Прикладные математические методы в статистической радиотехнике. Федосов В.П. - 21 стр.

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(x,y) ɩɥɨɫɤɨɫɬɶɸ, ɩɚɪɚɥɥɟɥɶɧɨɣ ɩɥɨɫɤɨɫɬɢ x0y. ɋɟɱɟɧɢɟ ɩɪɨɣɞɟɬ ɩɨ ɤɪɢɜɨɣ Ʉ.
ɋɩɪɨɟɰɢɪɭɟɦ ɷɬɭ ɤɪɢɜɭɸ ɧɚ ɩɥɨɫɤɨɫɬɶ x0y. ɉɪɨɟɤɰɢɹ, ɨɩɪɟɞɟɥɹɟɦɚɹ ɭɪɚɜɧɟɧɢɟɦ
M(x,y) = z, ɪɚɡɞɟɥɢɬ ɩɥɨɫɤɨɫɬɶ x0y ɧɚ ɞɜɟ ɨɛɥɚɫɬɢ. Ⱦɥɹ ɨɞɧɨɣ ɢɡ ɧɢɯ ɜɵɩɨɥɧɹɟɬɫɹ
ɭɫɥɨɜɢɟ Z < z. Ɉɛɨɡɧɚɱɢɦ ɷɬɭ ɩɥɨɫɤɨɫɬɶ ɱɟɪɟɡ [. ȼɟɪɨɹɬɧɨɫɬɶ ɩɨɩɚɞɚɧɢɹ ɜ ɷɬɭ
ɩɥɨɫɤɨɫɬɶ ɢ ɟɫɬɶ ɢɧɬɟɝɪɚɥɶɧɚɹ ɮɭɧɤɰɢɹ ɪɚɫɩɪɟɞɟɥɟɧɢɹ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, G(z) =
P((X,Y)[)=

³³
[
f(x,y)dxdy. Ⱦɢɮɮɟɪɟɧɰɢɪɭɹ G(z) ɩɨ z, ɩɨɥɭɱɢɦ ɢɫɤɨɦɭɸ ɮɭɧɤɰɢɸ
g(z). Ɉɫɬɚɥɨɫɶ ɜɵɪɚɡɢɬɶ ɱɟɪɟɡ z ɩɪɟɞɟɥɵ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ ɢ ɧɚɩɢɫɚɬɶ ɜɵɪɚɠɟɧɢɟ
ɞɥɹ g(z).
ɉɊɂɆȿɊ. ɇɚɣɬɢ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɫɭɦɦɵ ɞɜɭɯ ɋȼ X ɢ Y, ɢɦɟɸɳɢɯ ɫɨɜɦɟɫɬ-
ɧɭɸ ɩɥɨɬɧɨɫɬɶ ɜɟɪɨɹɬɧɨɫɬɢ f(x,y). ɋɭɦɦɚ Z = X + Y. ɇɚɣɬɢ g(z).
ɇɚ ɩɥɨɫɤɨɫɬɢ x0y ɢɦɟɟɬɫɹ ɥɢɧɢɹ, ɭɪɚɜɧɟɧɢɟ ɤɨɬɨɪɨɣ x+y=z ɟɫɬɶ ɩɪɹɦɚɹ, ɨɬ-
ɫɟɤɚɸɳɚɹ ɧɚ ɨɫɹɯ ɨɬɪɟɡɤɢ, ɪɚɜɧɵɟ z. ɗɬɚ ɩɪɹɦɚɹ ɞɟɥɢɬ ɜɫɸ ɩɥɨɫɤɨɫɬɶ ɧɚ ɞɜɟ
ɱɚɫɬɢ. Ɍɚ ɱɚɫɬɶ, ɱɬɨ ɥɟɜɟɟ ɢ ɧɢɠɟ ɷɬɨɣ ɩɪɹɦɨɣ, ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɭɫɥɨɜɢɸ X+Y<Z. ȼ
ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɮɨɪɦɭɥɨɣ (5.1), ɝɞɟ ɩɥɨɫɤɨɫɬɶ [ ɟɫɬɶ ɩɥɨɫɤɨɫɬɶ, ɧɚ ɤɨɬɨɪɨɣ ɜɵ-
ɩɨɥɧɹɟɬɫɹ ɷɬɨ ɭɫɥɨɜɢɟ, ɢɦɟɟɦ:
G(z)=

³³
[
y)dxdyf(x,
=
³³
f
f
f
xz
y)dxdyf(x,
=
³
f
f
x)dx-zf(x,
.
ȿɫɥɢ ɋȼ X ɢ Y ɧɟɡɚɜɢɫɢɦɵ, ɬɨ ɧɟɬɪɭɞɧɨ ɡɚɦɟɬɢɬɶ, ɱɬɨ ɢɧɬɟɝɪɚɥ ɩɪɢ ɪɚɜɟɧ-
ɫɬɜɟ f(x,y) = f(x) f(y) ɩɪɟɜɪɚɳɚɟɬɫɹ ɜ ɢɧɬɟɝɪɚɥ ɫɜɟɪɬɤɢ. Ɍɚɤ, ɫɜɟɪɬɤɚ ɞɜɭɯ ɩɪɹɦɨ-
ɭɝɨɥɶɧɵɯ ɡɚɤɨɧɨɜ ɩɥɨɬɧɨɫɬɢ ɜɟɪɨɹɬɧɨɫɬɢ ɞɚɟɬ ɡɚɤɨɧ ɋɢɦɩɫɨɧɚ.
5.4. ɋɭɦɦɚ ɞɜɭɯ ɧɨɪɦɚɥɶɧɵɯ ɋȼ
ȿɫɥɢ ɩɨɞɫɬɚɜɢɬɶ ɜ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɢɧɬɟɝɪɚɥɚ ɫɜɟɪɬɤɢ ɞɜɚ ɧɨɪɦɚɥɶɧɵɯ ɨɞ-
ɧɨɦɟɪɧɵɯ ɡɚɤɨɧɚ ɞɥɹ ɋȼ X ɢɋȼY, ɬɨ ɩɨɫɥɟ ɧɟɫɥɨɠɧɵɯ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ ɩɨɥɭ-
ɱɢɦ ɫɧɨɜɚ ɧɨɪɦɚɥɶɧɭɸ ɩɥɨɬɧɨɫɬɶ ɜɟɪɨɹɬɧɨɫɬɢ ɋȼ Z, ɭ ɤɨɬɨɪɨɣ ɆɈ ɪɚɜɧɨ ɫɭɦɦɟ
ɆɈ ɤɚɠɞɨɣ ɋȼ ɢ ɞɢɫɩɟɪɫɢɹ ɬɚɤɠɟ ɪɚɜɧɚ ɫɭɦɦɟ ɞɢɫɩɟɪɫɢɣ, ɟɫɥɢ ɋȼ ɧɟɤɨɪɪɟɥɢ-
ɪɨɜɚɧɵ, ɚ ɟɫɥɢ ɤɨɪɪɟɥɢɪɨɜɚɧɵ, ɬɨ V
z
2
=V
x
2
+V
y
2
r2RV
x
V
y
(+, ɟɫɥɢ ɤɨɷɮɮɢɰɢɟɧɬ
ɤɨɪɪɟɥɹɰɢɢ ɛɨɥɶɲɟ ɧɭɥɹ ɢ ɧɚɨɛɨɪɨɬ).
Ɂɚɞɚɧɵ: ɩɥɨɬɧɨɫɬɶ ɜɟɪɨɹɬɧɨɫɬɢ f(x) ɋȼ-ɚɪɝɭɦɟɧɬɚ, ɮɭɧɤɰɢɹ y = M(x), ɫɜɹɡɵ-
ɜɚɸɳɚɹ ɋȼ-ɚɪɝɭɦɟɧɬ ɢ ɋȼ-ɮɭɧɤɰɢɸ. Ɉɩɪɟɞɟɥɢɬɶ ɩɥɨɬɧɨɫɬɶ ɜɟɪɨɹɬɧɨɫɬɢ f(y)
ɋȼ-ɮɭɧɤɰɢɢ. Ⱦɥɹ ɨɩɪɟɞɟɥɟɧɢɹ f(y) ɜ ɨɛɳɟɦ ɜɢɞɟ ɦɨɠɧɨ ɢɫɩɨɥɶɡɨɜɚɬɶ ɪɚɜɟɧɫɬɜɨ
ɷɥɟɦɟɧɬɚɪɧɵɯ ɜɟɪɨɹɬɧɨɫɬɟɣ, ɧɟɢɡɦɟɧɹɸɳɢɯɫɹ ɩɪɢ ɥɸɛɵɯ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹɯ:
f(x)dx = f(y)dy. Ɉɬɫɸɞɚ f(y) = f(x)/(dy/dx).
Ⱦɥɹ ɪɟɲɟɧɢɹ ɫɮɨɪɦɭɥɢɪɨɜɚɧɧɨɣ ɡɚɞɚɱɢ ɧɟɨɛɯɨɞɢɦɨ ɧɚɣɬɢ ɮɭɧɤɰɢɸ, ɨɛ-
ɪɚɬɧɭɸ M(x). Ɂɚɬɟɦ ɨɩɪɟɞɟɥɢɬɶ ɹɤɨɛɢɚɧ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ J = |dM(x)/dx| ɢɩɨɞɫɬɚ-
ɜɢɬɶ ɜɦɟɫɬɨ y ɨɛɪɚɬɧɭɸ ɮɭɧɤɰɢɸ x=M
-1
(y) ɞɥɹ ɜɫɟɯ ɡɧɚɱɟɧɢɣ x, ɨɩɪɟɞɟɥɹɟɦɵɯ
ɧɟɨɞɧɨɡɧɚɱɧɨɫɬɶɸ ɨɛɪɚɬɧɨɣ ɮɭɧɤɰɢɢ M
-1
(y). ɉɨɬɨɦ ɜɫɟ ɩɨɥɭɱɟɧɧɵɟ ɜɵɪɚɠɟɧɢɹ
ɫɥɨɠɢɬɶ.

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