Прикладные математические методы в статистической радиотехнике. Федосов В.П. - 8 стр.

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Ɉɞɧɚɤɨ ɬɚɤɨɣ ɬɚɛɥɢɰɵ ɧɟɜɨɡɦɨɠɧɨ ɩɨɫɬɪɨɢɬɶ ɞɥɹ ɧɟɩɪɟɪɵɜɧɨɣ ɋȼ, ɩɨ-
ɫɤɨɥɶɤɭ ɞɥɹ ɧɟɟ ɤɚɠɞɨɟ ɨɬɞɟɥɶɧɨɟ ɡɧɚɱɟɧɢɟ ɧɟ ɨɛɥɚɞɚɟɬ ɨɬɥɢɱɧɨɣ ɨɬ ɧɭɥɹ ɜɟɪɨ-
ɹɬɧɨɫɬɶɸ.
3.2. Ɏɭɧɤɰɢɹ ɪɚɫɩɪɟɞɟɥɟɧɢɹ
Ⱦɥɹ ɤɨɥɢɱɟɫɬɜɟɧɧɨɣ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɢɫɩɨɥɶɡɭɸɬ ɡɚɜɢɫɢ-
ɦɨɫɬɶ 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) :
1) F(X) ɟɫɬɶ ɧɟɭɛɵɜɚɸɳɚɹ ɮɭɧɤɰɢɹ ɫɜɨɟɝɨ ɚɪɝɭɦɟɧɬɚ, ɬ.ɟ. ɩɪɢ x
2
>x
1
F(x
2
)
> F(x
1
) ;
2) ɧɚ - F(x) ɪɚɜɧɚ ɧɭɥɸ, ɬ.ɟ. F(-) = 0;
3) F(f) = 1.
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 ɜɟ-
ɥɢɱɢɧɚ 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
x
1
ɞɨ x
2
. ɗɬɨ ɫɨɛɵɬɢɟ ɧɚɡɵɜɚɟɬɫɹ "ɩɨɩɚɞɚɧɢɟɦ ɋȼ X ɧɚ ɭɱɚɫɬɨɤ ɨɬ x
1
ɞɨ x
2
". ȼɵ-
ɪɚɡɢɦ ɜɟɪɨɹɬɧɨɫɬɶ ɷɬɨɝɨ ɫɨɛɵɬɢɹ ɱɟɪɟɡ ɮɭɧɤɰɢɸ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɋȼ X. Ⱦɥɹ ɷɬɨɝɨ
ɪɚɫɫɦɨɬɪɢɦ ɬɪɢ ɫɨɛɵɬɢɹ:
x ɫɨɛɵɬɢɟ Ⱥ, ɫɨɫɬɨɹɳɟɟ ɜ ɬɨɦ, ɱɬɨ X < x
2
;
x ɫɨɛɵɬɢɟ ȼ, ɫɨɫɬɨɹɳɟɟ ɜ ɬɨɦ, ɱɬɨ X < x
1
;
x ɫɨɛɵɬɢɟ ɋ, ɫɨɫɬɨɹɳɟɟ ɜ ɬɨɦ, ɱɬɨ x
1
< X <x
2
.
ɍɱɢɬɵɜɚɹ, ɱɬɨ A = B + C, ɩɨ ɬɟɨɪɟɦɟ ɫɥɨɠɟɧɢɹ ɜɟɪɨɹɬɧɨɫɬɟɣ ɩɨɥɭɱɢɦ
P(X<x
2
)=P(X<x
1
)+P(x
1
<X<x
2
), ɢɥɢ F(x
2
) = F(x
1
) + P(x
1
<X<x
2
), ɨɬɤɭɞɚ
P(x
1
<X<x
2
) = F(x
2
)-F(x
1
), 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 ɜɟɪɨ-
ɹɬɧɨɫɬɶ ɪɚɜɧɚ ɩɪɢɪɚɳɟɧɢɸ ɮɭɧɤɰɢɢ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɧɚ ɷɬɨɦ ɭɱɚɫɬɤɟ. Ɋɚɫɫɦɨɬ-

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