Прикладные математические методы в статистической радиотехнике. Федосов В.П. - 15 стр.

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ɤɚ, ɤɨɝɞɚ ɨɛɚ ɪɚɡɦɟɪɚ ɩɪɹɦɨɭɝɨɥɶɧɢɤɚ ɫɬɪɟɦɹɬɫɹ ɤ ɧɭɥɸ. Ƚɟɨɦɟɬɪɢɱɟɫɤɢ f(x, y)
ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɤɚɤ ɧɟɤɨɬɨɪɭɸ ɩɨɜɟɪɯɧɨɫɬɶ. ȿɫɥɢ ɪɚɫɫɟɱɶ ɷɬɭ ɩɨɜɟɪɯɧɨɫɬɶ
ɩɥɨɫɤɨɫɬɶɸ, ɩɚɪɚɥɥɟɥɶɧɨɣ ɩɥɨɫɤɨɫɬɢ x0y, ɢ ɫɩɪɨɟɰɢɪɨɜɚɬɶ ɩɨɥɭɱɟɧɧɨɟ ɫɟɱɟɧɢɟ
ɧɚ ɩɥɨɫɤɨɫɬɶ x0y, ɬɨ ɩɨɥɭɱɢɬɫɹ ɤɪɢɜɚɹ, ɧɚɡɵɜɚɟɦɚɹ "ɤɪɢɜɨɣ ɪɚɜɧɨɣ ɩɥɨɬɧɨɫɬɢ
ɜɟɪɨɹɬɧɨɫɬɢ". ɂɧɨɝɞɚ ɭɞɨɛɧɨ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɫɟɦɟɣɫɬɜɚ ɤɪɢɜɵɯ ɪɚɜɧɨɣ ɩɥɨɬɧɨ-
ɫɬɢ ɩɪɢ ɪɚɡɧɵɯ ɭɪɨɜɧɹɯ ɫɟɱɟɧɢɹ.
Ʉɚɤ ɢ ɞɥɹ ɨɞɧɨɦɟɪɧɨɣ ɩɥɨɬɧɨɫɬɢ ɜɟɪɨɹɬɧɨɫɬɢ ɡɞɟɫɶ ɜɜɨɞɢɬɫɹ ɩɨɧɹɬɢɟ ɷɥɟ-
ɦɟɧɬɚ ɜɟɪɨɹɬɧɨɫɬɢ f(x, y)dxdy.
ȼɟɪɨɹɬɧɨɫɬɶ ɩɨɩɚɞɚɧɢɹ ɫɥɭɱɚɣɧɨɣ ɬɨɱɤɢ ɜ ɩɪɨɢɡɜɨɥɶɧɭɸ ɨɛɥɚɫɬɶ G ɨɩɪɟ-
ɞɟɥɹɟɬɫɹ ɞɜɭɦɟɪɧɵɦ ɢɧɬɟɝɪɚɥɨɦ ɨɬ f(x, y) ɩɨ ɷɬɨɣ ɨɛɥɚɫɬɢ. Ƚɟɨɦɟɬɪɢɱɟɫɤɢ ɷɬɨ
ɨɛɴɟɦ, ɨɝɪɚɧɢɱɟɧɧɵɣ f(x, y) ɢ ɨɛɥɚɫɬɶɸ G. ȿɫɥɢ G ɟɫɬɶ ɩɪɹɦɨɭɝɨɥɶɧɢɤ ɫ ɤɨɨɪɞɢ-
ɧɚɬɚɦɢ ɜɟɪɲɢɧ ɩɨ ɨɫɢ x D ɢ E, ɚɩɨɨɫɢy J ɢ G, ɬɨ ɜɟɪɨɹɬɧɨɫɬɶ ɩɨɩɚɞɚɧɢɹ ɜ ɷɬɨɬ
ɩɪɹɦɨɭɝɨɥɶɧɢɤ ɨɩɪɟɞɟɥɹɟɬɫɹ ɢɧɬɟɝɪɚɥɨɦ
P((X,Y)R)=
³³
E
D
G
J
f(x,y)dxdy.
ɋɜɨɣɫɬɜɚ ɞɜɭɦɟɪɧɨɣ ɩɥɨɬɧɨɫɬɢ ɜɟɪɨɹɬɧɨɫɬɢ:
- f(x, y) ɟɫɬɶ ɧɟɨɬɪɢɰɚɬɟɥɶɧɚɹ ɜɟɥɢɱɢɧɚ;
- ɫɜɨɣɫɬɜɨ ɧɨɪɦɢɪɨɜɤɢ ɚɧɚɥɨɝɢɱɧɨ ɨɞɧɨɦɟɪɧɨɣ ɩɥɨɬɧɨɫɬɢ ɜɟɪɨɹɬɧɨɫɬɢ, ɧɨ
ɩɪɢ ɞɜɭɦɟɪɧɨɦ ɢɧɬɟɝɪɢɪɨɜɚɧɢɢ ɜ ɛɟɫɤɨɧɟɱɧɵɯ ɩɪɟɞɟɥɚɯ.
4.3. ɍɫɥɨɜɧɵɟ ɡɚɤɨɧɵ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɨɬɞɟɥɶɧɵɯ ɋȼ,
ɜɯɨɞɹɳɢɯ ɜ ɫɢɫɬɟɦɭ ɋȼ
ɂɦɟɹ ɡɚɤɨɧ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɫɢɫɬɟɦɵ ɞɜɭɯ ɋȼ, ɜɫɟɝɞɚ ɦɨɠɧɨ ɨɩɪɟɞɟɥɢɬɶ ɡɚ-
ɤɨɧɵ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɨɬɞɟɥɶɧɵɯ ɋȼ, ɜɯɨɞɹɳɢɯ ɜ ɫɢɫɬɟɦɭ. ɇɚɩɪɢɦɟɪ, F(x)=F(x,f)
ɢ F(y) = F(f,y). ȿɫɥɢ ɢɡɜɟɫɬɧɚ ɩɥɨɬɧɨɫɬɶ ɜɟɪɨɹɬɧɨɫɬɢ f(x,y), ɬɨ F(x) = F(x,f) =
³³
f
f
f
x
f(x,y)dxdy.
Ⱥɧɚɥɨɝɢɱɧɨ ɨɩɪɟɞɟɥɹɟɬɫɹ F(y).
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɡɧɚɹ ɞɜɭɦɟɪɧɭɸ ɩɥɨɬɧɨɫɬɶ ɜɟɪɨɹɬɧɨɫɬɢ, ɜɫɟɝɞɚ ɦɨɠɧɨ ɨɩ-
ɪɟɞɟɥɢɬɶ ɨɞɧɨɦɟɪɧɭɸ ɩɥɨɬɧɨɫɬɶ ɜɟɪɨɹɬɧɨɫɬɢ. Ɉɛɪɚɬɧɭɸ ɡɚɞɚɱɭ ɜ ɨɛɳɟɦ ɫɥɭɱɚɟ
ɪɟɲɢɬɶ ɧɟɜɨɡɦɨɠɧɨ. ȿɟ ɦɨɠɧɨ ɪɟɲɢɬɶ, ɟɫɥɢ ɢɡɜɟɫɬɧɵ ɭɫɥɨɜɧɵɟ ɩɥɨɬɧɨɫɬɢ ɜɟ-
ɪɨɹɬɧɨɫɬɢ ɢɥɢ ɮɭɧɤɰɢɢ ɪɚɫɩɪɟɞɟɥɟɧɢɹ.
ɍɫɥɨɜɧɵɦ ɡɚɤɨɧɨɦ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɋȼ, ɜɯɨɞɹɳɟɣ ɜ ɫɢɫɬɟɦɭ ɋȼ, ɧɚɡɵɜɚɟɬɫɹ
ɟɟ ɡɚɤɨɧ ɪɚɫɩɪɟɞɟɥɟɧɢɹ, ɨɩɪɟɞɟɥɟɧɧɵɣ ɩɪɢ ɭɫɥɨɜɢɢ, ɱɬɨ ɞɪɭɝɚɹ ɋȼ ɩɪɢɧɹɥɚ ɨɩ-
ɪɟɞɟɥɟɧɧɨɟ ɡɧɚɱɟɧɢɟ: f(x/y). ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɦɨɠɧɨ ɧɚɣɬɢ ɞɜɭɦɟɪɧɭɸ ɩɥɨɬɧɨɫɬɶ
ɜɟɪɨɹɬɧɨɫɬɢ ɩɨ ɮɨɪɦɭɥɟ f(x, y) = f(x)f(y/x). ɗɬɚ ɮɨɪɦɭɥɚ ɧɚɡɵɜɚɟɬɫɹ ɬɟɨɪɟɦɨɣ
ɭɦɧɨɠɟɧɢɹ ɡɚɤɨɧɨɜ ɪɚɫɩɪɟɞɟɥɟɧɢɹ. Ɇɨɠɧɨ ɟɟ ɡɚɩɢɫɚɬɶ ɢ ɜ ɞɪɭɝɨɦ ɜɢɞɟ: f(x,y) =
=f(y)f(x/y). ɂɡ ɷɬɢɯ ɜɵɪɚɠɟɧɢɣ ɫɥɟɞɭɟɬ:
f(y/x)=f(x/y)/f(x), f(x/y)=f(y/x)/f(y).

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