Введение в теорию вероятностей. Учебное пособие
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Ɋɚɜɟɧɫɬɜɚ ɜ ɢ ɝ ɜɵɬɟɤɚɸɬ ɢɡ ɫɜɨɣɫɬɜɚ ɛ ɢ ɧɚɩɪɢɦɟɪ
x
³ f x dx F x F f F x F x
f
ПРИМЕР. ɉɥɨɬɧɨɫɬɶ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɧɟɤɨɬɨɪɨɣ ɧɟɩɪɟɪɵɜɧɨɣ ɫɥɭɱɚɣ
ɧɨɣ ɜɟɥɢɱɢɧɵ ɏ ɢɦɟɟɬ ɜɢɞ |
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ɝɞɟ ɫ ± ɩɨɫɬɨɹɧɧɚɹ ɜɟɥɢɱɢɧɚ ɇɚɣɬɢ ɡɧɚɱɟɧɢɟ ɫ ɮɭɧɤɰɢɸ ɪɚɫɩɪɟɞɟɥɟɧɢɹ F x ɢ ɜɟɪɨɹɬɧɨɫɬɶ P X
Ɋɟɲɟɧɢɟ ɋɨɝɥɚɫɧɨ ɫɜɨɣɫɬɜɭ ɝ ɧɨɪɦɢɪɨɜɚɧɧɨɫɬɢ ɩ ɢɦɟɟɦ
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ɨɬɤɭɞɚ c Ɍɟɩɟɪɶ |
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ɇɚɣɞɟɦ F x ɢɫɩɨɥɶɡɭɹ ɫɜɨɣɫɬɜɨ ɜ ɩ ɜ ɪɚɫɫɦɨɬɪɢɦ ɩɪɢ ɷɬɨɦ ɜɫɟ |
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ɜɨɡɦɨɠɧɵɟ ɡɞɟɫɶ ɫɥɭɱɚɢ x d x x t ɂɦɟɟɦ |
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ɩɪɢ x d |
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ɩɪɢ x
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F x ³ dx ³ x dx x
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F x ³ dx ³ x dx ³ dx |
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Ɍɚɤɢɦ ɨɛɪɚɡɨɦ ɫɦ ɪɢɫ
F x
< 
x
°®x d x °¯ x t
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Ðèñ. 3.1. Функция распределения
Ⱦɥɹ ɧɚɯɨɠɞɟɧɢɹ ɜɟɪɨɹɬɧɨɫɬɢ P X ɡɚɦɟɬɢɦ ɱɬɨ
> f >
Ɍɨɝɞɚ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɧɚɣɞɟɧɧɵɦ ɜɢɞɨɦ F x ɛɭɞɟɦ ɢɦɟɬɶ
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4. Числовые характеристики непрерывных случайных величин
ɉɭɫɬɶ ɏ± ɧɟɩɪɟɪɵɜɧɚɹ ɫɥɭɱɚɣɧɚɹ ɜɟɥɢɱɢɧɚ ɇɋȼ ɢ f x ± ɟɟ ɩɥɨɬ ɧɨɫɬɶ ɪɚɫɩɪɟɞɟɥɟɧɢɹ Ȼɭɞɟɦ ɧɚɡɵɜɚɬɶ ɦɚɬɟɦɚɬɢɱɟɫɤɢɦ ɨɠɢɞɚɧɢɟɦ ɢ ɞɢɫɩɟɪɫɢ ɟɣ ɏ ɡɧɚɱɟɧɢɹ ɫɥɟɞɭɸɳɢɯ ɢɧɬɟɝɪɚɥɨɜ
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M X ³ x f x dx |
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ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɉɪɢ ɷɬɨɦ ɦɵ ɩɪɟɞɩɨɥɚɝɚɟɦ ɫɯɨɞɢɦɨɫɬɶ ɧɟɫɨɛɫɬɜɟɧɧɵɯ ɢɧɬɟ ɝɪɚɥɨɜ ɢ ɩɪɢɱɟɦ ɜ ɫɥɭɱɚɟ ɬɪɟɛɭɟɦ ɫɯɨɞɢɦɨɫɬɶ ɚɛɫɨɥɸɬɧɭɸ
ɑɢɫɥɚ ɢ ɹɜɥɹɸɬɫɹ ɟɫɬɟɫɬɜɟɧɧɵɦɢ ɚɧɚɥɨɝɚɦɢ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɱɢɫɥɨɜɵɯ ɯɚɪɚɤɬɟɪɢɫɬɢɤ ɢ ɞɢɫɤɪɟɬɧɵɯ ɫɥɭɱɚɣɧɵɯ ɜɟɥɢɱɢɧ
ȼ ɬɨɦ ɫɥɭɱɚɟ ɤɨɝɞɚ ɜɫɟ ɜɨɡɦɨɠɧɵɟ ɡɧɚɱɟɧɢɹ ɏ ɪɚɫɩɨɥɨɠɟɧɵ ɧɚ ɨɬɪɟɡɤɟ >a b@ ɢɧɬɟɝɪɢɪɨɜɚɧɢɟ ɜ ɢ ɞɨɥɠɧɨ ɛɵɬɶ ɨɱɟɜɢɞɧɨ ɡɚɦɟɧɟɧɨ ɢɧɬɟɝɪɢ ɪɨɜɚɧɢɟɦ ɩɨ >a b@
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ɂɡ ɨɩɪɟɞɟɥɟɧɢɹ ɞɢɫɩɟɪɫɢɢ ɢ ɫɜɨɣɫɬɜɚ ɧɟɨɬɪɢɰɚɬɟɥɶɧɨɫɬɢ f x
ɜɵɬɟɤɚɟɬ ɨɰɟɧɤɚ D X t
ɋɪɟɞɧɟɟ ɤɜɚɞɪɚɬɢɱɟɫɤɨɟ ɨɬɤɥɨɧɟɧɢɟ ɧɟɩɪɟɪɵɜɧɨɣ ɫɥɭɱɚɣɧɨɣ ɜɟɥɢɱɢɧɵ
ɨɩɪɟɞɟɥɹɟɬɫɹ ɜ ɜɢɞɟ
V X D X
Ⱦɥɹ ɜɵɱɢɫɥɟɧɢɹ ɞɢɫɩɟɪɫɢɢ ɦɨɠɧɨ ɜɨɫɩɨɥɶɡɨɜɚɬɶɫɹ ɬɚɤɠɟ ɮɨɪɦɭɥɨɣ
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Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ ɩɪɨɜɨɞɢɬɫɹ ɫ ɩɨɦɨɳɶɸ ɪɚɫɫɭɠɞɟɧɢɣ ɩɨɞɨɛɧɵɯ ɩ ɧɨ ɞɟɣɫɬɜɢɹ ɫ ɫɭɦɦɚɦɢ ɡɚɦɟɧɹɸɬɫɹ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɦɢ ɞɟɣɫɬɜɢɹɦɢ ɢɧɬɟ ɝɪɚɥɚɦɢ ɢ ɢɫɩɨɥɶɡɭɟɬɫɹ ɫɜɨɣɫɬɜɨ ɧɨɪɦɢɪɨɜɚɧɧɨɫɬɢ f x ɩ
ПРИМЕР ɇɟɩɪɟɪɵɜɧɚɹ ɫɥɭɱɚɣɧɚɹ ɜɟɥɢɱɢɧɚ ɏ ɡɚɞɚɧɚ ɮɭɧɤɰɢɟɣ ɪɚɫ ɩɪɟɞɟɥɟɧɢɹ
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ɇɚɣɬɢ ɩɥɨɬɧɨɫɬɶ ɪɚɫɩɪɟɞɟɥɟɧɢɹ I x ɦɚɬɟɦɚɬɢɱɟɫɤɨɟ ɨɠɢɞɚɧɢɟ Ɇ ɏ ɢ ɞɢɫɩɟɪɫɢɸ D X
Ɋɟɲɟɧɢɟ ɉɥɨɬɧɨɫɬɶ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɧɚɣɞɺɦ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɟɺ ɨɩɪɟɞɟɥɟ ɧɢɟɦ ɤɚɤ ɩɪɨɢɡɜɨɞɧɭɸ ɨɬ F x
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ɉɨɥɟɡɧɨ ɩɨɫɬɪɨɢɬɶ ɝɪɚɮɢɤɢ y F x ɢ y I x ɫɦ ɪɢɫ
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Ðèñ. 4.1. Графики функции и плотности распределения |
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Ⱦɚɥɟɟ ɫɨɝɥɚɫɧɨ ɢ ɛɭɞɟɦ ɢɦɟɬɶ |
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M X ³ x x dx |
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5. Специальные виды дискретных распределений
Ɋɚɜɧɨɦɟɪɧɨɟ ɪɚɫɩɪɟɞɟɥɟɧɢɟ Ⱦɋȼ ȼɵɲɟ ɜ ɩ ɛɵɥɚ ɜɜɟɞɟɧɚ ɪɚɜ ɧɨɦɟɪɧɨ ɪɚɫɩɪɟɞɟɥɺɧɧɚɹ ɞɢɫɤɪɟɬɧɚɹ ɫɥɭɱɚɣɧɚɹ ɜɟɥɢɱɢɧɚ ɏ ȿɺ ɱɢɫɥɨɜɵɟ ɯɚɪɚɤ ɬɟɪɢɫɬɢɤɢ ɢɦɟɸɬ ɜɢɞ
Ɇ ɏ |
ɯ ɯ ɯɩ |
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ɯ ɯ ɯQ |
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ɂɬɚɤ ɦɚɬɟɦɚɬɢɱɟɫɤɨɟ ɨɠɢɞɚɧɢɟ ɪɚɜɧɨɦɟɪɧɨ ɪɚɫɩɪɟɞɟɥɟɧɧɨɣ ɜɟɥɢɱɢɧɵ ɪɚɜɧɨ ɜ ɬɨɱɧɨɫɬɢ ɫɪɟɞɧɟɦɭ ɚɪɢɮɦɟɬɢɱɟɫɤɨɦɭ ɜɫɟɯ ɡɧɚɱɟɧɢɣ ɫɥɭɱɚɣɧɨɣ ɜɟɥɢɱɢ ɧɵ ɷɬɨɬ ɮɚɤɬ ɭɠɟ ɨɬɦɟɱɚɥɫɹ ɚ ɞɢɫɩɟɪɫɢɹ ɪɚɜɧɚ ɫɪɟɞɧɟɦɭ ɤɜɚɞɪɚɬɨɜ ɡɧɚɱɟɧɢɣ ɜɟɥɢɱɢɧɵ ɦɢɧɭɫ ɤɜɚɞɪɚɬ ɟɟ ɫɪɟɞɧɟɝɨ ɡɧɚɱɟɧɢɹ
Ƚɢɩɟɪɝɟɨɦɟɬɪɢɱɟɫɤɨɟ ɪɚɫɩɪɟɞɟɥɟɧɢɟ Ⱦɋȼ Ɋɚɫɫɦɨɬɪɢɦ ɡɚɞɚɱɭ ɨ ɜɵ ɛɨɪɤɟ ɩ ɝɥɚɜɵ ɂɡ N ɨɛɴɟɤɬɨɜ ɫɪɟɞɢ ɤɨɬɨɪɵɯ M ɦɟɱɟɧɵɯ d Ɇ d 1 ɫɥɭɱɚɣɧɵɦ ɨɛɪɚɡɨɦ ɢɡɜɥɟɤɚɟɬɫɹ k ɨɛɴɟɤɬɨɜ d k d N ȼ ɤɚɱɟɫɬɜɟ ɫɥɭɱɚɣ ɧɨɣ ɜɟɥɢɱɢɧɵ ɏ ɪɚɫɫɦɨɬɪɢɦ ɱɢɫɥɨ ɦɟɱɟɧɵɯ ɨɛɴɟɤɬɨɜ ɜ ɢɡɜɥɟɱɟɧɧɨɣ ɜɵɛɨɪɤɟ ȿɟ ɡɧɚɱɟɧɢɹ l ^ PLQ k M ` Ʉɚɤ ɩɨɤɚɡɚɧɨ ɜɵɲɟ
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Ƚɨɜɨɪɹɬ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɱɬɨ ɞɢɫɤɪɟɬɧɚɹ ɫɥɭɱɚɣɧɚɹ ɜɟɥɢɱɢɧɚ ɏ ɪɚɫɩɪɟɞɟɥɟɧɚ ɝɢɩɟɪɝɟɨɦɟɬɪɢɱɟɫɤɢɦ ɨɛɪɚɡɨɦ Ɇɨɠɧɨ ɞɨɤɚɡɚɬɶ ɱɬɨ
54
M X |
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Ȼɢɧɨɦɢɚɥɶɧɨɟ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɉɭɫɬɶ ɞɢɫɤɪɟɬɧɚɹ ɫɥɭɱɚɣɧɚɹ ɜɟɥɢɱɢɧɚ ɏ ɩɪɢɧɢɦɚɟɬ ɡɧɚɱɟɧɢɹ ɪɚɜɧɵɟ ɤɨɥɢɱɟɫɬɜɭ ɩɨɹɜɥɟɧɢɣ ɫɨɛɵɬɢɹ Ⱥ ɜ ɩ ɢɫɩɵɬɚɧɢ ɹɯ ɩɪɢ ɭɫɥɨɜɢɢ ɱɬɨ ɜ ɤɚɠɞɨɦ ɨɩɵɬɟ ɜɟɪɨɹɬɧɨɫɬɶ ɪ Ɋ Ⱥ ɨɞɧɚ ɢ ɬɚ ɠɟ q p ɫɯɟɦɚ Ȼɟɪɧɭɥɥɢ ȿɟ ɡɚɤɨɧ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɧɚɡɵɜɚɟɬɫɹ ɛɢɧɨɦɢɚɥɶɧɵɦ
ȼ ɩɟɪɜɨɣ ɫɬɪɨɤɟ ɪɹɞɚ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɜɟɥɢɱɢɧɵ ɏ ɛɭɞɭɬ ɡɚɩɢɫɚɧɵ ɜɨɡɦɨɠɧɵɟ
ɡɧɚɱɟɧɢɹ ɏ ɏ ɏ |
ɩ ɜɨ ɜɬɨɪɨɣ ɫɬɪɨɤɟ ± ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɜɟɪɨɹɬɧɨ |
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ɫɬɢ Ȼɟɪɧɭɥɥɢ |
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ɇɚɡɜɚɧɢɟ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɨɛɴɹɫɧɹɟɬɫɹ ɬɟɦ ɱɬɨ ɫɭɦɦɚ ɜɫɟɯ ɜɟɪɨɹɬɧɨɫɬɟɣ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɫɭɦɦɭ ɱɥɟɧɨɜ ɪɚɡɥɨɠɟɧɢɹ ɛɢɧɨɦɚ ɇɶɸɬɨɧɚ
p q n
Ȼɢɧɨɦɢɚɥɶɧɵɣ ɡɚɤɨɧ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɲɢɪɨɤɨ ɢɫɩɨɥɶɡɭɟɬɫɹ ɩɪɢ ɫɬɚɬɢɫɬɢɱɟ ɫɤɨɦ ɤɨɧɬɪɨɥɟ ɤɚɱɟɫɬɜɚ ɩɪɨɞɭɤɰɢɢ ɩɪɢ ɨɩɢɫɚɧɢɢ ɮɭɧɤɰɢɨɧɢɪɨɜɚɧɢɹ ɫɢɫɬɟɦ ɦɚɫɫɨɜɨɝɨ ɨɛɫɥɭɠɢɜɚɧɢɹ ɜ ɬɟɨɪɢɢ ɫɬɪɟɥɶɛɵ ɢ ɬ ɞ
ɑɢɫɥɨɜɵɟ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɛɢɧɨɦɢɚɥɶɧɨɣ ɫɥɭɱɚɣɧɨɣ ɜɟɥɢɱɢɧɵ ɏ ɢɦɟɸɬ ɜɢɞ
M X np D X npq
ɍɫɬɚɧɨɜɢɦ ɧɚɩɪɢɦɟɪ ɩɟɪɜɨɟ ɢɡ ɫɨɨɬɧɨɲɟɧɢɣ ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɪɹɞɨɦ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɛɭɞɟɦ ɢɦɟɬɶ
n
Ɇ ; ¦N&nk Sk Tn k
k
Ɂɚɦɟɬɢɦ ɱɬɨ ɫɭɦɦɢɪɨɜɚɧɢɟ ɮɚɤɬɢɱɟɫɤɢ ɩɪɨɢɡɜɨɞɢɬɫɹ ɩɨ k n ɬɚɤ ɤɚɤ ɩɪɢ k ɫɥɚɝɚɟɦɨɟ ɨɛɪɚɳɚɟɬɫɹ ɜ ɧɨɥɶ ɉɪɟɨɛɪɚɡɭɟɦ ɩɨɥɭɱɟɧɧɭɸ ɫɭɦɦɭ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ
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ɨɛɳɢɣ ɦɧɨɠɢɬɟɥɶ ɩɪ ɜɵɧɟɫɟɧ ɡɚ ɡɧɚɤ ɫɭɦɦɵ ɉɟɪɟɧɭɦɟɪɨɜɚɜ ɱɥɟɧɵ ɬɨ ɟɫɬɶ ɡɚɦɟɧɢɜ ɢɧɞɟɤɫ ɫɭɦɦɢɪɨɜɚɧɢɹ k ɧɚ k ɩɨɥɭɱɚɟɦ
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ɱɬɨ ɢ ɭɬɜɟɪɠɞɚɥɨɫɶ ɋɥɟɞɭɸɳɢɟ ɞɜɚ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɨɬɧɨɫɹɬɫɹ ɤ ɫɥɭɱɚɸ ɞɢɫɤɪɟɬɧɵɯ ɫɥɭɱɚɣɧɵɯ
ɜɟɥɢɱɢɧ ɫ ɛɟɫɤɨɧɟɱɧɵɦ ɩɟɪɟɱɧɟɦ ɡɧɚɱɟɧɢɣ
Ɋɚɫɩɪɟɞɟɥɟɧɢɟ ɉɭɚɫɫɨɧɚ Ƚɨɜɨɪɹɬ ɱɬɨ ɞɢɫɤɪɟɬɧɚɹ ɫɥɭɱɚɣɧɚɹ ɜɟɥɢɱɢ ɧɚ ɏ ɪɚɫɩɪɟɞɟɥɟɧɚ ɩɨ ɡɚɤɨɧɭ ɉɭɚɫɫɨɧɚ ɫ ɩɚɪɚɦɟɬɪɨɦ Ȝ ɟɫɥɢ ɨɧɚ ɩɪɢɧɢɦɚɟɬ ɡɧɚ ɱɟɧɢɹ ^ m ` ɫ ɜɟɪɨɹɬɧɨɫɬɹɦɢ
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Ɂɚɤɨɧ ɉɭɚɫɫɨɧɚ ɦɨɠɧɨ ɩɨɧɢɦɚɬɶ ɤɚɤ ©ɩɪɟɞɟɥɶɧɵɣ ɫɥɭɱɚɣª ɛɢɧɨɦɢɚɥɶɧɨɝɨ |
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ɡɚɤɨɧɚ ɜ ɤɨɬɨɪɨɦ n of ɢ O |
np |
ɫɨQVW Ɂɚɩɢɲɟɦ ɪɹɞ ɪɚɫɩɪɟɞɟɥɟɧɢɹ |
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Ʉɨɪɪɟɤɬɧɨɫɬɶ ɪɚɫɫɦɨɬɪɟɧɢɹ ©ɩɪɟɞɟɥɶɧɨɝɨª ɪɹɞɚ ɛɭɞɟɬ ɩɨɞɬɜɟɪɠɞɟɧɚ ɟɫɥɢ ɦɵ ɭɫɬɚɧɨɜɢɦ ɱɬɨ ɫɭɦɦɚ ɜɫɟɯ ɜɟɪɨɹɬɧɨɫɬɟɣ ɨɫɬɚɟɬɫɹ ɪɚɜɧɨɣ ɟɞɢɧɢɰɟ ɂɬɚɤ ɜɵɱɢɫɥɢɦ ɫɭɦɦɭ ɪɹɞɚ
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ɂɫɩɨɥɶɡɨɜɚɜ ɪɚɡɥɨɠɟɧɢɟ ɷɤɫɩɨɧɟɧɬɵ |
ɟO ɩɨ ɫɬɟɩɟɧɹɦ O ɪɹɞ Ɇɚɤɥɨɪɟɧɚ |
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ɩɨɥɭɱɚɟɦ ɫɭɦɦɭ ɜ ɜɢɞɟ |
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ɱɬɨ ɢ ɬɪɟɛɨɜɚɥɨɫɶ ɭɫɬɚɧɨɜɢɬɶ Ʉɚɤ ɨɤɚɡɵɜɚɟɬɫɹ ɦɚɬɟɦɚɬɢɱɟɫɤɨɟ ɨɠɢɞɚɧɢɟ ɢ ɞɢɫɩɟɪɫɢɹ ɫɥɭɱɚɣɧɨɣ ɜɟɥɢɱɢ
ɧɵ ɪɚɫɩɪɟɞɟɥɟɧɧɨɣ ɩɨ ɡɚɤɨɧɭ ɉɭɚɫɫɨɧɚ ɫɨɜɩɚɞɚɸɬ ɦɟɠɞɭ ɫɨɛɨɸ ɢ ɢ ɪɚɜɧɵ ɩɚɪɚɦɟɬɪɭ Ȝ ɷɬɨɝɨ ɪɚɫɩɪɟɞɟɥɟɧɢɹ
M X O D X O
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Ɍɚɤ ɧɚɩɪɢɦɟɪ
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Ƚɟɨɦɟɬɪɢɱɟɫɤɨɟ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɉɭɫɬɶ ɫɧɨɜɚ ɢɦɟɟɬɫɹ ɫɯɟɦɚ Ȼɟɪɧɭɥ |
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ɥɢ ɜ ɤɨɬɨɪɨɣ ɜɟɪɨɹɬɧɨɫɬɶ ɪ ɪ Ⱥ |
S ɫɨɛɵɬɢɹ Ⱥ ɨɫɬɚɺɬɫɹ ɧɟɢɡɦɟɧɧɨɣ |
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ɢ q p ɇɨ ɬɟɩɟɪɶ ɪɟɱɶ ɩɨɣɞɺɬ ɨ ɫɥɭɱɚɣɧɨɣ ɜɟɥɢɱɢɧɟ ɏ ɩɪɟɞɫɬɚɜɥɹɸɳɟɣ ɫɨɛɨɣ ɱɢɫɥɨ ɢɫɩɵɬɚɧɢɣ ɩɪɨɜɟɞɟɧɧɵɯ ɞɨ ɩɟɪɜɨɝɨ ɩɨɹɜɥɟɧɢɹ Ⱥ Ɉɛɨɡɧɚɱɢɦ ɱɟ ɪɟɡ pm ɜɟɪɨɹɬɧɨɫɬɶ ɧɚɫɬɭɩɥɟɧɢɹ ɫɨɛɵɬɢɹ Ⱥ ɜ m ɨɦ ɨɩɵɬɟ ɬɨɝɞɚ ɤɚɤ ɜ ɩɟɪɜɵɯ m± ɢɫɩɵɬɚɧɢɹɯ Ⱥ ɧɟ ɩɪɨɢɡɨɲɥɨ
ɪ ɪ ɫɨɛɵɬɢɟ Ⱥ ɧɚɫɬɭɩɢɥɨ ɭɠɟ ɜ ɩɟɪɜɨɦ ɨɩɵɬɟ
B
p ɪ Ⱥ Ⱥ TS ɫɨɛɵɬɢɟ Ⱥ ɧɟ ɧɚɫɬɭɩɢɥɨ ɜ ɩɟɪɜɨɦ ɨɩɵɬɟɧɨ ɧɚɫɬɭɩɢɥɨ ɜɨ ɜɬɨɪɨɦ
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Ɍɚɤɢɦ ɨɛɪɚɡɨɦ ɩɨɥɭɱɚɟɦ ɪɹɞ ɝɟɨɦɟɬɪɢɱɟɫɤɨɝɨ ɪɚɫɩɪɟɞɟɥɟɧɢɹ |
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ɇɚɡɜɚɧɢɟ ɨɛɴɹɫɧɹɟɬɫɹ ɬɟɦ ɱɬɨ ɜɟɪɨɹɬɧɨɫɬɢ pm ɨɛɪɚɡɭɸɬ ɛɟɫɤɨɧɟɱɧɭɸ ɝɟɨ |
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ɦɟɬɪɢɱɟɫɤɭɸ ɩɪɨɝɪɟɫɫɢɸ ɫ ɩɟɪɜɵɦ ɱɥɟɧɨɦ p ɢ ɡɧɚɦɟɧɚɬɟɥɟɦ q |
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Ⱦɨɤɚɠɟɦ ɱɬɨ ɦɚɬɟɦɚɬɢɱɟɫɤɨɟ ɨɠɢɞɚɧɢɟ ɫɥɭɱɚɣɧɨɣ ɜɟɥɢɱɢɧɵ ɏ ɟɫɬɶ ɱɢɫ |
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ɥɨ ɨɛɪɚɬɧɨɟ ɜɟɪɨɹɬɧɨɫɬɢ ɩɨɹɜɥɟɧɢɹ ɫɨɛɵɬɢɹ ɜ ɨɞɧɨɦ ɢɫɩɵɬɚɧɢɢ |
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ɩɪɢ ɜɵɱɢɫɥɟɧɢɢ ɢɫɩɨɥɶɡɨɜɚɧɚ ɜɨɡɦɨɠɧɨɫɬɶ ɩɨɱɥɟɧɧɨɝɨ ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɹ ɫɬɟɩɟɧɧɨɝɨ ɪɹɞɚ ɫ ɨɛɳɢɦ ɱɥɟɧɨɦ qm q ɋɨɨɬɧɨɲɟɧɢɟ ɞɨɤɚɡɚɧɨ
Ɇɨɠɧɨ ɬɚɤɠɟ ɞɨɤɚɡɚɬɶ ɱɬɨ ɝɟɨɦɟɬɪɢɱɟɫɤɢ ɪɚɫɩɪɟɞɟɥɟɧɧɚɹ ɫɥɭɱɚɣɧɚɹ ɜɟ ɥɢɱɢɧɚ ɏ ɢɦɟɟɬ ɞɢɫɩɟɪɫɢɸ
57
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ПРИМЕР 1. Ʉɨɦɩɚɧɢɹ ɩɪɨɢɡɜɨɞɢɬ ɢɡɞɟɥɢɹ ɢɡ ɤɨɬɨɪɵɯ ɢɦɟɸɬ ɨɬɤɥɨɧɟɧɢɟ ɨɬ ɫɬɚɧɞɚɪɬɚ Ⱦɥɹ ɤɨɧɬɪɨɥɹ ɤɚɱɟɫɬɜɚ ɨɬɛɢɪɚɸɬ ɢɡɞɟɥɢɣ ɇɚɣɬɢ ɨɠɢɞɚɟɦɨɟ ɤɨɥɢɱɟɫɬɜɨ ɢɡɞɟɥɢɣ ɫ ɨɬɤɥɨɧɟɧɢɹɦɢ ɨɬ ɫɬɚɧɞɚɪɬɚ
Ɋɟɲɟɧɢɟ ɉɭɫɬɶ ɏ ± ɱɢɫɥɨ ɨɛɧɚɪɭɠɟɧɧɵɯ ɧɟɫɬɚɧɞɚɪɬɧɵɯ ɢɡɞɟɥɢɣ ȼɟɥɢɱɢ ɧɚ ɏ ɪɚɫɩɪɟɞɟɥɟɧɚ ɨɱɟɜɢɞɧɨ ɛɢɧɨɦɢɚɥɶɧɵɦ ɨɛɪɚɡɨɦ ɫ ɱɢɫɥɨɦ ɨɩɵɬɨɜ ɩ ɢ ɜɟɪɨɹɬɧɨɫɬɶɸ ɨɛɧɚɪɭɠɟɧɢɹ ɧɟɫɬɚɧɞɚɪɬɧɨɫɬɢ ɜɫ ɤɚɠɞɨɦ ɟɞɢɧɢɱɧɨɦ ɨɩɵɬɟ ɪ Ɉɠɢɞɚɟɦɨɟ ɫɪɟɞɧɟɟ ɤɨɥɢɱɟɫɬɜɨ ɧɟɫɬɚɧɞɚɪɬɧɵɯ ɢɡɞɟɥɢɣ ɟɫɬɶ ɬɨɝɞɚ
ɦɚɬɟɦɚɬɢɱɟɫɤɨɟ ɨɠɢɞɚɧɢɟ Ɇ ɏ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɱɢɫɥɨɜɵɦɢ ɯɚɪɚɤɬɟɪɢɫɬɢ
ɤɚɦɢ ɛɢɧɨɦɢɚɥɶɧɨɝɨ ɪɚɫɩɪɟɞɟɥɟɧɢɹ Ɇ ɏ ɩɪ Ɂɚɦɟɬɢɦ ɱɬɨ ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɨɟ ɜɵɱɢɫɥɟɧɢɟ ɦɚɬɟɦɚɬɢɱɟɫɤɨɝɨ ɨɠɢɞɚɧɢɹ ɩɨ
ɮɨɪɦɭɥɟ ɛɵɥɨ ɛɵ ɫɜɹɡɚɧɨ F ɩɨɫɬɪɨɟɧɢɟɦ ɪɹɞɚ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɫɨɞɟɪɠɚɳɟ ɝɨ ɡɧɚɱɟɧɢɟ ɫɥɭɱɚɣɧɨɣ ɜɟɥɢɱɢɧɵ ɢ ɜɵɱɢɫɥɟɧɢɟɦ ɜɟɪɨɹɬɧɨɫɬɢ
ПРИМЕР 2. ɇɟɨɩɵɬɧɵɣ ɮɭɬɛɨɥɢɫɬ ɛɶɺɬ ɬɪɢ ɩɟɧɚɥɶɬɢ ȼɟɪɨɹɬɧɨɫɬɶ ɩɨ
ɪɚɠɟɧɢɹ ɜɨɪɨɬ ɜɫɹɤɢɣ ɪɚɡ ɪ Ʉɚɤɨɜɚ ɜɟɪɨɹɬɧɨɫɬɶ ɬɨɝɨ ɱɬɨ ɨɧ ɩɨɪɚɡɢɬ ɜɨɪɨ
ɬɚ ɫ ɬɪɟɬɶɟɝɨ ɪɚɡɚ" Ʉɚɤɨɜɨ ɫɪɟɞɧɟɟ ɱɢɫɥɨ ɩɟɧɚɥɶɬɢ ɤɨɬɨɪɵɟ ɧɭɠɧɨ ɩɪɨɛɢɬɶ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɞɥɹ ɩɨɩɚɞɚɧɢɹ ɜ ɜɨɪɨɬɚ"
Ɋɟɲɟɧɢɟ Ɋɚɫɫɦɨɬɪɢɦ ɫɥɭɱɚɣɧɭɸ ɜɟɥɢɱɢɧɭ ɏ ± ɱɢɫɥɨ ɜɵɩɨɥɧɟɧɧɵɯ ɭɞɚɪɨɜ
ɩɨ ɜɨɪɨɬɚɦ ɂɦɟɟɦ ɝɟɨɦɟɬɪɢɱɟɫɤɨɟ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɫ ɜɟɪɨɹɬɧɨɫɬɶɸ ɪ
ɧɚɫɬɭɩɥɟɧɢɹ ɫɨɛɵɬɢɹ ɜ ɟɞɢɧɢɱɧɨɦ ɨɩɵɬɟ ɢ ɱɢɫɥɨɦ ɨɩɵɬɨɜ ɬ ɋɨɝɥɚɫɧɨ ɪɟ
ɡɭɥɶɬɚɬɚɦ ɩ |
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ɋɪɟɞɧɟɟ ɱɢɫɥɨ ɭɞɚɪɨɜ ɩɨ ɜɨɪɨɬɚɦ ɟɫɬɶ ɦɚɬɟɦɚɬɢɱɟɫɤɨɟ ɨɠɢɞɚɧɢɟ |
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ПРИМЕР 3. ȼ ɤɚɪɲɟɪɢɧɝɨɜɨɣ ɤɨɦɩɚɧɢɢ ɢɦɟɸɬɫɹ ɜɫɟɝɨ ɞɜɚ ɫɜɨɛɨɞɧɵɯ ɚɜɬɨɦɨɛɢɥɹ ɂɡɜɟɫɬɧɨ ɱɬɨ ɟɠɟɞɧɟɜɧɵɣ ɫɩɪɨɫ ɩɨɞɱɢɧɹɟɬɫɹ ɪɚɫɩɪɟɞɟɥɟɧɢɸ ɉɭɚɫɫɨɧɚ ɢ ɜ ɫɪɟɞɧɟɦ ɫɨɫɬɚɜɥɹɟɬ ɦɚɲɢɧɵ ɜ ɞɟɧɶ ɩɪɢ ɷɬɨɦ ɚɪɟɧɞɚɬɨɪ ɧɟ ɨɬɞɚɺɬ ɤɚɤɢɯ ɥɢɛɨ ɩɪɟɞɩɨɱɬɟɧɢɣ ɧɢ ɨɞɧɨɣ ɢɡ ɧɢɯ Ʉɚɤɨɜɚ ɜɟɪɨɹɬɧɨɫɬɶ ɱɬɨ ɜ ɥɸɛɨɣ ɢɡ ɞɧɟɣ
ɧɢ ɨɞɢɧ ɚɜɬɨɦɨɛɢɥɶ ɧɟ ɛɭɞɟɬ ɜɨɫɬɪɟɛɨɜɚɧɛɭɞɭɬ ɜɨɫɬɪɟɛɨɜɚɧɵ ɨɛɟ ɦɚɲɢɧɵ
58
Ɋɟɲɟɧɢɟ ɉɨ ɭɫɥɨɜɢɸ ɡɚɞɚɱɢ ɫɥɭɱɚɣɧɚɹ ɜɟɥɢɱɢɧɚ ɏ ɟɫɬɶ ɱɢɫɥɨ ɡɚɤɚɡɨɜ ɜ ɞɟɧɶ ɢ ɨɧɚ ɪɚɫɩɪɟɞɟɥɟɧɚ ɩɨ ɡɚɤɨɧɭ ɉɭɚɫɫɨɧɚ ɩɪɢ ɷɬɨɦ ɤɨɥɢɱɟɫɬɜɨ k ɡɚɤɚɡɨɜ ɦɨɠɟɬ ɛɵɬɶ ɜɨɨɛɳɟ ɝɨɜɨɪɹ ɧɟɨɝɪɚɧɢɱɟɧɧɵɦ ȼɟɪɨɹɬɧɨɫɬɶ ɩɨɫɬɭɩɥɟɧɢɹ ɪɨɜɧɨ k ɡɚɤɚɡɨɜ ɜɵɱɢɫɥɹɟɬɫɹ ɩɨ ɮɨɪɦɭɥɟ ɉɭɚɫɫɨɧɚ ɉɪɢ ɷɬɨɦ ɩɚɪɚɦɟɬɪ O ɟɫɬɶ ɦɚɬɟ ɦɚɬɢɱɟɫɤɨɟ ɨɠɢɞɚɧɢɟ ɜɟɥɢɱɢɧɵ ɏ ɫɦ ɩ ɢɥɢ ɟɟ ɫɪɟɞɧɟɟ ɡɧɚɱɟɧɢɟ ɤɨɬɨɪɨɟ ɩɨ ɭɫɥɨɜɢɸ ɪɚɜɧɨ ɋɥɟɞɨɜɚɬɟɥɶɧɨ
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ȿɫɥɢ ɠɟ ɤɚɤ ɦɢɧɢɦɭɦ ɧɚ ɨɛɚ ɚɜɬɨɦɨɛɢɥɹ ɩɨɫɬɭɩɢɥɢ ɡɚɤɚɡɵ ɬɨ ɱɢɫɥɨ ɡɚ ɤɚɡɨɜ X t ɢ
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6. Равномерное и показательное распределения непрерывных случайных величин
ɇɟɩɪɟɪɵɜɧɚɹ ɫɥɭɱɚɣɧɚɹ ɜɟɥɢɱɢɧɚ ɏ ɧɚɡɵɜɚɟɬɫɹ ɪɚɫɩɪɟɞɟɥɟɧɧɨɣ ɩɨ ɪɚɜɧɨɦɟɪɧɨɦɭ ɡɚɤɨɧɭ ɧɚ ɨɬɪɟɡɤɟ >a b@ ɟɫɥɢ ɟɟ ɩɥɨɬɧɨɫɬɶ ɜɟɪɨɹɬɧɨɫɬɢ ɩɨɫɬɨ
ɹɧɧɚ ɧɚ ɷɬɨɦ ɨɬɪɟɡɤɟ |
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ɂɡ ɫɜɨɣɫɬɜɚ ɧɨɪɦɢɪɨɜɚɧɧɨɫɬɢ ɩɥɨɬɧɨɫɬɢ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɛɭɞɟɬ ɜɵɬɟɤɚɬɶ ɱɬɨ
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ɇɚɣɞɟɦ ɬɟɩɟɪɶ ɮɭɧɤɰɢɸ ɪɚɫɩɪɟɞɟɥɟɧɢɹ |
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59
ɂɦɟɟɦ ɨɱɟɜɢɞɧɨ ɩɪɢ x d a ɡɧɚɱɟɧɢɹ F x |
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ɇɚɤɨɧɟɰ ɩɪɢ x ! b
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³ dt |
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Ƚɪɚɮɢɤɢ ɮɭɧɤɰɢɣ ɢ ɢɡɨɛɪɚɠɟɧɵ ɧɚ ɪɢɫɭɧɤɟ
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Ðèñ. 6.1. Плотность и функция равномерного распределения
Ⱦɨɤɚɠɟɦ ɱɬɨ ɞɥɹ ɪɚɜɧɨɦɟɪɧɨ ɪɚɫɩɪɟɞɟɥɟɧɧɨɣ ɧɚ ɨɬɪɟɡɤɟ >a b@ ɫɥɭɱɚɣɧɨɣ ɜɟɥɢɱɢɧɵ ɏ
M X a b ɢ D X b a
ɂɦɟɟɦ
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M X ³ |
xf x dx ³ |
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Ɏɨɪɦɭɥɚ ɞɥɹ ɞɢɫɩɟɪɫɢɢ ɞɨɤɚɡɵɜɚɟɬɫɹ ɚɧɚɥɨɝɢɱɧɨ
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D X ³ x |
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f x dx >M X @ |
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a ab b |
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60 |
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