смод

ɆɂɇɂɋɌȿɊɋɌȼɈ ɈȻɓȿȽɈ ɂ ɉɊɈɎȿɋɋɂɈɇȺɅɖɇɈȽɈ ɈȻɊȺɁɈȼȺɇɂə ɊɈɋɋɂɃɋɄɈɃ ɎȿȾȿɊȺɐɂɂ

ɌȺȽȺɇɊɈȽɋɄɂɃ ȽɈɋɍȾȺɊɋɌȼȿɇɇɕɃ ɊȺȾɂɈɌȿɏɇɂɑȿɋɄɂɃ ɍɇɂȼȿɊɋɂɌȿɌ

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ȼ.ɉ.Ɏɟɞɨɫɨɜ

ɉɊɂɄɅȺȾɇɕȿ ɆȺɌȿɆȺɌɂɑȿɋɄɂȿ ɆȿɌɈȾɕ ȼ ɋɌȺɌɂɋɌɂɑȿɋɄɈɃ ɊȺȾɂɈɌȿɏɇɂɄȿ

ɍɱɟɛɧɨɟ ɩɨɫɨɛɢɟ

Ɍɚɝɚɧɪɨɝ 1998

ɍȾɄ 621.37.01 (075.8)

Ɏɟɞɨɫɨɜ ȼ.ɉ. ɉɪɢɤɥɚɞɧɵɟ ɦɚɬɟɦɚɬɢɱɟɫɤɢɟ ɦɟɬɨɞɵ ɜ ɫɬɚɬɢɫɬɢɱɟɫɤɨɣ ɪɚɞɢɨɬɟɯɧɢɤɟ. ɍɱɟɛɧɨɟ ɩɨɫɨɛɢɟ.-Ɍɚɝɚɧɪɨɝ: ɂɡɞ-ɜɨ ɌɊɌɍ, 1998. 74 c.

ɉɪɢɜɨɞɹɬɫɹ ɤɪɚɬɤɢɟ ɫɜɟɞɟɧɢɹ ɨ ɫɥɭɱɚɣɧɵɯ ɜɟɥɢɱɢɧɚɯ ɢ ɫɥɭɱɚɣɧɵɯ ɩɪɨɰɟɫɫɚɯ ɜ ɫɬɚɬɢɫɬɢɱɟɫɤɨɣ ɪɚɞɢɨɬɟɯɧɢɤɟ. ɂɡɥɚɝɚɸɬɫɹ ɨɫɧɨɜɵ ɬɟɨɪɢɢ ɫɬɚɬɢɫɬɢɱɟɫɤɢɯ ɝɢɩɨɬɟɡ, ɬɟɨɪɢɢ ɨɰɟɧɨɤ ɩɪɢɦɟɧɢɬɟɥɶɧɨ ɤ ɫɢɧɬɟɡɭ ɨɩɬɢɦɚɥɶɧɵɯ ɚɥɝɨɪɢɬɦɨɜ ɢɡɦɟɪɟɧɢɹ ɷɧɟɪɝɟɬɢɱɟɫɤɢɯ ɢ ɧɟɷɧɟɪɝɟɬɢɱɟɫɤɢɯ ɩɚɪɚɦɟɬɪɨɜ ɷɯɨɫɢɝɧɚɥɨɜ ɜ ɥɨɤɚɰɢɢ. Ⱦɚɧɵ ɩɨɧɹɬɢɹ ɢɧɬɟɪɜɚɥɶɧɵɯ ɨɰɟɧɨɤ ɢ ɤɪɢɬɟɪɢɟɜ ɫɨɝɥɚɫɢɹ, ɦɟɬɨɞɵ ɮɢɥɶɬɪɚɰɢɢ ɫɥɭɱɚɣɧɵɯ ɩɪɨɰɟɫɫɨɜ ɢ ɷɥɟɦɟɧɬɵ ɬɟɨɪɢɢ ɤɥɚɫɫɢɮɢɤɚɰɢɢ. ɍɱɟɛɧɨɟ ɩɨɫɨɛɢɟ ɩɪɟɞɧɚɡɧɚɱɟɧɨ ɞɥɹ ɫɬɭɞɟɧɬɨɜ, ɨɛɭɱɚɸɳɢɯɫɹ ɩɨ ɫɩɟɰɢɚɥɶɧɨɫɬɹɦ ɧɚɩɪɚɜɥɟɧɢɹ "Ɋɚɞɢɨɬɟɯɧɢɤɚ".

Ɍɚɛɥ.2. ɂɥ.6. Ȼɢɛɥɢɨɝɪ.: 9 ɧɚɡɜ.

ɉɟɱɚɬɚɟɬɫɹ ɩɨ ɪɟɲɟɧɢɸ ɪɟɞɚɤɰɢɨɧɧɨ-ɢɡɞɚɬɟɥɶɫɤɨɝɨ ɫɨɜɟɬɚ Ɍɚɝɚɧɪɨɝɫɤɨɝɨ ɝɨɫɭɞɚɪɫɬɜɟɧɧɨɝɨ ɪɚɞɢɨɬɟɯɧɢɱɟɫɤɨɝɨ ɭɧɢɜɟɪɫɢɬɟɬɚ.

Ɋ ɟ ɰ ɟ ɧ ɡ ɟ ɧ ɬ ɵ:

ȼɵɱɢɫɥɢɬɟɥɶɧɵɣ ɰɟɧɬɪ Ɇɨɫɤɨɜɫɤɨɣ ɝɨɫɭɞɚɪɫɬɜɟɧɧɨɣ ɤɨɧɫɟɪɜɚɬɨɪɢɢ, ɡɚɜ. ȼɐ Ⱥ.ȼ. ɏɚɪɭɬɨ, ɤɚɧɞ. ɬɟɯɧ. ɧɚɭɤ.

ȼ.ɂ. Ɇɚɪɱɭɤ, ɤɚɧɞ. ɬɟɯɧ. ɧɚɭɤ, ɞɨɰɟɧɬ ɤɚɮɟɞɪɵ “Ɋɚɞɢɨɬɟɯɧɢɤɚ” Ⱦɨɧɫɤɨɣ ɝɨɫɭɞɚɪɫɬɜɟɧɧɨɣ ɚɤɚɞɟɦɢɢ ɫɟɪɜɢɫɚ.

¤Ɍɚɝɚɧɪɨɝɫɤɢɣ ɝɨɫɭɞɚɪɫɬɜɟɧɧɵɣ ɪɚɞɢɨɬɟɯɧɢɱɟɫɤɢɣ ɭɧɢɜɟɪɫɢɬɟɬ, 1998.

¤Ɏɟɞɨɫɨɜ ȼ.ɉ., 1998.

ɉ Ɋ ȿ Ⱦ ɂ ɋ Ʌ Ɉ ȼ ɂ ȿ

ɍɱɟɛɧɨɟ ɩɨɫɨɛɢɟ ɩɪɟɞɧɚɡɧɚɱɟɧɨ ɞɥɹ ɫɬɭɞɟɧɬɨɜ ɞɧɟɜɧɵɯ ɮɚɤɭɥɶɬɟɬɨɜ, ɢɡɭɱɚɸɳɢɯ ɢɡɛɪɚɧɧɵɟ ɪɚɡɞɟɥɵ ɦɚɬɟɦɚɬɢɤɢ "ɉɪɢɤɥɚɞɧɵɟ ɦɚɬɟɦɚɬɢɱɟɫɤɢɟ ɦɟɬɨɞɵ ɜ ɫɬɚɬɢɫɬɢɱɟɫɤɨɣ ɪɚɞɢɨɬɟɯɧɢɤɟ".

ȼ ɧɚɱɚɥɟ ɭɱɟɛɧɨɝɨ ɩɨɫɨɛɢɹ ɩɪɢɜɨɞɹɬɫɹ ɤɪɚɬɤɢɟ ɫɜɟɞɟɧɢɹ ɨ ɫɥɭɱɚɣɧɵɯ ɜɟɥɢɱɢɧɚɯ ɢ ɩɪɨɰɟɫɫɚɯ, ɱɬɨ ɩɨɡɜɨɥɹɟɬ ɫɬɭɞɟɧɬɚɦ ɜɨɫɫɬɚɧɨɜɢɬɶ ɜ ɩɚɦɹɬɢ ɦɚɬɟɪɢɚɥ, ɢɡɭɱɚɜɲɢɣɫɹ ɢɦɢ ɪɚɧɟɟ. Ɂɚɬɟɦ ɞɚɸɬɫɹ ɨɛɳɢɟ ɫɜɟɞɟɧɢɹ ɨɛ ɨɰɟɧɤɟ ɜɟɪɨɹɬɧɨɫɬɧɵɯ ɯɚɪɚɤɬɟɪɢɫɬɢɤ ɫɥɭɱɚɣɧɵɯ ɩɪɨɰɟɫɫɨɜ ɤɚɤ ɬɪɚɞɢɰɢɨɧɧɵɦɢ ɫɩɨɫɨɛɚɦɢ, ɬɚɤ ɢ ɦɟɬɨɞɚɦɢ ɷɤɫɩɪɟɫɫ-ɚɧɚɥɢɡɚ.

ɂɡɥɚɝɚɸɬɫɹ ɨɫɧɨɜɵ ɬɟɨɪɢɢ ɫɬɚɬɢɫɬɢɱɟɫɤɢɯ ɪɟɲɟɧɢɣ ɤɚɤ ɜ ɱɚɫɬɢ ɫɬɚɬɢɫɬɢɱɟɫɤɢɯ ɝɢɩɨɬɟɡ, ɬɚɤ ɢ ɨɰɟɧɨɤ ɩɪɢɦɟɧɢɬɟɥɶɧɨ ɤ ɡɚɞɚɱɚɦ ɫɬɚɬɢɫɬɢɱɟɫɤɨɣ ɪɚɞɢɨɬɟɯɧɢɤɢ.

Ⱦɚɧɵ ɩɪɢɦɟɪɵ ɫɢɧɬɟɡɚ ɨɩɬɢɦɚɥɶɧɵɯ ɚɥɝɨɪɢɬɦɨɜ ɨɛɧɚɪɭɠɟɧɢɹ ɢ ɨɰɟɧɤɢ ɩɚɪɚɦɟɬɪɨɜ ɫɢɝɧɚɥɨɜ ɧɚ ɮɨɧɟ ɲɭɦɨɜ. Ɋɚɫɫɦɨɬɪɟɧɵ ɪɚɡɥɢɱɧɵɟ ɜɚɪɢɚɧɬɵ ɩɪɟɨɞɨɥɟɧɢɹ ɚɩɪɢɨɪɧɨɣ ɧɟɨɩɪɟɞɟɥɟɧɧɨɫɬɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɩɚɪɚɦɟɬɪɨɜ ɢ ɯɚɪɚɤɬɟɪɢɫɬɢɤ ɫɢɝɧɚɥɨɜ ɢ ɩɨɦɟɯ.

Ɋɚɫɫɦɨɬɪɟɧɵ ɢɧɬɟɪɜɚɥɶɧɵɟ ɨɰɟɧɤɢ ɢ ɤɪɢɬɟɪɢɢ ɫɨɝɥɚɫɢɹ, ɢɡɥɚɝɚɸɬɫɹ ɷɥɟɦɟɧɬɵ ɬɟɨɪɢɢ ɤɥɚɫɫɢɮɢɤɚɰɢɢ.

1. ȼȼȿȾȿɇɂȿ ȼ ɌȿɈɊɂɘ ȼȿɊɈəɌɇɈɋɌɂ

1.1. Ɉɫɧɨɜɧɵɟ ɩɨɧɹɬɢɹ ɬɟɨɪɢɢ ɜɟɪɨɹɬɧɨɫɬɢ

ɉɨɥɧɚɹ ɝɪɭɩɩɚ ɫɨɛɵɬɢɣ: ɧɟɫɤɨɥɶɤɨ ɫɨɛɵɬɢɣ ɨɛɪɚɡɭɸɬ ɩɨɥɧɭɸ ɝɪɭɩɩɭ, ɟɫɥɢ ɜ ɪɟɡɭɥɶɬɚɬɟ ɨɩɵɬɚ ɧɟɩɪɟɦɟɧɧɨ ɞɨɥɠɧɨ ɩɨɹɜɢɬɶɫɹ ɯɨɬɹ ɛɵ ɨɞɧɨ ɢɯ ɧɢɯ.

ɇɟɫɨɜɦɟɫɬɧɵɟ ɫɨɛɵɬɢɹ: ɧɟɫɤɨɥɶɤɨ ɫɨɛɵɬɢɣ ɧɚɡɵɜɚɸɬɫɹ ɧɟɫɨɜɦɟɫɬɧɵɦɢ ɜ ɞɚɧɧɨɦ ɨɩɵɬɟ, ɟɫɥɢ ɧɢɤɚɤɢɟ ɞɜɚ ɢɡ ɧɢɯ ɧɟ ɦɨɝɭɬ ɩɨɹɜɢɬɶɫɹ ɜɦɟɫɬɟ.

Ɋɚɜɧɨɜɨɡɦɨɠɧɵɟ ɫɨɛɵɬɢɹ: ɧɟɫɤɨɥɶɤɨ ɫɨɛɵɬɢɣ ɧɚɡɵɜɚɸɬɫɹ ɪɚɜɧɨɜɨɡɦɨɠɧɵɦɢ, ɟɫɥɢ ɟɫɬɶ ɨɫɧɨɜɚɧɢɟ ɫɱɢɬɚɬɶ, ɱɬɨ ɧɢ ɨɞɧɨ ɢɡ ɧɢɯ ɧɟ ɹɜɥɹɟɬɫɹ ɩɪɟɞɩɨɱɬɢɬɟɥɶɧɵɦ ɩɨ ɫɪɚɜɧɟɧɢɸ ɫ ɞɪɭɝɢɦ.

ɑɚɫɬɨɬɚ ɫɨɛɵɬɢɹ: ɟɫɥɢ ɩɪɨɢɡɜɨɞɢɬɫɹ ɫɟɪɢɹ ɢɡ N ɨɩɵɬɨɜ, ɜ ɤɚɠɞɨɣ ɢɡ ɤɨɬɨɪɵɯ ɦɨɝɥɨ ɩɨɹɜɢɬɶɫɹ ɢɥɢ ɧɟ ɩɨɹɜɢɬɶɫɹ ɧɟɤɨɬɨɪɨɟ ɫɨɛɵɬɢɟ Ⱥ, ɬɨ ɱɚɫɬɨɬɨɣ ɫɨɛɵɬɢɹ Ⱥ ɜ ɞɚɧɧɨɣ ɫɟɪɢɢ ɨɩɵɬɨɜ ɧɚɡɵɜɚɟɬɫɹ ɨɬɧɨɲɟɧɢɟ ɱɢɫɥɚ ɨɩɵɬɨɜ, ɜ ɤɨɬɨɪɵɯ ɩɨɹɜɢɥɨɫɶ ɫɨɛɵɬɢɟ Ⱥ, ɤ ɨɛɳɟɦɭɱɢɫɥɭɩɪɨɢɡɜɟɞɟɧɧɵɯ ɨɩɵɬɨɜ.

ɑɚɫɬɨɬɭ ɫɨɛɵɬɢɹ ɱɚɫɬɨ ɧɚɡɵɜɚɸɬ ɫɬɚɬɢɫɬɢɱɟɫɤɨɣ ɜɟɪɨɹɬɧɨɫɬɶɸ ɢ ɜɵɱɢɫɥɹɸɬ ɧɚ ɨɫɧɨɜɚɧɢɢ ɪɟɡɭɥɶɬɚɬɨɜ ɨɩɵɬɚ ɩɨ ɮɨɪɦɭɥɟ P*(Ⱥ)=m/N, ɝɞɟ m-ɱɢɫɥɨ ɩɨɹɜɥɟɧɢɣ ɫɨɛɵɬɢɹ Ⱥ.

ɋɥɟɞɭɟɬ ɨɬɦɟɬɢɬɶ, ɱɬɨ ɩɪɢ ɧɟɛɨɥɶɲɨɦ ɱɢɫɥɟ ɨɩɵɬɨɜ N ɱɚɫɬɨɬɚ ɦɨɠɟɬ ɦɟɧɹɬɶɫɹ ɨɬ ɨɞɧɨɣ ɫɟɪɢɢ ɨɩɵɬɨɜ ɤ ɞɪɭɝɨɣ ɢɡ-ɡɚ ɫɥɭɱɚɣɧɨɫɬɢ ɫɨɛɵɬɢɣ. Ɉɞɧɚɤɨ ɩɪɢ

ɛɨɥɶɲɨɦ ɱɢɫɥɟ ɨɩɵɬɨɜ ɨɧɚ ɧɨɫɢɬ ɭɫɬɨɣɱɢɜɵɣ ɯɚɪɚɤɬɟɪ ɢ ɫɬɪɟɦɢɬɫɹ ɤ ɡɧɚɱɟɧɢɸ,

ɤɨɬɨɪɨɟ ɧɚɡɵɜɚɸɬ ɜɟɪɨɹɬɧɨɫɬɶɸ ɫɨɛɵɬɢɹ.

1.2. ɋɥɭɱɚɣɧɚɹ ɜɟɥɢɱɢɧɚ

(ɋȼ) ɧɚɡɵɜɚɟɬɫɹ ɜɟɥɢɱɢɧɚ, ɤɨɬɨɪɚɹ ɜ ɪɟɡɭɥɶɬɚɬɟ ɨɩɵɬɚ ɦɨɠɟɬ ɩɪɢɧɹɬɶ ɬɨ ɢɥɢ ɢɧɨɟ ɡɧɚɱɟɧɢɟ, ɩɪɢɱɟɦ ɧɟɢɡɜɟɫɬɧɨ ɡɚɪɚɧɟɟ, ɤɚɤɨɟ ɢɦɟɧɧɨ.

ɉɊɂɆȿɊɕ: ɱɢɫɥɨ ɩɨɩɚɞɚɧɢɣ ɩɪɢ ɨɝɪɚɧɢɱɟɧɧɨɦ ɱɢɫɥɟ ɜɵɫɬɪɟɥɨɜ; ɱɢɫɥɨ ɜɵɡɨɜɨɜ ɩɨ ɬɟɥɟɮɨɧɭ ɜ ɟɞɢɧɢɰɭ ɜɪɟɦɟɧɢ, ɤɨɥɢɱɟɫɬɜɨ ɧɟɤɨɧɞɢɰɢɨɧɧɵɯ ɬɪɚɧɡɢɫɬɨɪɨɜ ɜ ɩɚɪɬɢɢ ɜɵɩɭɫɤɚɟɦɵɯ ɢɡɞɟɥɢɣ ɢ ɬ.ɞ.

ɋɥɭɱɚɣɧɵɟ ɜɟɥɢɱɢɧɵ, ɩɪɢɧɢɦɚɸɳɢɟ ɬɨɥɶɤɨ ɨɬɞɟɥɶɧɵɟ ɡɧɚɱɟɧɢɹ, ɤɨɬɨɪɵɟ ɦɨɠɧɨ ɩɟɪɟɱɢɫɥɢɬɶ, ɧɚɡɵɜɚɸɬɫɹ ɞɢɫɤɪɟɬɧɵɦɢ ɫɥɭɱɚɣɧɵɦɢ ɜɟɥɢɱɢɧɚɦɢ.

ɋɭɳɟɫɬɜɭɸɬ ɋȼ ɞɪɭɝɨɝɨ ɬɢɩɚ: ɡɧɚɱɟɧɢɟ ɲɭɦɨɜɨɝɨ ɞɚɜɥɟɧɢɹ, ɢɡɦɟɪɟɧɧɨɟ ɜ ɪɚɡɥɢɱɧɵɟ ɦɨɦɟɧɬɵ ɜɪɟɦɟɧɢ; ɜɟɫ ɛɭɥɤɢ ɯɥɟɛɚ, ɩɪɨɞɚɜɚɟɦɨɝɨ ɜ ɦɚɝɚɡɢɧɟ ɢ ɬ.ɞ.

ɇɚɡɵɜɚɸɬ ɢɯ ɧɟɩɪɟɪɵɜɧɵɦɢ ɫɥɭɱɚɣɧɵɦɢ ɜɟɥɢɱɢɧɚɦɢ.

2. ɈɋɇɈȼɇɕȿ ɌȿɈɊȿɆɕ ɌȿɈɊɂɂ ȼȿɊɈəɌɇɈɋɌɂ

2.1.ɋɭɦɦɚ ɢ ɩɪɨɢɡɜɟɞɟɧɢɟ ɫɨɛɵɬɢɣ

ɋɭɦɦɨɣ ɞɜɭɯ ɫɨɛɵɬɢɣ Ⱥ ɢ Ȼ ɧɚɡɵɜɚɟɬɫɹ ɫɨɛɵɬɢɟ ɋ, ɫɨɫɬɨɹɳɟɟ ɜ ɜɵɩɨɥɧɟɧɢɢ ɫɨɛɵɬɢɹ Ⱥ ɢɥɢ ɫɨɛɵɬɢɹ Ȼ ɢɥɢ ɨɛɨɢɯ ɜɦɟɫɬɟ.

ɇɚɩɪɢɦɟɪ, ɟɫɥɢ ɫɨɛɵɬɢɟ A - ɩɨɩɚɞɚɧɢɟ ɜ ɦɢɲɟɧɶ ɩɪɢ ɩɟɪɜɨɦ ɜɵɫɬɪɟɥɟ, ɫɨɛɵɬɢɟ Ȼ - ɩɨɩɚɞɚɧɢɟ ɜ ɦɢɲɟɧɶ ɩɪɢ ɜɬɨɪɨɦ ɜɵɫɬɪɟɥɟ, ɬɨ ɫɨɛɵɬɢɟ ɋ=Ⱥ+Ȼ ɟɫɬɶ ɩɨɩɚɞɚɧɢɟ ɜ ɦɢɲɟɧɶ ɜɨɨɛɳɟ ɛɟɡɪɚɡɥɢɱɧɨ ɩɪɢ ɤɚɤɨɦ ɜɵɫɬɪɟɥɟ - ɩɪɢ ɩɟɪɜɨɦ, ɩɪɢ ɜɬɨɪɨɦ ɢɥɢ ɩɪɢ ɨɛɨɢɯ ɜɦɟɫɬɟ.

ɋɭɦɦɨɣ ɧɟɫɤɨɥɶɤɢɯ ɫɨɛɵɬɢɣ ɧɚɡɵɜɚɟɬɫɹ ɫɨɛɵɬɢɟ, ɫɨɫɬɨɹɳɟɟ ɜ ɩɨɹɜɥɟɧɢɢ ɯɨɬɹ ɛɵ ɨɞɧɨɝɨ ɢɡ ɷɬɢɯ ɫɨɛɵɬɢɣ.

ɉɪɨɢɡɜɟɞɟɧɢɟɦ ɞɜɭɯ ɫɨɛɵɬɢɣ Ⱥ ɢ Ȼ ɧɚɡɵɜɚɟɬɫɹ ɫɨɛɵɬɢɟ ɋ, ɫɨɫɬɨɹɳɟɟ ɜ ɫɨɜɦɟɫɬɧɨɦ ɜɵɩɨɥɧɟɧɢɢ ɫɨɛɵɬɢɹ Ⱥ ɢ ɫɨɛɵɬɢɹ Ȼ.

ȿɫɥɢ ɩɪɨɢɡɜɨɞɢɬɫɹ ɞɜɚ ɜɵɫɬɪɟɥɚ ɩɨ ɦɢɲɟɧɢ ɢ ɟɫɥɢ ɫɨɛɵɬɢɟ Ⱥ ɟɫɬɶ ɩɨɩɚɞɚɧɢɟ ɩɪɢ ɩɟɪɜɨɦ ɜɵɫɬɪɟɥɟ, ɚ ɫɨɛɵɬɢɟ Ȼ - ɩɨɩɚɞɚɧɢɟ ɩɪɢ ɜɬɨɪɨɦ ɜɵɫɬɪɟɥɟ, ɬɨ ɋ = ȺȻ ɟɫɬɶ ɩɨɩɚɞɚɧɢɟ ɩɪɢ ɨɛɨɢɯ ɜɵɫɬɪɟɥɚɯ.

ɉɪɨɢɡɜɟɞɟɧɢɟɦ ɧɟɫɤɨɥɶɤɢɯ ɫɨɛɵɬɢɣ ɧɚɡɵɜɚɟɬɫɹ ɫɨɛɵɬɢɟ, ɫɨɫɬɨɹɳɟɟ ɜ ɫɨ-

ɜɦɟɫɬɧɨɦ ɩɨɹɜɥɟɧɢɢ ɜɫɟɯ ɷɬɢɯ ɫɨɛɵɬɢɣ.

2.2. Ɍɟɨɪɟɦɚ ɫɥɨɠɟɧɢɹ ɜɟɪɨɹɬɧɨɫɬɟɣ

ȼɟɪɨɹɬɧɨɫɬɶ ɫɭɦɦɵ ɞɜɭɯ ɧɟɫɨɜɦɟɫɬɧɵɯ ɫɨɛɵɬɢɣ ɪɚɜɧɚ ɫɭɦɦɟ ɜɟɪɨɹɬɧɨɫɬɟɣ ɷɬɢɯ ɫɨɛɵɬɢɣ : P(Ⱥ+Ȼ)=P(Ⱥ)+P(Ȼ).

ɉɭɫɬɶ ɜɨɡɦɨɠɧɵɟ ɢɫɯɨɞɵ ɨɩɵɬɚ ɫɜɨɞɹɬɫɹ ɤ ɫɨɜɨɤɭɩɧɨɫɬɢ ɫɥɭɱɚɟɜ, ɤɨɬɨɪɵɟ ɦɵ ɞɥɹ ɧɚɝɥɹɞɧɨɫɬɢ ɩɪɟɞɫɬɚɜɥɟɧɵ ɜ ɜɢɞɟ N ɫɢɦɜɨɥɨɜ ɧɚ ɪɢɫ.2.1.

Ɋɢɫ.2.1

ɉɪɟɞɩɨɥɨɠɢɦ, ɱɬɨ ɢɡ ɷɬɢɯ ɫɥɭɱɚɟɜ m ɛɥɚɝɨɩɪɢɹɬɧɵ ɫɨɛɵɬɢɸ Ⱥ, ɚ k - ɫɨɛɵɬɢɸ Ȼ. Ɍɨɝɞɚ

P(A)=m/N; P(Ȼ)=k/N.

Ɍɚɤ ɤɚɤ ɫɨɛɵɬɢɹ Ⱥ ɢ Ȼ ɧɟɫɨɜɦɟɫɬɧɵ, ɬɨ ɧɟɬ ɫɥɭɱɚɟɜ, ɤɨɬɨɪɵɟ ɛɥɚɝɨɩɪɢɹɬɧɵ ɫɨɛɵɬɢɹɦ ɢ Ⱥ ɢ Ȼ ɜɦɟɫɬɟ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɫɨɛɵɬɢɸ Ⱥ+Ȼ ɛɥɚɝɨɩɪɢɹɬɧɵ m+k ɫɥɭɱɚɟɜ ɢ

P(A+Ȼ)=(m+k)/N.

ɉɨɞɫɬɚɜɥɹɹ ɩɨɥɭɱɟɧɧɵɟ ɜɵɪɚɠɟɧɢɹ ɜ ɮɨɪɦɭɥɭ ɞɥɹ ɜɟɪɨɹɬɧɨɫɬɢ ɫɭɦɦɵ ɞɜɭɯ ɫɨɛɵɬɢɣ, ɩɨɥɭɱɢɦ ɬɨɠɞɟɫɬɜɨ. Ɍɟɨɪɟɦɚ ɞɨɤɚɡɚɧɚ.

ɋɥɟɞɫɬɜɢɟ 1. ȿɫɥɢ ɫɨɛɵɬɢɹ A1,A2,...,AN ɨɛɪɚɡɭɸɬ ɩɨɥɧɭɸ ɝɪɭɩɩɭ ɧɟɫɨɜɦɟɫɬɧɵɯ ɫɨɛɵɬɢɣ, ɬɨ ɫɭɦɦɚ ɢɯ ɜɟɪɨɹɬɧɨɫɬɟɣ ɪɚɜɧɚ ɟɞɢɧɢɰɟ.

ɋɥɟɞɫɬɜɢɟ 2. ɋɭɦɦɚ ɜɟɪɨɹɬɧɨɫɬɟɣ ɩɪɨɬɢɜɨɩɨɥɨɠɧɵɯ ɫɨɛɵɬɢɣ ɪɚɜɧɚ ɟɞɢɧɢɰɟ.

2.3. Ɍɟɨɪɟɦɚ ɭɦɧɨɠɟɧɢɹ ɜɟɪɨɹɬɧɨɫɬɟɣ

ɉɟɪɟɞ ɬɟɨɪɟɦɨɣ ɧɟɨɛɯɨɞɢɦɨ ɜɜɟɫɬɢ ɩɨɧɹɬɢɹ ɧɟɡɚɜɢɫɢɦɵɯ ɢ ɡɚɜɢɫɢɦɵɯ ɫɨɛɵɬɢɣ.

ɋɨɛɵɬɢɟ Ⱥ ɧɚɡɵɜɚɟɬɫɹ ɧɟɡɚɜɢɫɢɦɵɦ ɨɬ ɫɨɛɵɬɢɹ Ȼ, ɟɫɥɢ ɜɟɪɨɹɬɧɨɫɬɶ ɫɨɛɵɬɢɹ Ⱥ ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɬɨɝɨ, ɩɪɨɢɡɨɲɥɨ ɫɨɛɵɬɢɟ Ȼ ɢɥɢ ɧɟɬ.

ɋɨɛɵɬɢɟ Ⱥ ɧɚɡɵɜɚɟɬɫɹ ɡɚɜɢɫɢɦɵɦ ɨɬ ɫɨɛɵɬɢɹ Ȼ, ɟɫɥɢ ɜɟɪɨɹɬɧɨɫɬɶ ɫɨɛɵɬɢɹ Ⱥ ɦɟɧɹɟɬɫɹ ɨɬ ɬɨɝɨ, ɩɪɨɢɡɨɲɥɨ ɫɨɛɵɬɢɟ Ȼ ɢɥɢ ɧɟɬ.

ȼɟɪɨɹɬɧɨɫɬɶ ɫɨɛɵɬɢɹ Ⱥ, ɜɵɱɢɫɥɟɧɧɚɹ ɩɪɢ ɭɫɥɨɜɢɢ, ɱɬɨ ɢɦɟɥɨ ɦɟɫɬɨ ɞɪɭɝɨɟ ɫɨɛɵɬɢɟ ȼ, ɧɚɡɵɜɚɟɬɫɹ ɭɫɥɨɜɧɨɣ ɜɟɪɨɹɬɧɨɫɬɶɸ ɫɨɛɵɬɢɹ Ⱥ ɢ ɨɛɨɡɧɚɱɚɟɬɫɹ

Ɋ(Ⱥ/ȼ).

Ɍɟɨɪɟɦɚ ɭɦɧɨɠɟɧɢɹ ɜɟɪɨɹɬɧɨɫɬɟɣ: ɜɟɪɨɹɬɧɨɫɬɶ ɩɪɨɢɡɜɟɞɟɧɢɹ ɞɜɭɯ ɫɨɛɵɬɢɣ ɪɚɜɧɚ ɩɪɨɢɡɜɟɞɟɧɢɸ ɜɟɪɨɹɬɧɨɫɬɢ ɨɞɧɨɝɨ ɢɡ ɧɢɯ ɧɚ ɭɫɥɨɜɧɭɸ ɜɟɪɨɹɬɧɨɫɬɶ ɞɪɭɝɨɝɨ, ɜɵɱɢɫɥɟɧɧɭɸ ɩɪɢ ɭɫɥɨɜɢɢ, ɱɬɨ ɩɟɪɜɨɟ ɢɦɟɥɨ ɦɟɫɬɨ:

P(AȻ)=P(Ⱥ)P(Ȼ/A).

ɉɭɫɬɶ ɜɨɡɦɨɠɧɵɟ ɢɫɯɨɞɵ ɨɩɵɬɚ ɫɜɨɞɹɬɫɹ ɤ N ɫɥɭɱɚɹɦ, ɤɨɬɨɪɵɟ ɞɥɹ ɧɚɝɥɹɞɧɨɫɬɢ ɞɚɧɵ ɜ ɜɢɞɟ ɫɢɦɜɨɥɨɜ ɧɚ ɪɢɫ.2.2.

Ɋɢɫ.2.2

ɉɪɟɞɩɨɥɨɠɢɦ, ɱɬɨ ɫɨɛɵɬɢɸ Ⱥ ɛɥɚɝɨɩɪɢɹɬɧɵ m ɫɥɭɱɚɟɜ, ɚ ɫɨɛɵɬɢɸ Ȼ ɛɥɚɝɨɩɪɢɹɬɧɵ k ɫɥɭɱɚɟɜ. Ɍɚɤ ɤɚɤ ɦɵ ɧɟ ɩɪɟɞɩɨɥɚɝɚɥɢ ɫɨɛɵɬɢɹ Ⱥ ɢ Ȼ ɫɨɜɦɟɫɬɧɵɦɢ, ɬɨ ɫɭɳɟɫɬɜɭɸɬ ɫɥɭɱɚɢ, ɛɥɚɝɨɩɪɢɹɬɧɵɟ ɢ ɫɨɛɵɬɢɸ Ⱥ ɢ ɫɨɛɵɬɢɸ Ȼ ɨɞɧɨɜɪɟɦɟɧɧɨ. ɉɭɫɬɶ ɱɢɫɥɨ ɬɚɤɢɯ ɫɥɭɱɚɟɜ l. Ɍɨɝɞɚ Ɋ(ȺȻ) = l/N; Ɋ(Ⱥ) = m/N. ȼɵɱɢɫɥɢɦ Ɋ(Ȼ/Ⱥ), ɬ.ɟ. ɭɫɥɨɜɧɭɸ ɜɟɪɨɹɬɧɨɫɬɶ ɫɨɛɵɬɢɹ Ȼ ɜ ɩɪɟɞɩɨɥɨɠɟɧɢɢ, ɱɬɨ Ⱥ ɢɦɟɥɨ ɦɟɫɬɨ. ȿɫɥɢ ɢɡɜɟɫɬɧɨ, ɱɬɨ ɫɨɛɵɬɢɟ Ⱥ ɩɪɨɢɡɨɲɥɨ, ɬɨ ɢɡ ɪɚɧɟɟ ɩɪɨɢɡɨɲɟɞɲɢɯ N ɫɥɭɱɚɟɜ ɨɫɬɚɸɬɫɹ ɜɨɡɦɨɠɧɵɦɢ ɬɨɥɶɤɨ ɬɟ m, ɤɨɬɨɪɵɟ ɛɥɚɝɨɩɪɢɹɬɫɬɜɨɜɚɥɢ ɫɨɛɵɬɢɸ Ⱥ. ɂɡ ɧɢɯ l ɫɥɭɱɚɟɜ ɛɥɚɝɨɩɪɢɹɬɧɵ ɫɨɛɵɬɢɸ Ȼ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, P(Ȼ/Ⱥ) = l/m. ɉɨɞɫɬɚɜɥɹɹ ɜɵɪɚɠɟɧɢɹ Ɋ(ȺȻ), Ɋ(Ⱥ) ɢ Ɋ(Ȼ/Ⱥ) ɜ ɮɨɪɦɭɥɭ ɜɟɪɨɹɬɧɨɫɬɢ ɩɪɨɢɡɜɟɞɟɧɢɹ ɞɜɭɯ ɫɨɛɵɬɢɣ, ɩɨɥɭɱɢɦ ɬɨɠɞɟɫɬɜɨ. Ɍɟɨɪɟɦɚ ɞɨɤɚɡɚɧɚ.

ɋɥɟɞɫɬɜɢɟ 1. ȿɫɥɢ ɫɨɛɵɬɢɟ Ⱥ ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɫɨɛɵɬɢɹ Ȼ, ɬɨ ɢ ɫɨɛɵɬɢɟ Ȼ ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɫɨɛɵɬɢɹ Ⱥ.

ɋɥɟɞɫɬɜɢɟ 2. ȼɟɪɨɹɬɧɨɫɬɶ ɩɪɨɢɡɜɟɞɟɧɢɹ ɞɜɭɯ ɧɟɡɚɜɢɫɢɦɵɯ ɫɨɛɵɬɢɣ ɪɚɜɧɚ ɩɪɨɢɡɜɟɞɟɧɢɸ ɷɬɢɯ ɫɨɛɵɬɢɣ.

2.4. Ɏɨɪɦɭɥɚ ɩɨɥɧɨɣ ɜɟɪɨɹɬɧɨɫɬɢ

Ɏɨɪɦɭɥɚ ɩɨɥɧɨɣ ɜɟɪɨɹɬɧɨɫɬɢ ɹɜɥɹɟɬɫɹ ɫɥɟɞɫɬɜɢɟɦ ɨɛɟɢɯ ɬɟɨɪɟɦ - ɬɟɨɪɟɦɵ ɫɥɨɠɟɧɢɹ ɜɟɪɨɹɬɧɨɫɬɟɣ ɢ ɬɟɨɪɟɦɵ ɭɦɧɨɠɟɧɢɹ ɜɟɪɨɹɬɧɨɫɬɟɣ.

ɉɭɫɬɶ ɬɪɟɛɭɟɬɫɹ ɨɩɪɟɞɟɥɢɬɶ ɜɟɪɨɹɬɧɨɫɬɶ ɧɟɤɨɬɨɪɨɝɨ ɫɨɛɵɬɢɹ Ⱥ, ɤɨɬɨɪɨɟ ɦɨɠɟɬ ɩɪɨɢɡɨɣɬɢ ɜɦɟɫɬɟ ɫ ɨɞɧɢɦ ɢɡ ɫɨɛɵɬɢɣ H1,H2,...,HN, ɨɛɪɚɡɭɸɳɢɯ ɩɨɥɧɭɸ ɝɪɭɩɩɭɧɟɫɨɜɦɟɫɬɧɵɯ ɫɨɛɵɬɢɣ. Ȼɭɞɟɦ ɷɬɢ ɫɨɛɵɬɢɹ ɧɚɡɵɜɚɬɶ ɝɢɩɨɬɟɡɚɦɢ. Ɍɨɝɞɚ

N

P(A)= ¦P(Hi)P(A/Hi),

i 1

ɬ.ɟ. ɜɟɪɨɹɬɧɨɫɬɶ ɫɨɛɵɬɢɹ Ⱥ ɜɵɱɢɫɥɹɟɬɫɹ ɤɚɤ ɫɭɦɦɚ ɩɪɨɢɡɜɟɞɟɧɢɣ ɜɟɪɨɹɬɧɨɫɬɢ ɤɚɠɞɨɣ ɝɢɩɨɬɟɡɵ ɧɚ ɜɟɪɨɹɬɧɨɫɬɶ ɫɨɛɵɬɢɹ ɩɪɢ ɷɬɨɣ ɝɢɩɨɬɟɡɟ.

ɗɬɚ ɮɨɪɦɭɥɚ ɧɨɫɢɬ ɧɚɡɜɚɧɢɟ ɮɨɪɦɭɥɵ ɩɨɥɧɨɣ ɜɟɪɨɹɬɧɨɫɬɢ.

Ɍɚɤ ɤɚɤ ɝɢɩɨɬɟɡɵ H1,H2,...,HN ɨɛɪɚɡɭɸɬ ɩɨɥɧɭɸ ɝɪɭɩɩɭ ɫɨɛɵɬɢɣ, ɬɨ ɫɨɛɵɬɢɟ Ⱥ ɦɨɠɟɬ ɩɨɹɜɢɬɶɫɹ ɬɨɥɶɤɨ ɜ ɤɨɦɛɢɧɚɰɢɢ ɫ ɤɚɤɨɣ-ɥɢɛɨ ɢɡ ɷɬɢɯ ɝɢɩɨɬɟɡ: A=H1Ⱥ+H2Ⱥ+...+HNȺ. Ɍɚɤ ɤɚɤ ɝɢɩɨɬɟɡɵ H1,H2,...,HN ɧɟɫɨɜɦɟɫɬɧɵ, ɬɨ ɢ ɤɨɦɛɢɧɚɰɢɢ H1Ⱥ,H2Ⱥ,...,HNȺ ɬɚɤɠɟ ɧɟɫɨɜɦɟɫɬɧɵ. ɉɪɢɦɟɧɹɹ ɤ ɧɢɦ ɬɟɨɪɟɦɭ ɫɥɨɠɟɧɢɹ, ɩɨɥɭɱɢɦ:

N

P(A)=P(H1Ⱥ)+P(H2Ⱥ)+...+P(HNȺ)=¦P(HiȺ).

i 1

ɉɪɢɦɟɧɹɹ ɤ ɫɨɛɵɬɢɸ HiȺ ɬɟɨɪɟɦɭ ɫɥɨɠɟɧɢɹ, ɩɨɥɭɱɢɦ ɢɫɤɨɦɭɸ ɮɨɪɦɭɥɭ. ɑɬɨ ɢ ɬɪɟɛɨɜɚɥɨɫɶ ɞɨɤɚɡɚɬɶ.

2.5. Ɍɟɨɪɟɦɚ ɝɢɩɨɬɟɡ (ɮɨɪɦɭɥɚ Ȼɚɣɟɫɚ)

ɂɦɟɟɬɫɹ ɩɨɥɧɚɹ ɝɪɭɩɩɚ ɧɟɫɨɜɦɟɫɬɧɵɯ ɝɢɩɨɬɟɡ H1,H2,...,HN. ȼɟɪɨɹɬɧɨɫɬɢ ɷɬɢɯ ɝɢɩɨɬɟɡ ɞɨ ɨɩɵɬɚ ɢɡɜɟɫɬɧɵ ɢ ɪɚɜɧɵ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ P(H1), P(H2), ...,P(HN). ɉɪɨɢɡɜɟɞɟɦ ɨɩɵɬ, ɜ ɪɟɡɭɥɶɬɚɬɟ ɤɨɬɨɪɨɝɨ ɛɭɞɟɬ ɧɚɛɥɸɞɚɬɶɫɹ ɩɨɹɜɥɟɧɢɟ ɧɟɤɨɬɨɪɨɝɨ ɫɨɛɵɬɢɹ Ⱥ. ɋɩɪɚɲɢɜɚɟɬɫɹ, ɤɚɤ ɫɥɟɞɭɟɬ ɢɡɦɟɧɢɬɶ ɜɟɪɨɹɬɧɨɫɬɢ ɝɢɩɨɬɟɡ ɜ ɫɜɹɡɢ ɫ ɩɨɹɜɥɟɧɢɟɦ ɷɬɨɝɨ ɫɨɛɵɬɢɹ?

Ɂɞɟɫɶ, ɩɨ ɫɭɳɟɫɬɜɭ, ɢɞɟɬ ɪɟɱɶ ɨ ɬɨɦ, ɤɚɤ ɧɚɣɬɢ ɭɫɥɨɜɧɭɸ ɜɟɪɨɹɬɧɨɫɬɶ P(Hi/A) ɞɥɹ ɤɚɠɞɨɣ ɝɢɩɨɬɟɡɵ ɩɨɫɥɟ ɩɪɨɜɟɞɟɧɢɹ ɷɤɫɩɟɪɢɦɟɧɬɚ.

ɗɬɚ ɮɨɪɦɭɥɚ ɢ ɧɨɫɢɬ ɧɚɡɜɚɧɢɟ ɮɨɪɦɭɥɵ Ȼɚɣɟɫɚ ɢɥɢ ɬɟɨɪɟɦɵ ɝɢɩɨɬɟɡ. ɂɫɩɨɥɶɡɭɟɬɫɹ ɨɧɚ ɜ ɬɟɨɪɢɢ ɩɪɨɜɟɪɤɢ ɫɬɚɬɢɫɬɢɱɟɫɤɢɯ ɝɢɩɨɬɟɡ.

3.ɋɅɍɑȺɃɇɕȿ ȼȿɅɂɑɂɇɕ ɂ ɂɏ ɁȺɄɈɇɕ ɊȺɋɉɊȿȾȿɅȿɇɂə

3.1.Ɂɚɤɨɧ ɪɚɫɩɪɟɞɟɥɟɧɢɹ. Ɋɹɞ ɪɚɫɩɪɟɞɟɥɟɧɢɹ

Ɂɚɤɨɧɨɦ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɋȼ ɧɚɡɵɜɚɟɬɫɹ ɜɫɹɤɨɟ ɫɨɨɬɧɨɲɟɧɢɟ, ɭɫɬɚɧɚɜɥɢɜɚɸɳɟɟ ɫɜɹɡɶ ɦɟɠɞɭ ɜɨɡɦɨɠɧɵɦɢ ɡɧɚɱɟɧɢɹɦɢ ɫɥɭɱɚɣɧɨɣ ɜɟɥɢɱɢɧɵ ɢ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɦɢ ɢɦ ɜɟɪɨɹɬɧɨɫɬɹɦɢ. ɉɪɨ ɫɥɭɱɚɣɧɭɸ ɜɟɥɢɱɢɧɭ ɛɭɞɟɦ ɝɨɜɨɪɢɬɶ, ɱɬɨ ɨɧɚ ɩɨɞɱɢɧɟɧɚ ɞɚɧɧɨɦɭɡɚɤɨɧɭɪɚɫɩɪɟɞɟɥɟɧɢɹ.

ɉɪɨɫɬɟɣɲɟɣ ɮɨɪɦɨɣ ɡɚɞɚɧɢɹ ɡɚɤɨɧɚ ɹɜɥɹɟɬɫɹ ɬɚɛɥ.3.1, ɜ ɤɨɬɨɪɨɣ ɩɟɪɟɱɢɫɥɟɧɵ ɜɨɡɦɨɠɧɵɟ ɡɧɚɱɟɧɢɹ ɋȼ ɢ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɢɦ ɜɟɪɨɹɬɧɨɫɬɢ:

Ɍɚɛɥɢɰɚ 3.1

Ɉɞɧɚɤɨ ɬɚɤɨɣ ɬɚɛɥɢɰɵ ɧɟɜɨɡɦɨɠɧɨ ɩɨɫɬɪɨɢɬɶ ɞɥɹ ɧɟɩɪɟɪɵɜɧɨɣ ɋȼ, ɩɨɫɤɨɥɶɤɭ ɞɥɹ ɧɟɟ ɤɚɠɞɨɟ ɨɬɞɟɥɶɧɨɟ ɡɧɚɱɟɧɢɟ ɧɟ ɨɛɥɚɞɚɟɬ ɨɬɥɢɱɧɨɣ ɨɬ ɧɭɥɹ ɜɟɪɨɹɬɧɨɫɬɶɸ.

3.2. Ɏɭɧɤɰɢɹ ɪɚɫɩɪɟɞɟɥɟɧɢɹ

Ⱦɥɹ ɤɨɥɢɱɟɫɬɜɟɧɧɨɣ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɢɫɩɨɥɶɡɭɸɬ ɡɚɜɢɫɢɦɨɫɬɶ ɜɟɪɨɹɬɧɨɫɬɢ ɫɨɛɵɬɢɹ X < x, ɝɞɟ x-ɧɟɤɨɬɨɪɚɹ ɬɟɤɭɳɚɹ ɩɟɪɟɦɟɧɧɚɹ, ɨɬ x. ɗɬɚ ɮɭɧɤɰɢɹ ɧɚɡɵɜɚɟɬɫɹ ɮɭɧɤɰɢɟɣ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɋȼ X ɢ ɨɛɨɡɧɚɱɚɟɬɫɹ F(x): F(x) = P(X < x). Ɏɭɧɤɰɢɸ ɪɚɫɩɪɟɞɟɥɟɧɢɹ F(x) ɢɧɨɝɞɚ ɧɚɡɵɜɚɸɬ ɢɧɬɟɝɪɚɥɶɧɨɣ ɮɭɧɤɰɢɟɣ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɢɥɢ ɢɧɬɟɝɪɚɥɶɧɵɦ ɡɚɤɨɧɨɦ ɪɚɫɩɪɟɞɟɥɟɧɢɹ.

Ɏɭɧɤɰɢɹ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɫɭɳɟɫɬɜɭɟɬ ɞɥɹ ɜɫɟɯ ɋȼ - ɤɚɤ ɧɟɩɪɟɪɵɜɧɵɯ, ɬɚɤ ɢ ɞɢɫɤɪɟɬɧɵɯ.

ɋɮɨɪɦɭɥɢɪɭɟɦ ɧɟɤɨɬɨɪɵɟ ɨɛɳɢɟ ɫɜɨɣɫɬɜɚ F(x) :

ɉɨ ɨɩɪɟɞɟɥɟɧɢɸ, F(x) ɩɪɢ ɧɟɤɨɬɨɪɨɦ x ɟɫɬɶ ɜɟɪɨɹɬɧɨɫɬɶ ɩɨɩɚɞɚɧɢɹ ɋȼ X ɜ ɢɧɬɟɪɜɚɥ ɨɬ f ɞɨ x.

Ⱦɥɹ ɞɢɫɤɪɟɬɧɨɣ ɋȼ ɪɚɫɩɪɟɞɟɥɟɧɢɟ F(x) ɢɦɟɟɬ ɫɬɭɩɟɧɱɚɬɵɣ ɜɢɞ, ɩɪɢɱɟɦ ɜɟɥɢɱɢɧɚ ɤɚɠɞɨɝɨ ɫɤɚɱɤɚ ɪɚɜɧɚ ɜɟɪɨɹɬɧɨɫɬɢ ɡɧɚɱɟɧɢɹ, ɩɪɢ ɤɨɬɨɪɨɦ ɢɦɟɟɬɫɹ ɫɤɚɱɨɤ

F(x).

ɉɪɢ ɪɟɲɟɧɢɢ ɩɪɚɤɬɢɱɟɫɤɢɯ ɡɚɞɚɱ ɱɚɫɬɨ ɧɟɨɛɯɨɞɢɦɨ ɜɵɱɢɫɥɹɬɶ ɜɟɪɨɹɬɧɨɫɬɶ ɬɨɝɨ, ɱɬɨ ɋȼ ɩɪɢɦɟɬ ɡɧɚɱɟɧɢɟ, ɡɚɤɥɸɱɟɧɧɨɟ ɜ ɧɟɤɨɬɨɪɵɯ ɩɪɟɞɟɥɚɯ, ɧɚɩɪɢɦɟɪ ɨɬ x1 ɞɨ x2 . ɗɬɨ ɫɨɛɵɬɢɟ ɧɚɡɵɜɚɟɬɫɹ "ɩɨɩɚɞɚɧɢɟɦ ɋȼ X ɧɚ ɭɱɚɫɬɨɤ ɨɬ x1 ɞɨ x2". ȼɵɪɚɡɢɦ ɜɟɪɨɹɬɧɨɫɬɶ ɷɬɨɝɨ ɫɨɛɵɬɢɹ ɱɟɪɟɡ ɮɭɧɤɰɢɸ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɋȼ X. Ⱦɥɹ ɷɬɨɝɨ ɪɚɫɫɦɨɬɪɢɦ ɬɪɢ ɫɨɛɵɬɢɹ:

x ɫɨɛɵɬɢɟ Ⱥ, ɫɨɫɬɨɹɳɟɟ ɜ ɬɨɦ, ɱɬɨ X < x2 ; x ɫɨɛɵɬɢɟ ȼ, ɫɨɫɬɨɹɳɟɟ ɜ ɬɨɦ, ɱɬɨ X < x1 ;

x ɫɨɛɵɬɢɟ ɋ, ɫɨɫɬɨɹɳɟɟ ɜ ɬɨɦ, ɱɬɨ x1 < X <x2 .

ɍɱɢɬɵɜɚɹ, ɱɬɨ A = B + C, ɩɨ ɬɟɨɪɟɦɟ ɫɥɨɠɟɧɢɹ ɜɟɪɨɹɬɧɨɫɬɟɣ ɩɨɥɭɱɢɦ

P(X<x2)=P(X<x1)+P(x1<X<x2), ɢɥɢ F(x2) = F(x1) + P(x1<X<x2), ɨɬɤɭɞɚ

P(x1<X<x2) = F(x2)-F(x1), ɬ.ɟ. ɜɟɪɨɹɬɧɨɫɬɶ ɩɨɩɚɞɚɧɢɹ ɋȼ ɧɚ ɡɚɞɚɧɧɵɣ ɭɱɚɫɬɨɤ ɪɚɜɧɚ ɩɪɢɪɚɳɟɧɢɸ ɮɭɧɤɰɢɢ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɧɚ ɷɬɨɦ ɭɱɚɫɬɤɟ.

3.3. ɉɥɨɬɧɨɫɬɶ ɜɟɪɨɹɬɧɨɫɬɢ

ɉɭɫɬɶ ɢɦɟɟɬɫɹ ɧɟɩɪɟɪɵɜɧɚɹ ɋȼ X ɫ ɮɭɧɤɰɢɟɣ ɪɚɫɩɪɟɞɟɥɟɧɢɹ F(x), ɤɨɬɨɪɭɸ ɫɱɢɬɚɟɦ ɧɟɩɪɟɪɵɜɧɨɣ ɢ ɞɢɮɮɟɪɟɧɰɢɪɭɟɦɨɣ. ȼɵɱɢɫɥɢɦ ɜɟɪɨɹɬɧɨɫɬɶ ɩɨɩɚɞɚɧɢɹ ɷɬɨɣ ɋȼ ɧɚ ɭɱɚɫɬɨɤ ɨɬ x ɞɨ x + 'x : P(x < X < x+'x)=F(x+'x)-F(x), ɬ.ɟ. ɷɬɚ ɜɟɪɨɹɬɧɨɫɬɶ ɪɚɜɧɚ ɩɪɢɪɚɳɟɧɢɸ ɮɭɧɤɰɢɢ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɧɚ ɷɬɨɦ ɭɱɚɫɬɤɟ. Ɋɚɫɫɦɨɬ-

Ɇɚɬɟɦɚɬɢɱɟɫɤɢɦ ɨɠɢɞɚɧɢɟɦ ɋȼ ɧɚɡɵɜɚɸɬ ɫɭɦɦɭ ɩɪɨɢɡɜɟɞɟɧɢɣ ɜɫɟɯ ɜɨɡɦɨɠɧɵɯ ɡɧɚɱɟɧɢɣ ɋȼ ɧɚ ɜɟɪɨɹɬɧɨɫɬɢ ɷɬɢɯ ɡɧɚɱɟɧɢɣ.

Ɇɚɬɟɦɚɬɢɱɟɫɤɨɟ ɨɠɢɞɚɧɢɟ ɋȼ X ɫɜɹɡɚɧɨ ɫɨ ɫɪɟɞɧɢɦ ɚɪɢɮɦɟɬɢɱɟɫɤɢɦ ɡɧɚɱɟɧɢɟɦ ɧɚɛɥɸɞɟɧɧɵɯ ɡɧɚɱɟɧɢɣ ɋȼ ɩɪɢ ɛɨɥɶɲɨɦ ɱɢɫɥɟ ɨɩɵɬɨɜ ɬɚɤ ɠɟ, ɤɚɤ ɢ ɜɟɪɨɹɬɧɨɫɬɶ ɫ ɱɚɫɬɨɬɨɣ ɫɨɛɵɬɢɹ, ɬ.ɟ. ɩɪɢ ɭɜɟɥɢɱɟɧɢɢ ɱɢɫɥɚ ɨɩɵɬɨɜ ɫɪɟɞɧɟɟ ɚɪɢɮɦɟɬɢɱɟɫɤɨɟ ɡɧɚɱɟɧɢɟ ɫɬɪɟɦɢɬɫɹ ɤ ɆɈ.

Ⱦɥɹ ɧɟɩɪɟɪɵɜɧɨɣ ɋȼ ɆɈ ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɨ ɮɨɪɦɭɥɟ

f

M(X) = ³ xf(x)dx.

f

Ɏɢɡɢɱɟɫɤɢ ɆɈ ɦɨɠɧɨ ɬɪɚɤɬɨɜɚɬɶ ɤɚɤ ɤɨɨɪɞɢɧɚɬɭ ɰɟɧɬɪɚ ɬɹɠɟɫɬɢ ɬɟɥɚ (ɩɥɨɬɧɨɫɬɢ ɜɟɪɨɹɬɧɨɫɬɟɣ). ȿɞɢɧɢɰɚ ɢɡɦɟɪɟɧɢɹ ɆɈ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɟɞɢɧɢɰɟ ɢɡɦɟɪɟɧɢɹ ɋȼ.

Ɇɨɦɟɧɬɵ. Ⱦɢɫɩɟɪɫɢɹ. ɋɪɟɞɧɟɤɜɚɞɪɚɬɢɱɟɫɤɨɟ ɨɬɤɥɨɧɟɧɢɟ. ɇɚɱɚɥɶɧɵɦ ɦɨɦɟɧɬɨɦ s-ɝɨ ɩɨɪɹɞɤɚ ɞɥɹ ɞɢɫɤɪɟɬɧɨɣ ɋȼ X ɧɚɡɵɜɚɟɬɫɹ

N

ɫɭɦɦɚ ɜɢɞɚ ms [X]=¦xispi. Ⱦɥɹ ɧɟɩɪɟɪɵɜɧɨɣ ɋȼ

i 1

f

ms [X]= ³ xsf(x)dx.

f

ɂɡ ɷɬɢɯ ɮɨɪɦɭɥ ɜɢɞɧɨ, ɱɬɨ ɆɈ ɟɫɬɶ ɧɟ ɱɬɨ ɢɧɨɟ, ɤɚɤ ɩɟɪɜɵɣ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɋȼ X. ɍɫɥɨɜɧɨ, ɢɫɩɨɥɶɡɭɹ ɡɧɚɤ ɆɈ, ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜɵɪɚɠɟɧɢɟ ɞɥɹ s-ɝɨ ɧɚɱɚɥɶɧɨɝɨ ɦɨɦɟɧɬɚ, ɬ.ɟ. ms [X]=M[X-s] ɧɚɱɚɥɶɧɵɦ ɦɨɦɟɧɬɨɦ s-ɝɨ ɩɨɪɹɞɤɚ ɋȼ X ɧɚɡɵɜɚɸɬ ɆɈ s-ɣ ɫɬɟɩɟɧɢ ɷɬɨɣ ɋȼ.

ɐɟɧɬɪɢɪɨɜɚɧɧɨɣ ɋȼ, ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɣ ɋȼ X, ɧɚɡɵɜɚɸɬ ɨɬɤɥɨɧɟɧɢɟ ɋȼ X

$

ɨɬ ɟɟ ɆɈ, ɬ.ɟ. X=X-mX. ɇɟɬɪɭɞɧɨ ɭɛɟɞɢɬɶɫɹ, ɱɬɨ ɆɈ ɰɟɧɬɪɢɪɨɜɚɧɧɨɣ ɋȼ ɪɚɜɧɨ ɧɭɥɸ. Ɇɨɦɟɧɬɵ ɰɟɧɬɪɢɪɨɜɚɧɧɨɣ ɋȼ ɧɚɡɵɜɚɸɬ ɰɟɧɬɪɚɥɶɧɵɦɢ ɦɨɦɟɧɬɚɦɢ. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɰɟɧɬɪɚɥɶɧɵɦ ɦɨɦɟɧɬɨɦ s-ɝɨ ɩɨɪɹɞɤɚ ɧɚɡɵɜɚɸɬ ɆɈ s-ɣ ɫɬɟɩɟɧɢ ɰɟɧɬɪɢɪɨɜɚɧɧɨɣ ɋȼ: Ps[X] = M[(X-mX)s]. Ⱦɥɹ ɧɟɩɪɟɪɵɜɧɨɣ ɋȼ s-ɣ ɰɟɧɬɪɚɥɶɧɵɣ ɦɨɦɟɧɬ ɜɵɪɚɠɚɸɬ ɢɧɬɟɝɪɚɥɨɦ:

f

Ps= ³(x-mX)sf(x)dx.

f

ȼɜɟɞɟɦ ɫɨɨɬɧɨɲɟɧɢɹ, ɫɜɹɡɵɜɚɸɳɢɟ ɰɟɧɬɪɚɥɶɧɵɟ ɢ ɧɚɱɚɥɶɧɵɟ ɦɨɦɟɧɬɵ

ɪɚɡɥɢɱɧɵɯ ɩɨɪɹɞɤɨɜ: P1=0; P2=m2-mX2; P3=m3-3mXm2+2mX3,...

ɂɡ ɜɫɟɯ ɦɨɦɟɧɬɨɜ ɱɚɳɟ ɜɫɟɝɨ ɩɪɢɦɟɧɹɸɬ ɆɈ ɢ ɜɬɨɪɵɟ ɦɨɦɟɧɬɵ-ɧɚɱɚɥɶɧɵɣ ɢ ɰɟɧɬɪɚɥɶɧɵɣ. ȼɬɨɪɨɣ ɰɟɧɬɪɚɥɶɧɵɣ ɦɨɦɟɧɬ ɧɚɡɵɜɚɸɬ ɞɢɫɩɟɪɫɢɟɣ ɋȼ X. Ⱦɥɹ ɧɟɟ ɜɜɨɞɹɬ ɫɩɟɰɢɚɥɶɧɨɟ ɨɛɨɡɧɚɱɟɧɢɟ: P2=D[X], ɢɥɢ DX.

Ⱦɢɫɩɟɪɫɢɹ ɯɚɪɚɤɬɟɪɢɡɭɟɬ ɫɬɟɩɟɧɶ ɪɚɡɛɪɨɫɚɧɧɨɫɬɢ (ɢɥɢ ɪɚɫɫɟɢɜɚɧɢɹ) ɋȼ X ɨɬɧɨɫɢɬɟɥɶɧɨ ɦɚɬɟɦɚɬɢɱɟɫɤɨɝɨ ɨɠɢɞɚɧɢɹ ɢ ɢɦɟɟɬ ɪɚɡɦɟɪɧɨɫɬɶ ɤɜɚɞɪɚɬɚ ɋȼ