Прикладные математические методы в статистической радиотехнике. Федосов В.П. - 14 стр.

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ɨɰɟɧɤɢ ɆɈ ɢ D
x
ɦɨɠɧɨ ɢɫɩɨɥɶɡɨɜɚɬɶ ɞɥɹ ɩɨɫɬɪɨɟɧɢɹ ɧɨɪɦɚɥɶɧɨɣ ɩɥɨɬɧɨɫɬɢ
ɜɟɪɨɹɬɧɨɫɬɢ.
4. ɋɂɋɌȿɆɕ ɋɅɍɑȺɃɇɕɏ ȼȿɅɂɑɂɇ
ɇɚ ɩɪɚɤɬɢɤɟ ɱɚɫɬɨ ɩɪɢɯɨɞɢɬɫɹ ɢɦɟɬɶ ɞɟɥɨ ɨɞɧɨɜɪɟɦɟɧɧɨ ɫ ɧɟɫɤɨɥɶɤɢɦɢ
ɋȼ, ɧɚɩɪɢɦɟɪ: ɩɨɥɨɠɟɧɢɟ ɬɨɱɤɢ ɩɚɞɟɧɢɹ ɨɛɥɨɦɤɚ ɡɞɚɧɢɹ ɩɪɢ ɟɝɨ ɪɚɡɪɭɲɟɧɢɢ
ɨɩɪɟɞɟɥɹɟɬɫɹ ɞɜɭɦɹ ɤɨɨɪɞɢɧɚɬɚɦɢ ɢ ɬ.ɞ. ɋɜɨɣɫɬɜɚ ɫɢɫɬɟɦɵ ɧɟɫɤɨɥɶɤɢɯ ɋȼ ɧɟ
ɢɫɱɟɪɩɵɜɚɟɬɫɹ ɫɜɨɣɫɬɜɚɦɢ ɨɬɞɟɥɶɧɨɣ ɋȼ, ɬɚɤ ɤɚɤ ɩɪɢ ɷɬɨɦ ɧɟɨɛɯɨɞɢɦɨ ɨɩɢɫɚ-
ɧɢɟ ɫɜɹɡɢ ɦɟɠɞɭ ɫɨɫɬɚɜɥɹɸɳɢɦɢ ɫɢɫɬɟɦɵ ɋȼ.
4.1. Ɏɭɧɤɰɢɢ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɫɢɫɬɟɦɵ ɢɡ ɞɜɭɯ ɋȼ
Ɏɭɧɤɰɢɟɣ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɫɢɫɬɟɦɵ ɢɡ ɞɜɭɯ ɋȼ (X,Y) ɧɚɡɵɜɚɟɬɫɹ ɜɟɪɨɹɬɧɨɫɬɶ
ɫɨɜɦɟɫɬɧɨɝɨ ɜɵɩɨɥɧɟɧɢɹ ɞɜɭɯ ɧɟɪɚɜɟɧɫɬɜ X < x ɢ Y < y:
F(x,y) = P((X < x)(Y < y)).
ɉɨ ɨɩɪɟɞɟɥɟɧɢɸ, ɮɭɧɤɰɢɹ ɪɚɫɩɪɟɞɟɥɟɧɢɹ F(x,y) ɟɫɬɶ ɜɟɪɨɹɬɧɨɫɬɶ ɩɨɩɚɞɚ-
ɧɢɹ ɫɥɭɱɚɣɧɨɣ ɬɨɱɤɢ (x,y) ɜ ɤɜɚɞɪɚɬ ɫ ɛɟɫɤɨɧɟɱɧɵɦɢ ɪɚɡɦɟɪɚɦɢ, ɪɚɫɩɨɥɨɠɟɧɧɵɣ
ɥɟɜɟɟ ɢ ɧɢɠɟ ɷɬɨɣ ɬɨɱɤɢ ɧɚ ɩɥɨɫɤɨɫɬɢ x0y. Ɉɬɞɟɥɶɧɨ ɞɥɹ ɤɚɠɞɨɣ ɋȼ X ɢ Y ɦɨɠɧɨ
ɨɩɪɟɞɟɥɢɬɶ ɨɞɧɨɦɟɪɧɭɸ ɮɭɧɤɰɢɸ ɪɚɫɩɪɟɞɟɥɟɧɢɹ, ɧɚɩɪɢɦɟɪ, F
1
(x) ɟɫɬɶ ɜɟɪɨɹɬ-
ɧɨɫɬɶ ɩɨɩɚɞɚɧɢɹ ɜ ɩɨɥɭɩɥɨɫɤɨɫɬɶ, ɪɚɫɩɨɥɨɠɟɧɧɭɸ ɥɟɜɟɟ ɬɨɱɤɢ x. Ɍɚɤɠɟ ɢ F
1
(y)
ɟɫɬɶ ɜɟɪɨɹɬɧɨɫɬɶ ɩɨɩɚɞɚɧɢɹ ɜ ɩɨɥɭɩɥɨɫɤɨɫɬɶ ɧɢɠɟ ɬɨɱɤɢ y.
ɋɜɨɣɫɬɜɚ F(x,y): 1)F(x,y) ɟɫɬɶ ɧɟɭɛɵɜɚɸɳɚɹ ɮɭɧɤɰɢɹ ɨɛɨɢɯ ɫɜɨɢɯ ɚɪɝɭɦɟɧ-
ɬɨɜ; 2)ɧɚ -f ɩɨ ɨɛɟɢɦ ɨɫɹɦ ɨɧɚ ɪɚɜɧɚ ɧɭɥɸ; 3)ɩɪɢ ɪɚɜɟɧɫɬɜɟ +f ɨɞɧɨɝɨ ɢɡ ɚɪɝɭ-
ɦɟɧɬɨɜ ɫɨɝɥɚɫɧɨ ɞɪɭɝɨɦɭ ɚɪɝɭɦɟɧɬɭ ɨɧɚ ɩɪɟɜɪɚɳɚɟɬɫɹ ɜ ɨɞɧɨɦɟɪɧɭɸ ɮɭɧɤɰɢɸ
ɪɚɫɩɪɟɞɟɥɟɧɢɹ; 4)ɟɫɥɢ ɨɛɚ ɚɪɝɭɦɟɧɬɚ ɪɚɜɧɵ +f, ɬɨ F(x,y) = 1.
ȼɟɪɨɹɬɧɨɫɬɶ ɩɨɩɚɞɚɧɢɹ ɫɥɭɱɚɣɧɨɣ ɬɨɱɤɢ ɜ ɤɜɚɞɪɚɬ R ɫ ɤɨɨɪɞɢɧɚɬɚɦɢ D, E
ɩɨ ɨɫɢ x ɢ J, G ɩɨ ɨɫɢ y ɪɚɜɧɚ
P((X,Y)R)=F(E,G)-F(D,G)-F(E,J)+F(D,J).
F(x,y) ɫɭɳɟɫɬɜɭɟɬ ɤɚɤ ɞɥɹ ɧɟɩɪɟɪɵɜɧɵɯ, ɬɚɤ ɢ ɞɥɹ ɞɢɫɤɪɟɬɧɵɯ ɋȼ.
4.2. Ⱦɜɭɦɟɪɧɚɹ ɩɥɨɬɧɨɫɬɶ ɜɟɪɨɹɬɧɨɫɬɢ
Ⱦɜɭɦɟɪɧɚɹ ɩɥɨɬɧɨɫɬɶ ɜɟɪɨɹɬɧɨɫɬɢ ɟɫɬɶ ɩɪɟɞɟɥ ɫɥɟɞɭɸɳɟɝɨ ɨɬɧɨɲɟɧɢɹ:
0,0
lim
o'o' yx
{P[(X,Y)R
'
]/('x'y)}.
ȿɫɥɢ F(x,y) ɧɟ ɬɨɥɶɤɨ ɧɟɩɪɟɪɵɜɧɚ, ɧɨ ɢ ɞɢɮɮɟɪɟɧɰɢɪɭɟɦɚ, ɬɨ ɞɜɭɦɟɪɧɚɹ
ɩɥɨɬɧɨɫɬɶ ɜɟɪɨɹɬɧɨɫɬɢ f(x,y) ɟɫɬɶ ɜɬɨɪɚɹ ɫɦɟɲɚɧɧɚɹ ɱɚɫɬɧɚɹ ɩɪɨɢɡɜɨɞɧɚɹ ɮɭɧɤ-
ɰɢɢ F(x,y) ɩɨ x ɢɩɨy. Ɋɚɡɦɟɪɧɨɫɬɶ f(x,y) ɨɛɪɚɬɧɚ ɩɪɨɢɡɜɟɞɟɧɢɸ ɪɚɡɦɟɪɧɨɫɬɟɣ
ɋȼ X ɢ Y.
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ ɞɜɭɦɟɪɧɚɹ ɩɥɨɬɧɨɫɬɶ ɜɟɪɨɹɬɧɨɫɬɢ ɟɫɬɶ ɩɪɟɞɟɥ ɨɬɧɨɲɟɧɢɹ
ɜɟɪɨɹɬɧɨɫɬɢ ɩɨɩɚɞɚɧɢɹ ɜ ɦɚɥɵɣ ɩɪɹɦɨɭɝɨɥɶɧɢɤ ɤ ɩɥɨɳɚɞɢ ɷɬɨɝɨ ɩɪɹɦɨɭɝɨɥɶɧɢ-

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