫɨɛɧɨɫɬɶ ɬɚɤɨɣ ɪɟɩɥɢɤɢ ɧɟɞɨɫɬɚɬɨɱɧɚ ɉɨɷɬɨ ɦɭ ɩɥɚɧ Ȼɨɤɫɚ ɩɪɢ n ɫ ɩɨɥɭɪɟɩɥɢɤɨɣ ɛɭɞɟɬ ɦɢɧɢɦɚɥɶɧɵɦ Ɉɛɳɟɟ
ɱɢɫɥɨ ɨɩɵɬɨɜ |
ɜ ɬɚɤɨɦ ɩɥɚɧɟ ɩɪɢ n ɪɚɜɧɨ N |
ɚ ɜ ɉɎɗ N |
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ɉɥɚɧ Ȼɨɤɫɚ ɞɥɹ ɞɚɧɧɨɝɨ ɤɨɥɢɱɟɫɬɜɚ ɮɚɤɬɨɪɨɜn ɧɚɡɵɜɚɟɬɫɹ ɦɢɧɢɦɚɥɶɧɵɦ ɟɫɥɢ ɢɡ ɜɫɟɯ ɜɨɡɦɨɠɧɵɯ ɩɥɚɧɨɜ Ȼɨɤɫɚ ɞɥɹ ɞɚɧɧɨɝɨ ɤɨɥɢɱɟɫɬɜɚ ɮɚɤɬɨɪɨɜ ɱɢɫɥɨ ɪɚɡɥɢɱɧɵɯ ɬɨɱɟɤ ɜ ɟɝɨ ɹɞɪɟ ɦɢɧɢɦɚɥɶɧɨ
ɉɪɢn ɢn ɜ ɤɚɱɟɫɬɜɟ ɹɞɪɚ ɩɥɚɧɚ Ȼɨɤɫɚ ɦɨɠɟɬ ɢɫɩɨɥɶɡɨɜɚɬɶɫɹ ɤɪɨɦɟ ɩɥɚɧɚ ɉɎɗ ɬɨɥɶɤɨ ɩɨɥɭɪɟɩɥɢɤɚ ɂɫɩɨɥɶɡɨɜɚɬɶ ɱɟɬɜɟɬɶɪɟɩɥɢɤɭ ɜ ɤɚɱɟɫɬɜɟ ɹɞɪɚ ɩɥɚɧɚ Ȼɨɤɫɚ ɧɟɥɶɡɹ ɬɚɤ ɤɚɤ ɪɚɡɪɟɲɚɸɳɚɹ ɫɩɨɫɨɛɧɨɫɬɶ ɜɫɟɯ ɱɟɬɜɟɪɬɶɪɟ ɩɥɢɤ ɫɬɪɨɝɨ ɦɟɧɶɲɟ
ɂɫɩɨɥɶɡɨɜɚɧɢɟ ɱɟɬɜɟɪɬɶɪɟɩɥɢɤɢ ɜ ɩɥɚɧɚɯ Ȼɨɤɫɚ ɛɭɞɟɬ ɜɨɡɦɨɠɧɨ ɩɥɚɧ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɨɤɚɡɵɜɚɟɬɫɹ ɦɢɧɢɦɚɥɶɧɵɦ
ɋ ɪɨɫɬɨɦ ɤɨɥɢɱɟɫɬɜɚ ɮɚɤɬɨɪɨɜ ɡɚɞɚɱɚ ɧɚɯɨɠɞɟɧɢɹ ɦɢɧɢɦɚɥɶɧɨɝɨ ɩɥɚɧɚ
ɭɫɥɨɠɧɹɟɬɫɹ
ɉɥɚɧɵ ɏɚɪɬɥɢ
ȼ ɫɥɭɱɚɟ ɤɨɝɞɚ ɱɢɫɥɨ ɨɩɵɬɨɜ ɩɨ ɭɫɥɨɜɢɹɦ ɩɪɨɜɟɞɟɧɢɹ ɷɤɫɩɟɪɢɦɟɧɬɚ ɨɝɪɚ ɧɢɱɟɧɨ ɤɨɝɞɚ ɱɚɫɬɶ ɷɮɮɟɤɬɨɜbi ɢɥɢbi j ɜ ɦɨɞɟɥɢ ɦɨɠɟɬ ɨɬɫɭɬɫɬɜɨɜɚɬɶ ɩɪɢɦɟ
ɧɹɸɬ ɩɥɚɧɵ ɏɚɪɬɥɢ ɐɄɉ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ ɧɚɡɵɜɚɟɬɫɹ ɩɥɚɧɨɦ ɏɚɪɬɥɢ,ɟɫɥɢ ɜ ɤɚɱɟɫɬɜɟ ɟɝɨ ɹɞɪɚ
ɢɫɩɨɥɶɡɭɟɬɫɹ ȾɎɗ n k ɜ ɤɨɬɨɪɨɦ ɧɟɤɨɬɨɪɵɟ ɩɚɪɧɵɟ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɪɚɜɧɵ ɩɨ ɦɨɞɭɥɸ ɥɢɧɟɣɧɵɦ ɩɟɪɟɦɟɧɧɵɦ ɬ ɟ ɭɫɥɨɜɢɟ
xi x j z rxl xs
ɜɵɩɨɥɧɹɟɬɫɹ ɧɨ ɞɥɹ ɧɟɤɨɬɨɪɵɯ ɩɚɪɧɵɯ ɜɡɚɢɦɨɞɟɣɫɬɜɢɣ ɞɨɩɭɫɤɚɟɬɫɹ
xi rx j xl j zl i j l n
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ ɩɪɨɢɡɜɨɥɶɧɵɣ ɐɄɉɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ ɛɭɞɟɬ ɥɢɛɨ ɩɥɚɧɨɦ Ȼɨɤɫɚ ɥɢɛɨ ɩɥɚɧɨɦ ɏɚɪɬɥɢ ɆɧɨɠɟɫɬɜɨCn ɐɄɉ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ ɞɥɹ ɮɢɤɫɢ ɪɨɜɚɧɧɨɝɨ ɱɢɫɥɚ ɩɟɪɟɦɟɧɧɵɯnɫɨɫɬɨɢɬ ɢɡ ɞɜɭɯ ɩɨɞɦɧɨɠɟɫɬɜH n ± ɩɨɞɦɧɨɠɟ ɫɬɜɚ ɩɥɚɧɨɜ ɏɚɪɬɥɢ ɢ Bn ± ɩɨɞɦɧɨɠɟɫɬɜɚ ɩɥɚɧɨɜ Ȼɨɤɫɚ ɬɨ ɟɫɬɶCn Hn Bn . ɉɪɢ ɷɬɨɦ H n Bn ɬɨ ɟɫɬɶ ɩɥɚɧ ɧɟ ɦɨɠɟɬ ɛɵɬɶ ɨɞɧɨɜɪɟɦɟɧɧɨ ɢ ɩɥɚɧɨɦ ɏɚɪɬɥɢ ɢ ɩɥɚɧɨɦ Ȼɨɤɫɚ
ɇɚɢɛɨɥɶɲɢɣ ɩɪɚɤɬɢɱɟɫɤɢɣ ɢɧɬɟɪɟɫ ɢɦɟɟɬ ɩɪɢɦɟɧɟɧɢɟ ɦɢɧɢɦɚɥɶɧɵɯ ɩɥɚ
ɧɨɜ ɏɚɪɬɥɢ ɉɨɧɹɬɢɟ ɦɢɧɢɦɚɥɶɧɨɝɨ ɩɥɚɧɚ ɏɚɪɬɥɢ ɜɜɨɞɢɬɫɹ ɚɧɚɥɨɝɢɱɧɨ ɩɨɧɹ
ɬɢɸ ɦɢɧɢɦɚɥɶɧɨɝɨ ɩɥɚɧɚ Ȼɨɤɫɚ ɉɥɚɧ ɏɚɪɬɥɢ ɞɥɹ ɞɚɧɧɨɝɨ ɤɨɥɢɱɟɫɬɜɚ ɮɚɤɬɨɪɨɜ n ɧɚɡɵɜɚɟɬɫɹ ɦɢɧɢɦɚɥɶ
ɧɵɦ ɟɫɥɢ ɢɡ ɜɫɟɯ ɜɨɡɦɨɠɧɵɯ ɩɥɚɧɨɜ ɏɚɪɬɥɢ ɞɥɹ ɞɚɧɧɨɝɨ ɤɨɥɢɱɟɫɬɜɚ ɮɚɤɬɨɪɨɜ
ɱɢɫɥɨ ɪɚɡɥɢɱɧɵɯ ɬɨɱɟɤ ɜ ɟɝɨ ɹɞɪɟ ɦɢɧɢɦɚɥɶɧɨ Ɍɚɤ ɤɚɤ ɜ ɩɥɚɧɚɯ ɏɚɪɬɥɢ ɞɨɩɭɫɤɚɟɬɫɹ xi rx j xl ɢɧɚɱɟ ɨɧ ɛɭɞɟɬ ɩɥɚɧɨɦ
Ȼɨɤɫɚ ɬɨ ɞɥɹ ɩɨɫɬɪɨɟɧɢɹ ɟɝɨ ɹɞɪɚ ɞɨɥɠɧɵ ɨɛɹɡɚɬɟɥɶɧɨ ɢɫɩɨɥɶɡɨɜɚɬɶɫɹ ɞɪɨɛ
ɧɵɟ ɪɟɩɥɢɤɢ ɫ ɪɚɡɪɟɲɚɸɳɟɣ ɫɩɨɫɨɛɧɨɫɬɶɸ ɪɚɜɧɨɣ ɉɨɫɤɨɥɶɤɭ ɹɞɪɨɦ ɩɥɚɧɚ ɏɚɪɬɥɢ ɹɜɥɹɟɬɫɹ ɞɪɨɛɧɚɹ ɪɟɩɥɢɤɚ n k ɡɚɞɚɱɚ ɩɨ
ɫɬɪɨɟɧɢɹ ɦɢɧɢɦɚɥɶɧɨɝɨ ɩɥɚɧɚ ɏɚɪɬɥɢ ɪɚɜɧɨɫɢɥɶɧɚ ɡɚɞɚɱɢ ɧɚɯɨɠɞɟɧɢɹ ɦɚɤɫɢ
ɦɚɥɶɧɨɝɨ |
ɡɧɚɱɟɧɢɹ ɞɪɨɛɧɨɫɬɢ |
k |
ɩɪɢ |
ɤɨɬɨɪɨɦ |
ɜɵɩɨɥɧɹɟɬɫɹ ɭɫɥɨɜɢɟ |
xi x j z rxl xs |
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ɉɨɫɬɪɨɢɦ ɩɥɚɧ ɏɚɪɬɥɢ ɞɥɹ n |
ɢn |
ȼ ɤɚɱɟɫɬɜɟ ɹɞɪɚ ɜɨɡɶɦɟɦ ɩɨɥɭ |
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ɪɟɩɥɢɤɭ ɬɢɩɚ ɫ ɝɟɧɟɪɢɪɭɸɳɢɦ ɫɨɨɬɧɨɲɟɧɢɟɦ x |
x x .Ɉɩɪɟɞɟɥɹɸɳɢɣ |
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ɤɨɧɬɪɚɫɬ |
x x x ɢ ɫɥɟɞɨɜɚɬɟɥɶɧɨ ɪɚɡɪɟɲɚɸɳɚɹ ɫɩɨɫɨɛɧɨɫɬɶ ɪɚɜɧɚ ȼɫɟ |
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ɩɚɪɧɵɟ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɧɟ ɫɜɹɡɚɧɵ ɦɟɠɞɭ ɫɨɛɨɣ |
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x x |
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x x x x x |
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x x |
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x x x x x |
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x x |
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x x x x x |
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ɑɢɫɥɨ ɨɩɵɬɨɜ ɜ ɷɬɨɦ ɩɥɚɧɟ N ɉɥɚɧ ɹɜɥɹɟɬɫɹ ɦɢɧɢ ɦɚɥɶɧɵɦ ɩɨɫɤɨɥɶɤɭ ɜ ɤɚɱɟɫɬɜɟ ɹɞɪɚ ɧɟɥɶɡɹ ɢɫɩɨɥɶɡɨɜɚɬɶ ɞɪɨɛɧɭɸ ɪɟɩɥɢ ɤɭ
ɉɨɫɬɪɨɢɦ ɹɞɪɨ ɩɥɚɧɚ ɏɚɪɬɥɢ ɞɥɹn əɞɪɨɦ ɩɥɚɧɚ ɦɨɠɟɬ ɛɵɬɶ ɩɨɥɭɪɟ ɩɥɢɤɚ ɫ ɝɟɧɟɪɢɪɭɸɳɢɦ ɫɨɨɬɧɨɲɟɧɢɟɦ x x x ɉɨɥɭɪɟɩɥɢɤɚ ɢɦɟɟɬ ɪɚɡ ɪɟɲɚɸɳɭɸ ɫɩɨɫɨɛɧɨɫɬɶ ɪɚɜɧɭɸ ɢ ɩɚɪɧɵɟ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɜ ɧɟɣ ɧɟ ɫɜɹɡɚɧɵ ɦɟɠɞɭ ɫɨɛɨɣ ɉɥɚɧ ɛɭɞɟɬ ɦɢɧɢɦɚɥɶɧɵɦ
Ⱦɟɣɫɬɜɢɬɟɥɶɧɨɞɥɹɞɪɨɛɧɵɯɪɟɩɥɢɤ ɧɟɤɨɬɨɪɵɟɩɚɪɧɵɟɜɡɚɢɦɨɞɟɣ ɫɬɜɢɹ ɪɚɜɧɵ ɩɨ ɦɨɞɭɥɸ ɞɪɭɝ ɞɪɭɝɭ ɉɭɫɬɶ ɪɟɩɥɢɤɚ ɡɚɞɚɟɬɫɹ ɝɟɧɟɪɢɪɭɸ
ɳɢɦɢɫɨɨɬɧɨɲɟɧɢɹɦɢ x |
x x ɢ x |
x x Ɍɨɝɞɚɨɛɨɛɳɟɧɧɵɣɨɩɪɟɞɟɥɹɸ |
ɳɢɣɤɨɧɬɪɚɫɬɢɦɟɟɬɜɢɞ |
x x x |
x x x x x x x ɢɩɚɪɧɵɟɜɡɚɢɦɨɞɟɣ |
ɫɬɜɢɹx x ɢx x ɪɚɜɧɵ Ɇɨɠɧɨ ɩɨɤɚɡɚɬɶ ɱɬɨ ɜɫɟ ɪɟɩɥɢɤɢ ɫ ɪɚɡɥɢɱɧɵɦɢ ɜɨɡɦɨɠɧɵɦɢ ɝɟɧɟɪɢɪɭɸɳɢɦɢ ɫɨɨɬɧɨɲɟɧɢɹɦɢ ɛɭɞɭɬ ɞɚɜɚɬɶ ɪɚɜɧɵɟ ɦɟɠɞɭ ɫɨ ɛɨɣ ɩɨ ɦɨɞɭɥɸ ɞɜɨɣɧɵɟ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ
ɂɬɚɤ ɩɪɢ n ɩɥɚɧɵ ɏɚɪɬɥɢ ɢ Ȼɨɤɫɚ ɫ ɩɨɥɭɪɟɩɥɢɤɨɣ ɛɭɞɭɬ ɦɢɧɢ ɦɚɥɶɧɵɦɢ ɢ ɩɪɢ ɨɞɢɧɚɤɨɜɵɯ ɡɧɚɱɟɧɢɹɯ n ɫɨɞɟɪɠɚɬɶ ɨɞɢɧɚɤɨɜɨɟ ɤɨɥɢɱɟ ɫɬɜɨ ɨɩɵɬɨɜ
ɋɪɚɜɧɟɧɢɟ ɦɢɧɢɦɚɥɶɧɵɯ ɩɨ ɱɢɫɥɭ ɨɩɵɬɨɜ ɩɥɚɧɨɜ Ȼɨɤɫɚ ɢ ɏɚɪɬɥɢ ɩɪɢɜɟ
ɞɟɧɨ ɜ ɬɚɛɥɢɰɟ
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Ɍ ɚ ɛ ɥ ɢ ɰ ɚ |
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ɋɪɚɜɧɟɧɢɟ ɦɢɧɢɦɚɥɶɧɵɯ ɩɨ ɱɢɫɥɭ ɨɩɵɬɨɜ ɩɥɚɧɨɜ Ȼɨɤɫɚ ɢ ɏɚɪɬɥɢ |
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ɑɢɫɥɨ |
Ɇɢɧɢɦɚɥɶɧɵɣ ɩɥɚɧ Ȼɨɤɫɚ |
Ɇɢɧɢɦɚɥɶɧɵɣ ɩɥɚɧ ɏɚɪɬɥɢ |
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ɮɚɤɬɨɪɨɜ |
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əɞɪɨ ɩɥɚɧɚ |
ɑɢɫɥɨ ɨɩɵɬɨɜ |
əɞɪɨ ɩɥɚɧɚ |
ɑɢɫɥɨ ɨɩɵɬɨɜ |
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ɉɎɗ |
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ȾɎɗ |
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ɂɡ ɬɚɛɥɢɰɵ ɜɢɞɧɨ ɱɬɨ ɦɢɧɢɦɚɥɶɧɵɟ ɩɥɚɧɵ ɏɚɪɬɥɢ ɬɪɟɛɭɸɬ ɦɟɧɶɲɟɝɨ
ɱɟɦ ɦɢɧɢɦɚɥɶɧɵɟ ɩɥɚɧɵ Ȼɨɤɫɚ ɱɢɫɥɚ ɨɩɵɬɨɜ ɞɥɹ ɨɰɟɧɢɜɚɧɢɹ ɤɨɷɮɮɢɰɢɟɧɬɨɜ
ɩɨɥɢɧɨɦɚ ɣ ɫɬɟɩɟɧɢ
Ɉɞɧɚɤɨ ɩɥɚɧɵ ɏɚɪɬɥɢ ɭɫɬɭɩɚɸɬ ɩɥɚɧɚɦ Ȼɨɤɫɚ ɜ ɬɨɱɧɨɫɬɢ ɨɰɟɧɢɜɚɧɢɹ ɷɬɢɯ
ɤɨɷɮɮɢɰɢɟɧɬɨɜ
Ɉɪɬɨɝɨɧɚɥɶɧɵɟ ɐɄɉ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ
Ɉɪɬɨɝɨɧɚɥɶɧɵɦ ɩɥɚɧɨɦ ɧɚɡɵɜɚɟɬɫɹ ɬɚɤɨɣ ɭ ɤɨɬɨɪɨɝɨ ɦɚɬɪɢɰɚ ɩɥɚɧɢɪɨɜɚ
ɧɢɹ ; ɫɬɪɨɢɬɫɹ ɬɚɤ ɱɬɨɛɵ ɦɚɬɪɢɰɚ & ;T ; ɨɤɚɡɚɥɚɫɶ ɞɢɚɝɨɧɚɥɶɧɨɣ ɂɫ ɩɨɥɶɡɭɟɦ ɷɬɨɬ ɩɨɞɯɨɞ ɢ ɩɪɢ ɩɨɫɬɪɨɟɧɢɢ ɩɥɚɧɨɜ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ ɈɐɄɉ ± ɨɪɬɨ ɝɨɧɚɥɶɧɵɣ ɰɟɧɬɪɚɥɶɧɵɣ ɤɨɦɩɨɡɢɰɢɨɧɧɵɣ ɩɥɚɧ ± ɩɥɚɧ ɤɨɬɨɪɵɣ ɹɜɥɹɟɬɫɹ
ɨɪɬɨɝɨɧɚɥɶɧɵɦ ± ɦɚɬɪɢɰɚ& ;T ;ɞɢɚɝɨɧɚɥɶɧɚɹ
ɰɟɧɬɪɚɥɶɧɵɦ ± ɜɫɟ ɬɨɱɤɢ ɪɚɫɩɨɥɨɠɟɧɵ ɫɢɦɦɟɬɪɢɱɧɨ ɨɬɧɨɫɢɬɟɥɶɧɨ ɰɟɧɬɪɚ ɩɥɚɧɚ
ɤɨɦɩɨɡɢɰɢɨɧɧɵɦ ± ɩɥɚɧ ɹɜɥɹɟɬɫɹ ɤɨɦɩɨɡɢɰɢɟɣ ɬɪɟɯ ɩɥɚɧɨɜəɞɪɨ ± ɩɥɚɧ ɉɎɗ ɫN n ɬɨɱɤɚɦɢ ɩɥɚɧɚ
©Ɂɜɟɡɞɧɵɣ ɩɥɚɧª ± ɫ n ɬɨɱɤɚɦɢ ɩɨ ɞɜɟ ©ɡɜɟɡɞɧɵɟª ɬɨɱɤɢ ɞɥɹ ɤɚɠɞɨɝɨ
ɮɚɤɬɨɪɚ xi |
rD i n Ɉɫɬɚɥɶɧɵɟ ɮɚɤɬɨɪɵ ɩɪɢ ɷɬɨɦ ɪɚɜɧɵ |
ɤɪɨɦɟx |
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ɐɟɧɬɪ ɩɥɚɧɚ ± ɨɞɧɚ ɰɟɧɬɪɚɥɶɧɚɹ ɬɨɱɤɚ ɩɥɚɧɚn
Ɉɛɳɟɟ ɤɨɥɢɱɟɫɬɜɨ ɬɨɱɟɤ ɜ ɩɥɚɧɟ ɈɐɄɉ ɫɨɫɬɚɜɥɹɟɬ
N n n
ɉɪɢ n ! ɜ ɈɐɄɉ ɨɤɚɡɵɜɚɟɬɫɹ ɦɟɧɶɲɟɟ ɤɨɥɢɱɟɫɬɜɨ ɬɨɱɟɤ ɱɟɦ ɜ ɩɥɚɧɟ ɉɎɗ n ɬɚɛɥɢɰɚ
Ɍ ɚ ɛ ɥ ɢ ɰ ɚ
ɑɢɫɥɨ ɬɨɱɟɤ ɜ ɩɥɚɧɚɯ ɈɐɄɉ ɢ ɉɎɗ n ɩɪɢ ɪɚɡɥɢɱɧɨɦ ɱɢɫɥɟ ɮɚɤɬɨɪɨɜn
ɑɢɫɥɨ |
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Ⱦɥɹ ɨɪɬɨɝɨɧɚɥɶɧɨɝɨ ɩɥɚɧɚ ɧɟɨɛɯɨɞɢɦɨ ɱɬɨɛɵ ɜɵɩɨɥɧɹɥɨɫɶ ɫɨɨɬɧɨɲɟɧɢɟ
N
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Ɍɚɤ ɤɚɤ x u ɬɨ ɞɥɹ ɫɬɨɥɛɰɨɜ j m ɞɨɥɠɧɨ ɜɵɩɨɥɧɹɬɶɫɹ ɭɫɥɨɜɢɟ
N
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ɨɡɧɚɱɚɸɳɟɟ ɧɟɨɛɯɨɞɢɦɨɫɬɶ ɜɵɩɨɥɧɟɧɢɹ ɬɪɟɛɨɜɚɧɢɹ ± ɫɭɦɦɚ ɷɥɟɦɟɧɬɨɜ ɥɸɛɨɝɨ ɫɬɨɥɛɰɚ ɤɪɨɦɟj ɜɤɥɸɱɚɹ ɫɬɨɥɛɰɵ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɤɜɚɞɪɚɬɚɦ ɮɚɤɬɨɪɨɜ ɞɨɥɠɧɚ ɛɵɬɶ ɪɚɜɧɚ ɧɭɥɸ ɗɬɨ ɜɨɡɦɨɠɧɨ ɟɫɥɢ ɱɥɟɧɵ ɫɬɨɥɛɰɨɜ ɫɨɨɬɜɟɬɫɬɜɭɸ ɳɢɯ ɤɜɚɞɪɚɬɚɦ ɮɚɤɬɨɪɨɜ ɩɪɟɨɛɪɚɡɨɜɚɧɵ ɢɧɚɱɟ ɫɭɦɦɚ ɤɜɚɞɪɚɬɨɜ ɮɚɤɬɨɪɨɜ ɧɟ ɦɨɠɟɬ ɛɵɬɶ ɪɚɜɧɚ ɧɭɥɸ
ɉɪɟɨɛɪɚɡɨɜɚɧɢɟ ɷɥɟɦɟɧɬɨɜ ɷɬɢɯ ɫɬɨɥɛɰɨɜ ɨɫɭɳɟɫɬɜɥɹɟɬɫɹ ɜ ɜɢɞɟ
xcju x ju a
ɝɞɟa ± ɜɟɥɢɱɢɧɚ ɡɚɜɢɫɹɳɚɹ ɨɬ ɱɢɫɥɚ ɮɚɤɬɨɪɨɜ
ɋɭɦɦɚ ɷɥɟɦɟɧɬɨɜ ɫɬɨɥɛɰɨɜ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɤɜɚɞɪɚɬɚɦ ɮɚɤɬɨɪɨɜ
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ɬɪɟɯ ɮɚɤɬɨɪɚɯ ɩɪɟɞɫɬɚɜɥɟɧ ɜ ɬɚɛɥɢɰɟ
ȼ ɈɐɄɉ ɤɚɠɞɵɣ ɮɚɤɬɨɪ ɮɢɤɫɢɪɭɟɬɫɹ ɜ ɨɛɳɟɦ ɫɥɭɱɚɟ ɧɚ ɩɹɬɢ ɭɪɨɜɧɹɯ
±D±D
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ɇɭɥɟɜɚɹ ɬɨɱɤɚ |
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ɉɨɫɥɟ ɩɪɨɫɬɟɣɲɢɯ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ ɫ ɭɱɟɬɨɦ ɬɨɝɨ ɱɬɨ ɨɛɳɟɟ ɱɢɫɥɨ ɨɩɵɬɨɜ ɜ ɩɥɚɧɟN N n n ɩɨɥɭɱɚɟɦ ɫɨɨɬɧɨɲɟɧɢɟ
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ɋɨɨɬɧɨɲɟɧɢɟ ɞɥɹɚɩɪɢj =ɢɥɢ ɦɨɠɟɬ ɛɵɬɶ ɡɚɩɢɫɚɧɨ ɤɚɤ
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ɉɨɞɫɬɚɜɢɜ ɟɝɨ ɜ ɩɨɫɥɟɞɧɟɟ ɭɪɚɜɧɟɧɢɟ ɩɨɥɭɱɚɟɦ |
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ɨɬɤɭɞɚ ɧɚɯɨɞɢɬɫɹ ɤɨɧɫɬɚɧɬɚa |
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Ɍɨɝɞɚ |
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ɢ ɩɥɟɱɨ ©ɡɜɟɡɞɧɵɯª ɬɨɱɟɤ
DNN N
ɇɚɩɪɢɦɟɪ ɞɥɹ ɈɐɄɉ ɬɚɛɥɢɰɚ ɩɪɢ ɱɢɫɥɟ ɮɚɤɬɨɪɨɜ n ɢɦɟɟɦ ɫɥɟ ɞɭɸɳɢɟ ɩɚɪɚɦɟɬɪɵ ɩɥɚɧɚ
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Ɉɱɟɜɢɞɧɨ ɱɬɨ ɩɥɚɧ ɹɜɥɹɟɬɫɹ ɨɪɬɨɝɨɧɚɥɶɧɵɦ ȼ ɨɬɥɢɱɢɟ ɨɬ ɩɥɚɧɨɜ ɉɎɗ
ɞɥɹ ɈɐɄɉ ɫɭɦɦɚ ɤɜɚɞɪɚɬɨɜ ɮɚɤɬɨɪɨɜ ɪɚɡɧɵɯ ɫɬɨɥɛɰɨɜ ɧɟ ɹɜɥɹɟɬɫɹ ɨɞɢɧɚɤɨɜɨɣ
Ɍ ɚ ɛ ɥ ɢ ɰ ɚ
ɉɚɪɚɦɟɬɪɵ ɈɐɄɉ ɞɥɹ ɬɪɟɯ ɮɚɤɬɨɪɨɜ
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ɉɨ ɪɟɡɭɥɶɬɚɬɚɦ ɨɩɵɬɨɜ ɩɥɚɧɚ ɮɨɪɦɢɪɭɟɬɫɹ ɩɨɥɢɧɨɦ
y b b x b x b x b x x b x x b x x b x x xb x a b x a b x a
Ʉɨɷɮɮɢɰɢɟɧɬɵ ɩɨɥɢɧɨɦɚb b b b ɨɩɪɟɞɟɥɹɸɬɫɹ ɤɚɤ
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ɝɞɟbc b b a b a b a
Ɋɚɫɫɦɨɬɪɢɦ ɩɨɞɪɨɛɧɨ ɩɪɨɫɬɨɣ ɩɪɢɦɟɪ ɜ ɤɨɬɨɪɨɦ ɯɨɪɨɲɨ ɜɢɞɟɧ ɩɨɫɥɟɞɨ
ɜɚɬɟɥɶɧɵɣ ɩɪɢɧɰɢɩ ɩɥɚɧɢɪɨɜɚɧɢɹ ɷɤɫɩɟɪɢɦɟɧɬɚ
ɉɭɫɬɶ ɬɪɟɛɭɟɬɫɹ ɩɨɫɬɪɨɢɬɶ ɩɨ ɪɟɡɭɥɶɬɚɬɚɦ ɢɡɦɟɪɟɧɢɣ ɦɨɞɟɥɶ ɜ ɜɢɞɟ ɡɚɜɢ
ɫɢɦɨɫɬɢ ɬɨɤɚ ɨɬ ɞɜɭɯ ɚɪɝɭɦɟɧɬɨɜ ɩɪɢɥɨɠɟɧɧɨɝɨ ɧɚɩɪɹɠɟɧɢɹ ɢ ɚɤɬɢɜɧɨɝɨ ɫɨ ɩɪɨɬɢɜɥɟɧɢɹ ɰɟɩɢI f U R
ȼɵɛɟɪɟɦ ɞɢɚɩɚɡɨɧɵ ɜɚɪɶɢɪɨɜɚɧɢɹ ɮɚɤɬɨɪɨɜ
ɇɚɩɪɹɠɟɧɢɟU ȼ ɞɟɣɫɬɜɭɸɳɟɟ ɡɧɚɱɟɧɢɟ ɧɚɩɪɹɠɟɧɢɹ
Ⱥɤɬɢɜɧɨɟ ɫɨɩɪɨɬɢɜɥɟɧɢɟR |
Ɉɦ |
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ȼ ɤɚɱɟɫɬɜɟ ɢɡɦɟɪɟɧɧɵɯ ɡɧɚɱɟɧɢɣ ɬɨɤɚ ɜɨɡɶɦɟɦ ɡɧɚɱɟɧɢɹ ɪɚɫɫɱɢɬɚɧɧɵɟ ɩɨ |
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ɢɡɜɟɫɬɧɨɣ ɬɟɨɪɟɬɢɱɟɫɤɨɣ ɦɨɞɟɥɢ I |
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ɋɨɫɬɚɜɢɦ ɦɚɬɪɢɰɭ ɩɥɚɧɚ ɉɎɗ |
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R |
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ɜ ɤɨɬɨɪɭɸ ɡɚɩɢɲɟɦ ɬɚɤɠɟ ɢɡɦɟɪɟɧɧɵɟ ɡɧɚɱɟɧɢɹ ɢ ɪɚɫɫɱɢɬɚɧɧɵɟ ɡɧɚɱɟɧɢɹ ɛɟɡ
ɭɱɟɬɚ ɢ ɫ ɭɱɟɬɨɦ ɞɜɨɣɧɵɯ ɫɨɱɟɬɚɧɢɣ ɮɚɤɬɨɪɨɜ ɬɚɛɥɢɰɚ
ɇɚɣɞɟɦ ɤɨɷɮɮɢɰɢɟɧɬɵ ɦɨɞɟɥɢ
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ȼɟɥɢɱɢɧɚ ɤɨɷɮɮɢɰɢɟɧɬɚb ɞɨɥɠɧɚ ɫɨɜɩɚɞɚɬɶ ɫɨ ɡɧɚɱɟɧɢɟɦ ɬɨɤɚ ɜ ɧɭɥɟɜɨɣ ɬɨɱɤɟ ɬɨ ɟɫɬɶ ɫɨ ɡɧɚɱɟɧɢɟɦ ɬɨɤɚ ɩɪɢ ɧɚɩɪɹɠɟɧɢɢ ȼ ɢ ɫɨɩɪɨɬɢɜɥɟɧɢɢ Ɉɦ
Ɍ ɚ ɛ ɥ ɢ ɰ ɚ
Ɋɚɫɲɢɪɟɧɧɚɹ ɦɚɬɪɢɰɚ ɩɥɚɧɚ ɉɎɗ
ɫ ɪɟɡɭɥɶɬɚɬɚɦɢ ɢɡɦɟɪɟɧɢɣ ɢ ɪɚɫɱɟɬɨɜ ɩɨ ɦɨɞɟɥɢ
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Ɋɚɫɫɱɢɬɚɧɧɵɟ ɡɧɚɱɟɧɢɹ ɬɨɤɚ |
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ɂɡɦɟɪɟɧɧɵɟ |
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ɩɨ ɦɨɞɟɥɢ |
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ɛɟɡ ɭɱɟɬɚ |
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ɫ ɭɱɟɬɨɦ ɞɜɨɣɧɨɝɨ |
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ɇɨɦɟɪ |
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ɞɜɨɣɧɨɝɨ |
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ɨɩɵɬɚ |
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ɫɨɱɟɬɚɧɢɹ |
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ɫɨɱɟɬɚɧɢɹ |
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y >Ⱥ@ |
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ɮɚɤɬɨɪɨɜ |
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ɮɚɤɬɨɪɨɜ |
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ycc >Ⱥ@ |
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yc >Ⱥ@ |
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ɉɪɨɜɟɪɹɟɦ |
ɬɨɤ |
ɩɪɢ |
ɧɭɥɟɜɵɯ |
ɩɪɢɜɟɞɟɧɧɵɯ |
ɡɧɚɱɟɧɢɹɯ ɮɚɤɬɨɪɨɜ |
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I
ɞɨɫɬɚɬɨɱɧɨ ɜɟɥɢɤɨ ȼɟɥɢɱɢɧɚ ɤɨɷɮɮɢɰɢɟɧɬɚb ɨɬɥɢɱɚɟɬɫɹ ɧɚ ɨɬ ɢɡɦɟ ɪɟɧɧɨɝɨ ɢɞɟɚɥɶɧɨɝɨ ɡɧɚɱɟɧɢɹ
ɉɨɥɨɠɢɬɟɥɶɧɵɣ ɡɧɚɤ ɤɨɷɮɮɢɰɢɟɧɬɚb ɝɨɜɨɪɢɬ ɨ ɬɨɦ ɱɬɨ ɫ ɪɨɫɬɨɦ ɮɚɤɬɨ ɪɚ x ɧɚɩɪɹɠɟɧɢɹɨɬɤɥɢɤy ɬɨɤɪɚɫɬɟɬɨɬɪɢɰɚɬɟɥɶɧɵɣɡɧɚɤɤɨɷɮɮɢɰɢɟɧ ɬɚb ± ɧɚɨɛɨɪɨɬ ɫ ɪɨɫɬɨɦ x ɫɨɩɪɨɬɢɜɥɟɧɢɹyɭɦɟɧɶɲɚɟɬɫɹ Ɉɬɪɢɰɚɬɟɥɶɧɵɣ ɡɧɚɤ ɤɨɷɮɮɢɰɢɟɧɬɚb ɨɡɧɚɱɚɟɬ ɱɬɨ ɩɪɢ ɨɞɧɨɜɪɟɦɟɧɧɨɦ ɞɟɣɫɬɜɢɢ ɞɜɭɯ ɮɚɤɬɨ ɪɨɜ ɨɬɤɥɢɤ ɭɦɟɧɶɲɚɟɬɫɹ ɩɪɢ ɷɬɨɦ ɜɥɢɹɧɢɟ ɤɚɠɞɨɝɨ ɮɚɤɬɨɪɚ ɜ ɨɬɞɟɥɶɧɨɫɬɢ ɭɦɟɧɶɲɚɟɬɫɹ ɢ ɜɬɨɪɨɣ ɮɚɤɬɨɪ x ɨɤɚɡɵɜɚɟɬ ɛɨɥɶɲɟɟ ɜɥɢɹɧɢɟ ɧɚ ɨɬɤɥɢɤ ɱɟɦ ɩɟɪɜɵɣ ɮɚɤɬɨɪx