Математика для юристов - Д.А. Ловцова
.pdfb |
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P(a X b) ³f(x)udx . |
(8.7) |
a |
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Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɜɟɪɨɹɬɧɨɫɬɶ ɩɨɩɚɞɚɧɢɹ ɫɥɭɱɚɣɧɨɣ ɜɟɥɢɱɢɧɵ ɜ ɡɚɞɚɧɧɵɣ ɞɢɚɩɚɡɨɧ ɪɚɜɧɚ ɩɥɨɳɚɞɢ ɩɨɞ ɤɪɢɜɨɣ f(x) ɧɚ ɢɧɬɟɪɜɚɥɟ ]a,b[.
ɉɪɢɦɟɪ. ɉɥɨɬɧɨɫɬɶ ɜɟɪɨɹɬɧɨɫɬɢ ɋȼ W ɡɚɞɚɧɚ ɮɨɪɦɭɥɨɣ (8.4). ȼɵɱɢɫɥɢɬɶ ɜɟɪɨɹɬɧɨɫɬɶ ɩɨɩɚɞɚɧɢɹ W ɜ ɞɢɚɩɚɡɨɧ ɨɬ a 1 b 3. Ɍɟɩɟɪɶ ɞɟɣɫɬɜɭɟɦ ɩɨ ɮɨɪɦɭɥɟ (8.7):
3
P(1 X 3) ³f(x)udx F(3) F(1) 0.32.
1
Ɍɟɩɟɪɶ (ɪɢɫ. 8.4) ɱɢɫɥɨ P(1 X 3) 0.32 – ɩɥɨɳɚɞɶ ɩɨɞ ɤɪɢɜɨɣ f(x) ɧɚ ɢɧɬɟɪɜɚɥɟ ]1,3[.
8.3. Числовые характеристики случайных величин
Ɂɚɤɨɧ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɬɨɣ ɢɥɢ ɢɧɨɣ ɫɥɭɱɚɣɧɨɣ ɜɟɥɢɱɢɧɵ ɨɩɢɫɵɜɚɟɬ ɟɟ ɩɨɥɧɨɫɬɶɸ ɫ ɜɟɪɨɹɬɧɨɫɬɧɨɣ ɬɨɱɤɢ ɡɪɟɧɢɹ. Ʌɸɛɵɟ ɡɚɞɚɱɢ, ɫɜɹɡɚɧɧɵɟ ɫɨ ɫɥɭɱɚɣɧɵɦɢ ɜɟɥɢɱɢɧɚɦɢ, ɦɨɝɭɬ ɛɵɬɶ ɪɟɲɟɧɵ ɫ ɩɨɦɨɳɶɸ ɡɚɤɨɧɨɜ ɪɚɫɩɪɟɞɟɥɟɧɢɹ. Ɉɞɧɚɤɨ ɞɚɥɟɤɨ ɧɟ ɜɫɟ ɡɚɞɚɱɢ ɩɨɞɨɛɧɨɝɨ ɪɨɞɚ ɬɪɟɛɭɸɬ ɞɥɹ ɢɯ ɪɟɲɟɧɢɹ ɬɚɤɨɣ ɬɹɠɟɥɨɣ ɚɪɬɢɥɥɟɪɢɢ. Ȼɵɜɚɟɬ ɞɨɫɬɚɬɨɱɧɨ ɨɩɟɪɢɪɨɜɚɬɶ ɫ ɤɨɦɩɚɤɬɧɵɦɢ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚɦɢ, ɨɬɪɚɠɚɸɳɢɦɢ ɫɚɦɵɟ ɫɭɳɟɫɬɜɟɧɧɵɟ ɨɫɨɛɟɧɧɨɫɬɢ ɫɥɭɱɚɣɧɵɯ ɜɟɥɢɱɢɧ. Ⱦɥɹ ɷɬɢɯ ɰɟɥɟɣ ɢ ɫɥɭɠɚɬ ɱɢɫɥɨɜɵɟ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɫɥɭɱɚɣɧɵɯ ɜɟɥɢɱɢɧ. ȼ ɩɟɪɜɭɸ ɨɱɟɪɟɞɶ, ɷɬɨ ɦɚɬɟɦɚɬɢɱɟɫɤɨɟ ɨɠɢɞɚɧɢɟ ɢ ɞɢɫɩɟɪɫɢɹ ɫɥɭɱɚɣɧɨɣ ɜɟɥɢɱɢɧɵ.
Ɇ ɚ ɬ ɟ ɦ ɚ ɬ ɢ ɱ ɟ ɫ ɤ ɨ ɟ ɨ ɠ ɢ ɞ ɚ ɧ ɢ ɟ ɆɈ ɯɚɪɚɤɬɟɪɢɡɭɟɬ ɦɟɫɬɨɩɨɥɨɠɟɧɢɟ ɫɥɭɱɚɣɧɨɣ ɜɟɥɢɱɢɧɵ ɧɚ ɱɢɫɥɨɜɨɣ ɨɫɢ. ɗɬɨ ɫɜɨɟɝɨ ɪɨɞɚ ɰɟɧɬɪ ɬɹɠɟɫɬɢ ɜɫɟɝɨ ɦɚɫɫɢɜɚ ɟɟ ɨɬɫɱɟɬɨɜ. Ɉɛɨɡɧɚɱɚɸɬ ɦɚɬɟɦɚɬɢɱɟɫɤɨɟ ɨɠɢɞɚɧɢɟ ɫɥɭɱɚɣɧɨɣ ɜɟɥɢɱɢɧɵ X ɤɚɤ M[X] ɥɢɛɨ ɤɚɤ mx. Ɇɚɬɟɦɚɬɢɱɟɫɤɨɟ ɨɠɢɞɚɧɢɟ ɫɥɭɱɚɣɧɨɣ ɜɟɥɢɱɢɧɵ X ɧɚɡɵɜɚɸɬ ɟɟ ɫɪɟɞɧɢɦ.
Ⱦ ɢ ɫ ɩ ɟ ɪ ɫ ɢ ɹ ɫɥɭɱɚɣɧɨɣ ɜɟɥɢɱɢɧɵ X ɯɚɪɚɤɬɟɪɢɡɭɟɬ ɪɚɡɛɪɨɫ (ɪɚɫɫɟɹɧɢɟ, ɪɚɫɩɪɟɞɟɥɟɧɢɟ) ɟɟ ɨɬɫɱɟɬɨɜ ɧɚ ɱɢɫɥɨɜɨɣ ɨɫɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɦɚɬɟɦɚɬɢɱɟɫɤɨɝɨ ɨɠɢɞɚɧɢɹ mx ɷɬɨɣ ɫɥɭɱɚɣɧɨɣ ɜɟɥɢɱɢɧɵ. Ɉɛɨɡɧɚɱɚɸɬ ɞɢɫɩɟɪɫɢɸ ɫɥɭɱɚɣɧɨɣ ɜɟɥɢɱɢɧɵ X ɤɚɤ D[X] ɢɥɢ ɤɚɤ Dx.
ɉɭɫɬɶ ɦɚɬɟɦɚɬɢɱɟɫɤɨɟ ɨɠɢɞɚɧɢɟ mx ɫɥɭɱɚɣɧɨɣ ɜɟɥɢɱɢɧɵ X ɡɚɞɚɧɨ. Ɍɨɝɞɚ ɞɢɫɩɟɪɫɢɹ ɫɥɭɱɚɣɧɨɣ ɜɟɥɢɱɢɧɵ ɜɵɱɢɫɥɹɟɬɫɹ ɬɚɤ:
D[X] M[X2] (mx)2, |
(8.8) |
ɚ ɢɦɟɧɧɨ, ɞɢɫɩɟɪɫɢɹ ɋȼ ɪɚɜɧɚ ɪɚɡɧɨɫɬɢ ɦɟɠɞɭ ɟɟ ɫɪɟɞɧɢɦ ɤɜɚɞɪɚɬɨɦ ɢ ɤɜɚɞɪɚɬɨɦ ɟɟ ɫɪɟɞɧɟɝɨ.
121
ɐɟɧɬɪɢɪɨɜɚɧɧɨɣ ɫɥɭɱɚɣɧɨɣ ɜɟɥɢɱɢɧɨɣ Xɐ, ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɣ X, ɧɚɡɵɜɚɟɬɫɹ ɨɬɤɥɨɧɟɧɢɟ X ɨɬ ɟɟ ɦɚɬɟɦɚɬɢɱɟɫɤɨɝɨ ɨɠɢɞɚɧɢɹ mx:
Xɐ X mx.
Ƚɟɨɦɟɬɪɢɱɟɫɤɢ ɩɟɪɟɯɨɞ ɨɬ X ɤ Xɐ ɨɡɧɚɱɚɟɬ ɩɟɪɟɧɨɫ ɧɚɱɚɥɚ ɤɨɨɪɞɢɧɚɬ ɧɚ ɱɢɫɥɨɜɨɣ ɨɫɢ ɜ ɬɨɱɤɭ mx. ɂɧɨɝɞɚ ɭɞɨɛɧɟɟ ɛɵɜɚɟɬ ɜɵɱɢɫɥɹɬɶ ɞɢɫɩɟɪɫɢɸ ɩɨ ɮɨɪɦɭɥɟ
D[X] Dx M[(X mx)2 M[(Xɐ)2], |
(8.9) |
ɬɨ ɟɫɬɶ ɞɢɫɩɟɪɫɢɟɣ ɫɥɭɱɚɣɧɨɣ ɜɟɥɢɱɢɧɵ X ɧɚɡɵɜɚɸɬ ɦɚɬɟɦɚɬɢɱɟɫɤɨɟ ɨɠɢɞɚɧɢɟ ɤɜɚɞɪɚɬɚ ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɣ ɟɣ ɰɟɧɬɪɢɪɨɜɚɧɧɨɣ ɫɥɭɱɚɣɧɨɣ ɜɟɥɢɱɢɧɵ Xɐ.
Ɉɬɦɟɬɢɦ ɫɭɳɟɫɬɜɟɧɧɵɣ ɮɚɤɬ. ȿɫɥɢ ɪɚɡɦɟɪɧɨɫɬɶ ɦɚɬɟɦɚɬɢɱɟɫɤɨɝɨ ɨɠɢɞɚɧɢɹ mx ɫɨɜɩɚɞɚɟɬ ɫ ɪɚɡɦɟɪɧɨɫɬɶɸ ɫɚɦɨɣ ɫɥɭɱɚɣɧɨɣ ɜɟɥɢɱɢɧɵ X, ɬɨ ɞɢɫɩɟɪɫɢɹ ɢɦɟɟɬ ɪɚɡɦɟɪɧɨɫɬɶ ɤɜɚɞɪɚɬɚ ɪɚɡɦɟɪɧɨɫɬɢ ɫɥɭɱɚɣɧɨɣ ɜɟɥɢɱɢɧɵ. ɍɞɨɛɧɟɟ ɛɵɥɨ ɛɵ ɨɩɟɪɢɪɨɜɚɬɶ ɫ ɱɢɫɥɨɜɵɦɢ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚɦɢ ɨɞɧɨɣ ɪɚɡɦɟɪɧɨɫɬɢ. Ⱦɥɹ ɷɬɨɝɨ ɢɡ ɞɢɫɩɟɪɫɢɢ ɢɡɜɥɟɤɚɸɬ ɤɨɪɟɧɶ ɤɜɚɞɪɚɬɧɵɣ. ɉɨɥɭɱɟɧɧɭɸ ɜɟɥɢɱɢɧɭ ɧɚɡɵɜɚɸɬ ɫɪɟɞɧɢɦ ɤɜɚɞɪɚɬɢɱɟɫɤɢɦ ɨɬɤɥɨɧɟɧɢɟɦ ɋɄɈ ɫɥɭɱɚɣɧɨɣ ɜɟɥɢɱɢɧɵ X ɢ ɨɛɨɡɧɚɱɚɸɬ ɤɚɤ Vx:
Vx Dx . |
(8.10) |
Ɋɚɡɦɟɪɧɨɫɬɶ ɋɄɈ ɫɨɜɩɚɞɚɟɬ ɫ ɪɚɡɦɟɪɧɨɫɬɶɸ ɫɥɭɱɚɣɧɨɣ ɜɟɥɢɱɢɧɵ.
Ɋɚɫɫɦɨɬɪɢɦ ɱɢɫɥɨɜɵɟ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɞ ɢ ɫ ɤ ɪ ɟ ɬ ɧ ɵ ɯ ɫɥɭɱɚɣɧɵɯ ɜɟɥɢɱɢɧ.
ɆɈ ɞ ɢ ɫ ɤ ɪ ɟ ɬ ɧ ɨ ɣ ɫɥɭɱɚɣɧɨɣ ɜɟɥɢɱɢɧɵ ɜɵɱɢɫɥɹɸɬ ɬɚɤ:
n |
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M[X] mx x0up0 x1up1 xnupn ¦xk upk . |
(8.11) |
k 0 |
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Ʉɚɤ ɜɢɞɢɦ, ɦɚɬɟɦɚɬɢɱɟɫɤɨɟ ɨɠɢɞɚɧɢɟ ɞɢɫɤɪɟɬɧɨɣ ɫɥɭɱɚɣɧɨɣ ɜɟɥɢɱɢɧɵ – ɷɬɨ ɜɡɜɟɲɟɧɧɚɹ ɫɭɦɦɚ ɟɟ ɨɬɫɱɟɬɨɜ, ɤɨɝɞɚ ɤɚɠɞɵɣ ɨɬɫɱɟɬ xk ɭɦɧɨɠɚɟɬɫɹ ɧɚ ɫɜɨɸ ɜɟɪɨɹɬɧɨɫɬɶ pk (ɧɚ ɫɜɨɣ ɜɟɫ), ɢ ɩɨɥɭɱɟɧɧɵɟ ɩɪɨɢɡɜɟɞɟɧɢɹ ɫɭɦɦɢɪɭɸɬɫɹ.
122
Ɍɚɛɥɢɰɚ 8.2
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Ɍɚɛɥɢɰɚ 8.3 |
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Ɋɢɫ. 8.6 |
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Ⱦɢɫɩɟɪɫɢɹ ɞ ɢ ɫ ɤ ɪ ɟ ɬ ɧ ɨ ɣ ɋȼ ɩɨ ɮɨɪɦɭɥɟ (8.8) ɜɵɱɢɫɥɹɟɬɫɹ ɬɚɤ:
n
Dx ¦xk2 upk m2x . k 0
ɇɚɩɪɢɦɟɪ, ɜ ɬɚɛɥ. 8.2 ɢ 8.3 ɡɚɞɚɧɵ ɡɚɤɨɧɵ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɞɢɫɤɪɟɬɧɵɯ ɜɟɥɢɱɢɧ Q ɢ R, ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ. ɇɚɣɞɟɦ ɱɢɫɥɨɜɵɟ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ
ɷɬɢɯ ɫɥɭɱɚɣɧɵɯ ɜɟɥɢɱɢɧ.
ɇɚ ɪɢɫ. 8.6 ɩɨɤɚɡɚɧɨ ɪɚɡɦɟɳɟɧɢɟ ɨɬɫɱɟɬɨɜ ɫɥɭɱɚɣɧɵɯ ɜɟɥɢɱɢɧ Q ɢ R ɧɚ ɱɢɫɥɨɜɨɣ ɩɪɹɦɨɣ. ɋɧɚɱɚɥɚ ɩɨ ɮɨɪɦɭɥɟ (8.11) ɜɵɱɢɫɥɹɟɦ ɦɚɬɟɦɚɬɢɱɟɫɤɢɟ ɨɠɢɞɚɧɢɹ ɞɥɹ ɫɥɭɱɚɣɧɵɯ ɜɟɥɢɱɢɧ Q ɢ R:
mq 1u0.3 2u0.5 5u0.4 7u0.1 4
,
mr 1u0.2 2u0.7 7u0.2 10u0.1 4.
Ʉɚɤ ɨɤɚɡɚɥɨɫɶ, Q ɢ R ɢɦɟɸɬ ɨɞɢɧɚɤɨɜɵɟ ɫɪɟɞɧɢɟ: ɢ mq 4, ɢ mr 4. ɇɨ ɥɟɝɤɨ ɡɚɦɟɬɢɬɶ (ɪɢɫ. 8.6), ɱɬɨ ɨɬɫɱɟɬɵ R ɨɬɧɨɫɢɬɟɥɶɧɨ mr ɪɚɡɛɪɨɫɚɧɵ ɫɢɥɶɧɟɟ, ɱɟɦ ɨɬɫɱɟɬɵ Q ɨɬɧɨɫɢɬɟɥɶɧɨ mq.
ɉɨ ɮɨɪɦɭɥɚɦ (8.12) ɢ (8.10) ɜɵɱɢɫɥɢɦ ɞɢɫɩɟɪɫɢɢ ɢ ɋɄɈ ɞɥɹ ɫɥɭɱɚɣɧɵɯ ɜɟɥɢɱɢɧ Q ɢ R:
Dq 12u0.3 22u0.5 52u0.4 72u0.1 42 1.2,
Vq 1.1,
Dr ( 1)2u0.2 22u0.7 72u0.2 102u0.1 42 6.8,
Vr 2.6.
Ʉɚɤ ɜɢɞɢɦ, ɛɨɥɶɲɟɦɭ ɪɚɡɛɪɨɫɭ ɨɬɫɱɟɬɨɜ ɫɥɭɱɚɣɧɨɣ ɜɟɥɢɱɢɧɵ ɨɬɜɟɱɚɸɬ ɛɨɥɶɲɢɟ ɞɢɫɩɟɪɫɢɹ ɢ ɋɄɈ.
ɉɪɢɦɟɪ. ɇɚɣɬɢ ɱɢɫɥɨɜɵɟ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɞɢɫɤɪɟɬɧɨɣ ɫɥɭɱɚɣɧɨɣ ɜɟɥɢɱɢɧɵ Z (ɬɚɛɥ. 8.1).
Ɋɟɲɟɧɢɟ. Ⱦɟɣɫɬɜɭɹ ɩɨ ɮɨɪɦɭɥɟ (8.11), ɧɚɯɨɞɢɦ ɆɈ ɞɥɹ ɋȼ Z:
M[z] mz 0u0.064 1u0.288 2u0.432 3u0.216 1.8.
Ɂɧɚɱɢɬ, ɰɟɧɬɪɨɦ ɬɹɠɟɫɬɢ ɞɥɹ ɬɨɱɟɤ z {0, 1, 2, 3} ɢɡ (ɬɚɛɥ. 8.1) ɛɭɞɟɬ ɬɨɱɤɚ mz 1.8.
Ⱦɟɣɫɬɜɭɟɦ ɩɨ ɮɨɪɦɭɥɚɦ (8.10) ɢ (8.10):
Dz 02u0.064 12u0.288 22u0.432 32u0.216 1.82 0.72.
123
Vz 0.85.
Ɋɚɫɫɦɨɬɪɢɦ ɱɢɫɥɨɜɵɟ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɧ ɟ ɩ ɪ ɟ ɪ ɵ ɜ ɧ ɵ ɯ ɫɥɭɱɚɣɧɵɯ ɜɟɥɢɱɢɧ.
Ɏɨɪɦɭɥɭ ɞɥɹ ɦɚɬɟɦɚɬɢɱɟɫɤɨɝɨ ɨɠɢɞɚɧɢɹ ɧ ɟ ɩ ɪ ɟ ɪ ɵ ɜ ɧ ɨ ɣ ɫɥɭɱɚɣɧɨɣ ɜɟɥɢɱɢɧɵ ɩɨɥɭɱɢɦ, ɟɫɥɢ ɜ ɫɨɨɬɧɨɲɟɧɢɢ (8.11) ɜɵɩɨɥɧɢɦ ɬɚɤɢɟ ɡɚɦɟɧɵ:
9ɨɬɫɱɟɬɵ xk ɧɚ ɩɟɪɟɦɟɧɧɭɸ x,
9ɜɟɪɨɹɬɧɨɫɬɶ pk ɧɚ ɷɥɟɦɟɧɬ ɜɟɪɨɹɬɧɨɫɬɢ f(x)udx,
9ɫɭɦɦɭ n ɫɥɚɝɚɟɦɵɯ – ɧɚ ɢɧɬɟɝɪɚɥ ɜ ɛɟɫɤɨɧɟɱɧɵɯ ɩɪɟɞɟɥɚɯ:
f |
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M[X] ³x uf(x)udx . |
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Ⱦɢɫɩɟɪɫɢɸ ɧɟɩɪɟɪɵɜɧɨɣ ɫɥɭɱɚɣɧɨɣ ɜɟɥɢɱɢɧɵ ɜɵɱɢɫɥɹɸɬ ɬɚɤ:
f
Dx ³ x mx 2 uf(x)udx .
f
ɇɚɩɪɢɦɟɪ, ɫɥɭɱɚɣɧɵɟ ɜɟɥɢɱɢɧɵ S ɢ T ɡɚɞɚɧɵ ɫɜɨɢɦɢ ɩɥɨɬɧɨɫɬɹɦɢ ɪɚɫɩɪɟɞɟɥɟɧɢɹ (ɪɢɫ. 8.7):
0 ɩɪɢs 1, f(s) °®0.5 ɩɪɢ1d s d 3, °¯0 ɩɪɢs ! 3.
0 ɩɪɢ t 0, f(t) °®0.25 ɩɪɢ0 d t d 4,
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ɩɪɢ t ! 4. |
¯0 |
f(s) (8.14)
0.5
f(t)
0.2
s
0 1 2 3 4 t
Ɋɢɫ. 8.7
ɉɥɨɬɧɨɫɬɢ ɪɚɫɩɪɟɞɟɥɟɧɢɹ f(s) ɢ f(t) ɨɬɜɟɱɚɸɬ ɫɜɨɣɫɬɜɭ 2: ɩɥɨɳɚɞɶ ɩɨɞ ɤɚɠɞɨɣ ɢɡ ɧɢɯ ɪɚɜɧɚ ɟɞɢɧɢɰɟ.
ɇɚɣɞɟɦ ɱɢɫɥɨɜɵɟ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɫɥɭɱɚɣɧɵɯ ɜɟɥɢɱɢɧ S ɢ T. ɉɨ ɮɨɪɦɭɥɟ (8.13) ɜɵɱɢɫɥɹɟɦ ɦɚɬɟɦɚɬɢɱɟɫɤɢɟ ɨɠɢɞɚɧɢɹ:
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³su0.5uds |
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Ⱥ ɬɟɩɟɪɶ ɞɥɹ S ɢ T ɜɵɱɢɫɥɹɟɦ ɞɢɫɩɟɪɫɢɢ ɩɨ ɮɨɪɦɭɥɟ (8.14) ɢ ɋɄɈ ɩɨ ɮɨɪɦɭɥɟ (8.10):
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ɂ ɜ ɞɚɧɧɨɦ ɫɥɭɱɚɟ ɫɥɭɱɚɣɧɚɹ ɜɟɥɢɱɢɧɚ, ɡɧɚɱɟɧɢɹ ɤɨɬɨɪɨɣ ɡɚɧɢɦɚɸɬ ɧɚ ɱɢɫɥɨɜɨɣ ɨɫɢ ɛɨɥɟɟ ɲɢɪɨɤɭɸ ɡɨɧɭ, ɢɦɟɟɬ ɛɨɥɶɲɢɟ ɞɢɫɩɟɪɫɢɸ ɢ ɋɄɈ.
ɉɪɢɦɟɪ. ȼɵɱɢɫɥɢɬɶ ɱɢɫɥɨɜɵɟ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɫɥɭɱɚɣɧɨɣ ɜɟɥɢɱɢɧɵ W, ɪɚɫɩɪɟɞɟɥɟɧɧɨɣ ɩɨ ɡɚɤɨɧɭ (8.4).
Ɋɟɲɟɧɢɟ. ɉɨɫɤɨɥɶɤɭ ɡɚɞɚɧɧɚɹ f(w) 0 ɩɪɢ w 0 ɧɢɠɧɢɣ ɩɪɟɞɟɥ ɢɧɬɟɝɪɚɥɚ (8.13) ɪɚɜɟɧ 0, ɢ
f
M[W] mw ³x ue x udx ¢Ƚɥ. 6, ɢɧɬɟɝɪɢɪɨɜɚɧɢɟ ɩɨ ɱɚɫɬɹɦ² 1.
0
Ⱦɟɣɫɬɜɭɟɦ ɩɨ ɮɨɪɦɭɥɟ (8.14), ɜ ɤɨɬɨɪɨɣ ɧɢɠɧɢɣ ɩɪɟɞɟɥ ɢɧɬɟɝɪɚɥɚ ɪɚɜɟɧ 0:
f
Dw ³ w 1 2 ue w udw ¢ɜɵɱɢɫɥɹɟɦ ɜ Mathcad² 1, Vw 1.
0
8.4.Канонические распределения случайных величин
ȼɬɟɨɪɢɢ ɜɟɪɨɹɬɧɨɫɬɟɣ ɨɩɟɪɢɪɭɸɬ ɫ ɛɨɥɶɲɢɦ ɤɨɥɢɱɟɫɬɜɨɦ ɪɚɫɩɪɟɞɟɥɟɧɢɣ ɫɥɭɱɚɣɧɵɯ ɜɟɥɢɱɢɧ. Ɇɧɨɝɢɟ ɢɡ ɧɢɯ ɫɱɢɬɚɸɬɫɹ ɤɚɧɨɧɢɱɟɫɤɢɦɢ. Ɇɵ ɢɡɭɱɢɦ ɨɞɧɨ ɢɡ ɤɚɧɨɧɢɱɟɫɤɢɯ ɪɚɫɩɪɟɞɟɥɟɧɢɣ ɞɥɹ ɞɢɫɤɪɟɬɧɵɯ ɫɥɭɱɚɣɧɵɯ ɜɟɥɢɱɢɧ ɢ ɨɞɧɨ – ɞɥɹ ɧɟɩɪɟɪɵɜɧɵɯ.
Ȼɢɧɨɦɢɚɥɶɧɨɟ ɪɚɫɩɪɟɞɟɥɟɧɢɟ. ɉɭɫɬɶ ɜɵɩɨɥɧɹɟɬɫɹ ɫɟɪɢɹ ɢɡ n
ɨɩɵɬɨɜ. ȼ ɤɚɠɞɨɦ ɨɩɵɬɟ ɫ ɜɟɪɨɹɬɧɨɫɬɶɸ p const ɩɪɨɢɫɯɨɞɢɬ (ɚ ɫ ɜɟɪɨɹɬɧɨɫɬɶɸ q 1 p ɧɟ ɩɪɨɢɫɯɨɞɢɬ) ɫɨɛɵɬɢɟ A. Ɉɩɵɬ ɫɱɢɬɚɟɬɫɹ ɭɞɚɱɧɵɦ, ɟɫɥɢ ɜ ɷɬɨɦ ɨɩɵɬɟ ɫɨɛɵɬɢɟ A ɫɥɭɱɢɥɨɫɶ. Ʉɨɥɢɱɟɫɬɜɨ ɭɞɚɱɧɵɯ ɨɩɵɬɨɜ ɜ ɬɚɤɨɣ ɫɟɪɢɢ ɟɫɬɶ ɫɥɭɱɚɣɧɚɹ ɞɢɫɤɪɟɬɧɚɹ ɜɟɥɢɱɢɧɚ X. Ɉɧɚ ɦɨɠɟɬ ɩɪɢɧɢɦɚɬɶ ɰɟɥɵɟ ɧɟɨɬɪɢɰɚɬɟɥɶɧɵɟ ɡɧɚɱɟɧɢɹ X {0, 1, 2, 3, , n}. ȼɟɪɨɹɬɧɨɫɬɶ ɬɨɝɨ, ɱɬɨ ɫɨɛɵɬɢɟ A ɩɪɨɢɡɨɣɞɟɬ ɜ ɫɟɪɢɢ ɢɡ n ɨɩɵɬɨɜ ɪɨɜɧɨ m ɪɚɡ, ɬɨ ɟɫɬɶ X ɩɪɢɦɟɬ ɡɧɚɱɟɧɢɟ, ɪɚɜɧɨɟ m, ɜɵɱɢɫɥɹɟɬɫɹ ɩɨ ɮɨɪɦɭɥɟ Ȼɟɪɧɭɥɥɢ:
P(X m) |
Pm |
Cm upm uqn m . |
(8.15) |
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Ɍɨɝɞɚ ɞɢɫɤɪɟɬɧɚɹ ɫɥɭɱɚɣɧɚɹ ɜɟɥɢɱɢɧɚ X ɪɚɫɩɪɟɞɟɥɟɧɚ ɩɨ ɛɢɧɨɦɢɚɥɶɧɨɦɭ ɡɚɤɨɧɭ ɫ ɩɚɪɚɦɟɬɪɚɦɢ n ɢ p. ɂɫɫɥɟɞɭɟɦ, ɤɚɤ ɜɟɞɟɬ ɫɟɛɹ ɮɭɧɤɰɢɹ (8.15) ɩɪɢ p const ɢ n var.
125
ɇɚ ɪɢɫ. 8.8 ɩɨɤɚɡɚɧɵ ɝɪɚɮɢɤɢ ɛɢɧɨɦɢɚɥɶɧɨɝɨ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɫ |
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ɩɚɪɚɦɟɬɪɚɦɢ: n |
{3, 6, 10, 13} |
var, p 0.6 |
const. Ʌɨɦɚɧɚɹ Pk |
ɧɚɡɵɜɚɟɬ- |
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ɫɹ ɦɧɨɝɨɭɝɨɥɶɧɢɤɨɦ ɪɚɫɩɪɟɞɟɥɟɧɢɹ. Ɇɧɨɝɨɭɝɨɥɶɧɢɤ ɪɚɫɩɪɟɞɟɥɟɧɢɹ – |
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ɟɳɟ ɨɞɢɧ ɫɩɨɫɨɛ ɡɚɞɚɬɶ ɡɚɤɨɧ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɞɢɫɤɪɟɬɧɨɣ ɫɥɭɱɚɣɧɨɣ |
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ɜɟɥɢɱɢɧɵ. |
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ɉɟɪɜɵɣ ɦɧɨɝɨɭɝɨɥɶɧɢɤ ɧɚ ɪɢɫ. 8.8 (ɫ ɤɪɭɝɥɵɦɢ ɦɚɪɤɟɪɚɦɢ) ɩɪɟɞ- |
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ɫɬɚɜɥɹɟɬ ɫɨɛɨɸ ɛɢɧɨɦɢɚɥɶɧɵɣ ɡɚɤɨɧ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɫɥɭɱɚɣɧɨɣ ɜɟɥɢ- |
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ɱɢɧɵ Z (ɬɚɛɥ. 8.1). ɉɚɪɚɦɟɬɪɵ ɷɬɨɝɨ ɪɚɫɩɪɟɞɟɥɟɧɢɹ n |
3, p |
0.6. |
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Ɇɚɬɟɦɚɬɢɱɟɫɤɨɟ ɨɠɢɞɚɧɢɟ, ɞɢɫɩɟɪɫɢɸ ɢ ɋɄɈ ɞɥɹ ɫɥɭɱɚɣɧɨɣ ɜɟ- |
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ɥɢɱɢɧɵ X, ɪɚɫɩɪɟɞɟɥɟɧɧɨɣ ɩɨ ɛɢɧɨɦɢɚɥɶɧɨɦɭ ɡɚɤɨɧɭ, ɜɵɱɢɫɥɹɸɬ ɩɨ |
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ɮɨɪɦɭɥɚɦ: |
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mx nup, Dx nupuq mxuq, Vx |
n up u q . |
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ɇɚɣɞɟɦ ɱɢɫɥɨɜɵɟ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɫɥɭɱɚɣɧɵɯ ɜɟɥɢɱɢɧ ɧɚ ɪɢɫ. |
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8.8: |
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0.3 |
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n 10 |
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n 13 |
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4 Vn 10 6 Vn 10 |
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mn 3 |
mn 6 |
mn 10 |
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Vn 3 Vn 3 |
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Ɋɢɫ.8.8 |
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mn 3 |
3u0.6 |
1.8, |
Dn 3 |
1.8u0.4 |
0.72, |
Vn 3 |
0.72 |
0.85. |
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mn 6 |
6u0.6 |
3.6, |
Dn 6 |
3.6u0.4 |
1.44, |
Vn 6 |
1.44 |
1.20. |
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mn 10 |
10u0.6 |
6.0, |
Dn 10 |
6.0u0.4 |
2.40, |
Vn 10 |
2.40 |
1.55. |
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mn 13 |
13u0.6 |
7.8, |
Dn 13 |
7.8u0.4 |
3.12, |
Vn 13 |
3.12 |
1.77. |
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ɉɨɥɭɱɟɧɧɵɟ ɡɧɚɱɟɧɢɹ ɆɈ ɧɚɧɟɫɟɧɵ ɧɚ ɨɫɶ 0k (ɪɢɫ. 8.8). ɉɪɢ ɷɬɨɦ ɦɚɪɤɟɪɵ ɧɚ ɦɧɨɝɨɭɝɨɥɶɧɢɤɟ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɫɥɭɱɚɣɧɨɣ ɜɟɥɢɱɢɧɵ ɢ ɟɟ ɦɚɬɟɦɚɬɢɱɟɫɤɨɝɨ ɨɠɢɞɚɧɢɹ ɫɨɜɩɚɞɚɸɬ, ɬɨɥɶɤɨ ɦɚɪɤɟɪ ɆɈ ɱɟɪɧɵɣ. Ɂɞɟɫɶ ɠɟ ɫɟɪɵɦɢ ɨɬɪɟɡɤɚɦɢ ɩɨɤɚɡɚɧɵ ɋɄɈ ɞɥɹ ɤɚɠɞɨɣ ɋȼ.
Ʉɚɤ ɜɢɞɢɦ, ɩɪɢ ɭɜɟɥɢɱɟɧɢɢ n ɦɧɨɝɨɭɝɨɥɶɧɢɤɢ ɛɢɧɨɦɢɚɥɶɧɨɝɨ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɫɦɟɳɚɸɬɫɹ ɜɩɪɚɜɨ, ɢ ɱɟɦ ɛɨɥɶɲɟ n, ɬɟɦ ɜɫɟ ɛɨɥɟɟ ɫɢɦ-
ɦɟɬɪɢɱɧɨɣ ɢ ɩɥɚɜɧɨɣ ɫɬɚɧɨɜɢɬɫɹ ɥɨɦɚɧɚɹ Pnk . ɉɪɢ ɷɬɨɦ ɦɚɤɫɢɦɭɦɵ
n
ɦɧɨɝɨɭɝɨɥɶɧɢɤɨɜ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɭɦɟɧɶɲɚɸɬɫɹ, ɩɨɫɤɨɥɶɤɭ ¦Pnk 1.
k 0
Ɂɧɚɱɟɧɢɹ ɆɈ ɛɥɢɡɤɢ ɤ ɚɛɫɰɢɫɫɚɦ ɦɚɤɫɢɦɭɦɨɜ ɦɧɨɝɨɭɝɨɥɶɧɢɤɨɜ ɪɚɫɩɪɟɞɟɥɟɧɢɹ. Ⱥ ɫ ɪɨɫɬɨɦ n ɪɚɫɬɟɬ ɢ ɋɄɈ (ɪɚɫɫɟɹɧɢɟ ɨɬɫɱɟɬɨɜ ɋȼ ɨɬɧɨɫɢɬɟɥɶɧɨ ɟɟ ɆɈ).
ɇɨɪɦɚɥɶɧɨɟ ɪɚɫɩɪɟɞɟɥɟɧɢɟ. ɇɟɩɪɟɪɵɜɧɚɹ ɫɥɭɱɚɣɧɚɹ ɜɟɥɢɱɢɧɚ X ɩɨɞɱɢɧɟɧɚ ɧɨɪɦɚɥɶɧɨɦɭ ɡɚɤɨɧɭ, ɟɫɥɢ ɩɥɨɬɧɨɫɬɶ ɟɟ ɜɟɪɨɹɬɧɨɫɬɢ ɨɩɢɫɵɜɚɟɬɫɹ ɫɨɨɬɧɨɲɟɧɢɟɦ:
f(x) |
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x m 2 |
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ue 2uV2 . |
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Vu |
2uS |
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ɉɚɪɚɦɟɬɪɵ ɧɨɪɦɚɥɶɧɨɝɨ ɪɚɫɩɪɟɞɟɥɟɧɢɹ m ɢ V — ɧɟ ɱɬɨ ɢɧɨɟ, ɤɚɤ ɱɢɫɥɨɜɵɟ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɫɥɭɱɚɣɧɨɣ ɜɟɥɢɱɢɧɵ X: m – ɟɟ ɦɚɬɟɦɚɬɢɱɟɫɤɨɟ ɨɠɢɞɚɧɢɟ, V2 – ɞɢɫɩɟɪɫɢɹ X, ɚ V – ɟɟ ɋɄɈ.
ɉɨ ɨɩɪɟɞɟɥɟɧɢɸ (8.5) ɮɭɧɤɰɢɹ ɧɨɪɦɚɥɶɧɨɝɨ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɢɦɟɟɬ ɜɢɞ:
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t m 2 |
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F(x) |
u ³e 2uV2 udt . |
(8.17) |
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Vu |
2uS |
f |
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ɉɥɨɬɧɨɫɬɶ ɜɟɪɨɹɬɧɨɫɬɢ (8.16) ɧɟ ɢɦɟɟɬ ɩɟɪɜɨɨɛɪɚɡɧɨɣ ɜ ɚɧɚɥɢɬɢɱɟɫɤɨɣ ɮɨɪɦɟ (ɝɨɜɨɪɹɬ, ɱɬɨ ɢɧɬɟɝɪɚɥ (8.17) ɧɟ ɛɟɪɟɬɫɹ, ɬɨ ɟɫɬɶ ɧɟ ɜɵɪɚɠɚɟɬɫɹ ɜ ɷɥɟɦɟɧɬɚɪɧɵɯ ɮɭɧɤɰɢɹɯ). ȼ ɦɚɬɟɦɚɬɢɤɟ ɪɚɡɪɚɛɨɬɚɧɵ ɷɮɮɟɤɬɢɜɧɵɟ ɚɥɝɨɪɢɬɦɵ ɬɚɛɥɢɱɧɨɝɨ ɩɪɟɞɫɬɚɜɥɟɧɢɹ ɮɭɧɤɰɢɢ ɧɨɪɦɚɥɶɧɨɝɨ ɪɚɫɩɪɟɞɟɥɟɧɢɹ. ɉɪɚɜɞɚ, ɬɚɛɥɢɰɵ ɫɬɪɨɢɥɢɫɶ ɞɥɹ ɧɨɪɦɢɪɨɜɚɧɧɵɯ ɩɚɪɚɦɟɬɪɨɜ ɧɨɪɦɚɥɶɧɨɝɨ ɪɚɫɩɪɟɞɟɥɟɧɢɹ: m 0 ɢ V 1. ȼ ɛɵɥɵɟ ɜɪɟɦɟɧɚ ɬɚɤɢɟ ɬɚɛɥɢɰɵ ɢɦɟɥɢɫɶ ɜɨ ɜɫɟɯ ɭɱɟɛɧɢɤɚɯ ɢ ɫɩɪɚɜɨɱɧɢɤɚɯ ɩɨ ɬɟɨɪɢɢ ɜɟɪɨɹɬɧɨɫɬɟɣ ɢ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɫɬɚɬɢɫɬɢɤɟ. ɂɯ ɢɫɩɨɥɶɡɨɜɚɧɢɟ ɞɥɹ ɪɟɲɟɧɢɹ ɤɨɧɤɪɟɬɧɵɯ ɡɚɞɚɱ ɫ ɪɚɡɦɟɪɧɵɦɢ x, m, V ɬɪɟɛɨɜɚɥɨ ɨɬ ɢɫɩɨɥɧɢɬɟɥɟɣ ɱɪɟɡɦɟɪɧɵɯ ɡɚɬɪɚɬ ɬɪɭɞɚ ɢ ɜɪɟɦɟɧɢ.
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ɋɟɝɨɞɧɹ ɷɬɢ ɚɥɝɨɪɢɬɦɵ ɪɟɚɥɢɡɨɜɚɧɵ ɜ ɫɢɫɬɟɦɚɯ ɤɨɦɩɶɸɬɟɪɧɨɣ ɦɚɬɟɦɚɬɢɤɢ. Ɍɚɤ, ɜ Mathcad ɮɨɪɦɭɥɚ (8.16) ɩɪɟɞɫɬɚɜɥɟɧɚ ɜɫɬɪɨɟɧɧɨɣ ɮɭɧɤɰɢɟɣ dnorm(x,m,V), ɚ ɮɨɪɦɭɥɚ (8.17) – ɜɫɬɪɨɟɧɧɨɣ ɮɭɧɤɰɢɟɣ pnorm(x,m,V). ɉɪɢ ɢɫɩɨɥɶɡɨɜɚɧɢɢ ɷɬɢɯ ɮɭɧɤɰɢɣ ɧɟɬ ɧɭɠɞɵ ɧɨɪɦɢɪɨɜɚɬɶ ɢɫɯɨɞɧɵɟ ɞɚɧɧɵɟ ɪɟɲɚɟɦɨɣ ɡɚɞɚɱɢ.
ɉɥɨɬɧɨɫɬɶ ɜɟɪɨɹɬɧɨɫɬɢ (8.16) ɢ ɡɚɤɨɧ ɪɚɫɩɪɟɞɟɥɟɧɢɹ (8.17) ɧɨɪɦɚɥɶɧɨ ɪɚɫɩɪɟɞɟɥɟɧɧɨɣ ɋȼ – ɮɭɧɤɰɢɢ ɞɜɭɯ ɩɚɪɚɦɟɬɪɨɜ: ɟɟ ɆɈ m ɢ ɟɟ ɋɄɈ V. ɇɚ ɪɢɫ. 8.9 ɩɪɢɜɟɞɟɧɵ ɝɪɚɮɢɤɢ ɩɥɨɬɧɨɫɬɢ ɜɟɪɨɹɬɧɨɫɬɢ f(x) (ɪɢɫ. 8.9,ɚ) ɢ ɮɭɧɤɰɢɢ ɪɚɫɩɪɟɞɟɥɟɧɢɹ F(x) (ɪɢɫ. 8.9,ɛ) ɧɨɪɦɚɥɶɧɨ ɪɚɫɩɪɟɞɟɥɟɧɧɨɣ ɋȼ. Ʉɚɤ ɜɢɞɢɦ, ɭɜɟɥɢɱɟɧɢɟ ɦɚɬɟɦɚɬɢɱɟɫɤɨɝɨ ɨɠɢɞɚɧɢɹ m ɩɪɢ ɩɨɫɬɨɹɧɧɨɦ ɋɄɈ V ɩɪɢɜɨɞɢɬ ɤ ɫɦɟɳɟɧɢɸ ɤɪɢɜɵɯ f(x) ɢ F(x) ɜɩɪɚɜɨ ɜɞɨɥɶ ɨɫɢ ɚɛɫɰɢɫɫ. Ⱥ ɛɨɥɶɲɟɦɭ ɡɧɚɱɟɧɢɸ V ɩɪɢ ɨɞɧɨɦ ɢ ɬɨɦ ɠɟ m ɨɬɜɟɱɚɸɬ ɛɨɥɟɟ ɩɨɥɨɝɢɟ ɤɪɢɜɵɟ f(x) ɢ F(x) (ɫ ɛɨɥɶɲɢɦ ɪɚɡɛɪɨɫɨɦ ɡɧɚɱɟɧɢɣ X).
f(x,m1,V) f(x,m2,V)
m2!m1
V2!V1
f(x,m2,V2)
x
ɚ) |
m1 |
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m2 |
1.0F(x,m1,V1) F(x,m2,V1)
F(x,m2,V2)
0.5
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ɛ) |
m1 |
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Ɋɢɫ. 8.9 |
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ȼ ɨɬɧɨɲɟɧɢɢ ɧɨɪɦɚɥɶɧɨ ɪɚɫɩɪɟɞɟɥɟɧɧɨɣ ɫɥɭɱɚɣɧɨɣ ɜɟɥɢɱɢɧɵ X ɞɟɣɫɬɜɭɟɬ «ɩ ɪ ɚ ɜ ɢ ɥ ɨ ɬ ɪ ɟ ɯ ɫ ɢ ɝ ɦ »:
ɫ ɜɟɪɨɹɬɧɨɫɬɶɸ, ɛɥɢɡɤɨɣ ɤ ɟɞɢɧɢɰɟ, ɡɧɚɱɟɧɢɹ ɧɨɪɦɚɥɶɧɨ ɪɚɫɩɪɟɞɟɥɟɧɧɨɣ ɫɥɭɱɚɣɧɨɣ ɜɟɥɢɱɢɧɵ X ɥɟɠɚɬ ɜ ɢɧɬɟɪɜɚɥɟ
]m 3uV, m 3uV[,
ɬɨ ɟɫɬɶ
P(m 3uVX m 3uV) 0.997#1.
Ⱦɪɭɝɢɦɢ ɫɥɨɜɚɦɢ, ɜɟɥɢɱɢɧɚ X ɩɪɚɤɬɢɱɟɫɤɢ ɧɟ ɨɬɤɥɨɧɹɟɬɫɹ ɨɬ ɟɟ ɦɚɬɟɦɚɬɢɱɟɫɤɨɝɨ ɨɠɢɞɚɧɢɹ m ɧɚ ɪɚɫɫɬɨɹɧɢɟ ɛɨɥɶɲɟ ɬɪɟɯ ɋɄɈ V.
f(x)
P( 3 X 3) 0.997
V 1
m 0
3uV 3uV
Ɋɢɫ. 8.10
«ɉɪɚɜɢɥɨ ɬɪɟɯ ɫɢɝɦ» ɞɥɹ ɧɨɪɦɢɪɨɜɚɧɧɵɯ ɩɚɪɚɦɟɬɪɨɜ ɧɨɪɦɚɥɶɧɨɝɨ ɪɚɫɩɪɟɞɟɥɟɧɢɹ m 0 ɢ V 1 ɢɥɥɸɫɬɪɢɪɭɟɬ ɪɢɫ. 8.10. (Ⱥ ɩɪɨɧɨɪɦɢɪɨɜɚɬɶ ɦɨɠɧɨ ɥɸɛɵɟ ɩɚɪɚɦɟɬɪɵ ɤɨɧɤɪɟɬɧɨɣ ɧɨɪɦɚɥɶɧɨ ɪɚɫɩɪɟɞɟɥɟɧɧɨɣ ɫɥɭɱɚɣɧɨɣ ɜɟɥɢɱɢɧɵ). Ɂɚɲɬɪɢɯɨɜɚɧɧɚɹ ɩɥɨɳɚɞɶ ɩɨɞ ɤɪɢɜɨɣ f(x) ɪɚɜɧɚ ɜɟɪɨɹɬɧɨɫɬɢ ɩɨɩɚɞɚɧɢɹ ɋȼ X ɜ ɞɢɚɩɚɡɨɧ ɨɬ 3 ɞɨ 3. Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ, ɫɨɛɵɬɢɟ ( 3 X 3) ɩɪɚɤɬɢɱɟɫɤɢ ɞɨɫɬɨɜɟɪɧɨɟ. Ⱥ ɜɟɪɨɹɬɧɨɫɬɶ ɜɵɯɨɞɚ ɟɟ ɡɧɚɱɟɧɢɣ ɡɚ ɩɪɟɞɟɥɵ ɷɬɨɝɨ ɞɢɚɩɚɡɨɧɚ ɪɚɜɧɚ ɧɟɡɚɲɬɪɢɯɨɜɚɧɧɨɣ ɩɥɨɳɚɞɢ ɩɨɞ ɤɪɢɜɨɣ f(x) ɨɬ f ɞɨ 3 ɢ ɨɬ 3 ɞɨ f, ɤɨɬɨɪɚɹ ɫɨɫɬɚɜɥɹɟɬ 1 0.997 0.003. Ɂɧɚɱɢɬ, ɫɨɛɵɬɢɟ º( 3 X 3) ɩɪɚɤɬɢɱɟɫɤɢ ɧɟɜɨɡɦɨɠɧɨɟ.
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Ⱦɨɤɚɠɟɦ «ɩɪɚɜɢɥɨ ɬɪɟɯ ɫɢɝɦ» ɞɥɹ ɧɨɪɦɢɪɨɜɚɧɧɵɯ ɩɚɪɚɦɟɬɪɨɜ ɧɨɪɦɚɥɶɧɨɝɨ ɪɚɫɩɪɟɞɟɥɟɧɢɹ m 0 ɢ V 1. ɂɦɟɟɦ:
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x2 |
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f(x) |
ue 2 . |
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2uS
ɉɨ ɨɩɪɟɞɟɥɟɧɢɸ
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x2 |
3 |
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x2 |
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21uS u ³e |
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21uS u³e |
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P( 3 X 3) |
2 2u |
2 2uJ. |
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ɂɧɬɟɝɪɚɥ J ɜɵɱɢɫɥɹɟɦ ɦɟɬɨɞɨɦ Ɋɭɧɝɟ-Ɋɨɦɛɟɪɝɚ ɩɨ ɮɨɪɦɭɥɟ ɬɪɚɩɟɰɢɣ, ɩɨɫɤɨɥɶɤɭ f(x) ɩɟɪɜɨɨɛɪɚɡɧɨɣ ɧɟ ɢɦɟɟɬ ɢ ɮɨɪɦɭɥɭ ɇɶɸɬɨ- ɧɚ–Ʌɟɣɛɧɢɰɚ ɞɥɹ ɜɵɱɢɫɥɟɧɢɹ J ɩɪɢɦɟɧɢɬɶ ɧɟɥɶɡɹ.
1. ɋɬɪɨɢɦ ɬɚɛɥɢɰɭ f(x) ɫ ɲɚɝɨɦ h.
n |
4. h |
b a |
0.75. xt0 a 0, xti 1 xti h, i |
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0,n 1, yti f(xti). |
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i |
0 |
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3 |
4 |
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xti |
0.00 |
0.75 |
1.50 |
2.25 |
3.00 |
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yti |
0.3989 |
0.3011 |
0.1295 |
0.0317 |
0.0044 |
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j0 
1 
2
ȼɵɱɢɫɥɹɟɦ ɡɧɚɱɟɧɢɟ J ɩɨ ɮɨɪɦɭɥɟ ɬɪɚɩɟɰɢɣ ɧɚ ɲɚɝɟ h.
§ |
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ytn |
n 1 |
· |
¨ yt0 |
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¸ |
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Fh ¨ |
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¦yti ¸uh 0.4981. |
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i 1 |
¹ |
2. Ɏɨɪɦɢɪɭɟɦ ɬɚɛɥɢɰɭ ɫ ɲɚɝɨɦ 2uh:
n2 |
n |
2, h2 2uh, yt2j, j |
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0,n2 1. |
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ȼɵɱɢɫɥɹɟɦ ɡɧɚɱɟɧɢɟ J ɩɨ ɮɨɪɦɭɥɟ ɬɪɚɩɟɰɢɣ ɧɚ ɲɚɝɟ 2uh:
§ |
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yt2n |
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¨ yt20 |
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F2h ¨ |
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2 |
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¨ |
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© |
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n2 1 |
· |
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¸ |
¦yt2j ¸uh2 0.4968. |
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j 1 |
¸ |
¹ |
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3. ȼɵɱɢɫɥɹɟɦ ɩɨɩɪɚɜɤɭ Ɋɭɧɝɟ:
PR Fh F2h 0.0004.
3
4. ȼɵɱɢɫɥɹɟɦ ɡɧɚɱɟɧɢɟ J ɩɨ ɮɨɪɦɭɥɟ Ɋɭɧɝɟ–Ɋɨɦɛɟɪɝɚ:
130