Прикладные математические методы в статистической радиотехнике. Федосов В.П. - 20 стр.

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2. Ɏɭɧɤɰɢɹ M(x) ɦɨɧɨɬɨɧɧɨ ɭɛɵɜɚɟɬ. Ɍɨɝɞɚ ɩɪɨɢɡɜɨɞɧɚɹ ɜ ɩɨɫɥɟɞɧɟɦ ɜɵɪɚ-
ɠɟɧɢɢ ɨɬɪɢɰɚɬɟɥɶɧɚ, ɧɨ ɩɥɨɬɧɨɫɬɶ ɜɟɪɨɹɬɧɨɫɬɢ ɧɟ ɦɨɠɟɬ ɛɵɬɶ ɨɬɪɢɰɚɬɟɥɶɧɨɣ.
ɉɪɢɦɟɧɹɹ ɡɧɚɤ ɦɨɞɭɥɹ, ɨɤɨɧɱɚɬɟɥɶɧɨ ɩɨɥɭɱɢɦ ɨɛɴɟɞɢɧɟɧɧɭɸ ɮɨɪɦɭɥɭ: g(y) = f(
I(y))|I'(y)|.
5.1. ɉɥɨɬɧɨɫɬɶ ɜɟɪɨɹɬɧɨɫɬɢ ɥɢɧɟɣɧɨɣ ɮɭɧɤɰɢɢ ɨɬ ɧɨɪɦɚɥɶɧɨɣ ɋȼ
ɋȼ X ɩɨɞɱɢɧɹɟɬɫɹ ɧɨɪɦɚɥɶɧɨɦɭ ɡɚɤɨɧɭ ɩɥɨɬɧɨɫɬɢ ɜɟɪɨɹɬɧɨɫɬɢ:
f(x)=(V
x
S
2 )
-1
exp[-(x-m
x
)
2
/(2V
x
2
)], ɚɋȼY ɫɜɹɡɚɧɚ ɫ ɧɟɣ ɥɢɧɟɣɧɨɣ ɡɚɜɢɫɢɦɨ-
ɫɬɶɸ Y = aX + b, ɝɞɟ a ɢ b - ɧɟɫɥɭɱɚɣɧɵɟ ɤɨɷɮɮɢɰɢɟɧɬɵ. Ɍɪɟɛɭɟɬɫɹ ɧɚɣɬɢ ɩɥɨɬ-
ɧɨɫɬɶ ɜɟɪɨɹɬɧɨɫɬɢ f(y).
ɉɨɫɤɨɥɶɤɭ ɡɚɞɚɧɧɚɹ ɮɭɧɤɰɢɹ ɦɨɧɨɬɨɧɧɚ, ɩɪɢɦɟɧɢɦ ɪɟɡɭɥɶɬɢɪɭɸɳɭɸ ɮɨɪ-
ɦɭɥɭ ɩɪɟɞɵɞɭɳɟɝɨ ɩɨɞɪɚɡɞɟɥɚ, ɜ ɤɨɬɨɪɨɣ I(y)=(y-b)/a,I'(y)= =1/a. ɉɨɞɫɬɚɜɢɜ
ɩɨɥɭɱɟɧɧɵɟ ɮɨɪɦɭɥɵ ɜ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɧɨɪɦɚɥɶɧɨɣ ɩɥɨɬɧɨɫɬɢ ɜɟɪɨɹɬɧɨɫɬɢ, ɩɨ-
ɥɭɱɢɦ
g(y) = [|a|V
x
S
2 ]
-1
exp{-[y-(am
x
+b)]
2
/(2|a|
2
V
x
2
)},
ɚ ɷɬɨ ɟɫɬɶ ɧɨɪɦɚɥɶɧɚɹ ɩɥɨɬɧɨɫɬɶ ɜɟɪɨɹɬɧɨɫɬɢ, ɭ ɤɨɬɨɪɨɣ ɆɈ m
Y
=am
X
+b, ɚɞɢɫ-
ɩɟɪɫɢɹ V
Y
2
= (a V
X
)
2
. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɥɢɧɟɣɧɚɹ ɮɭɧɤɰɢɹ ɨɬ ɧɨɪɦɚɥɶɧɨɝɨ ɚɪɝɭ-
ɦɟɧɬɚ ɬɚɤɠɟ ɩɨɞɱɢɧɹɟɬɫɹ ɧɨɪɦɚɥɶɧɨɦɭ ɡɚɤɨɧɭ. ɉɪɢ ɷɬɨɦ ɢɡɦɟɧɟɧɢɟ ɫɪɟɞɧɟɤɜɚɞ-
ɪɚɬɢɱɟɫɤɨɝɨ ɨɬɤɥɨɧɟɧɢɹ ɨɩɪɟɞɟɥɹɟɬɫɹ ɡɧɚɱɟɧɢɟɦ ɤɨɷɮɮɢɰɢɟɧɬɚ ɚ.
5.2. Ɋɚɫɩɪɟɞɟɥɟɧɢɟ ɧɟɦɨɧɨɬɨɧɧɨɣ ɮɭɧɤɰɢɢ ɨɞɧɨɣ ɋȼ
ɋɨɛɵɬɢɟ Y < y ɪɚɜɧɨɡɧɚɱɧɨ ɩɨ ɜɟɪɨɹɬɧɨɫɬɢ ɩɨɩɚɞɚɧɢɸ ɜɨ ɜɫɟ ɭɱɚɫɬɤɢ ɧɟ-
ɨɞɧɨɡɧɚɱɧɨɫɬɢ ɩɨ ɨɫɢ x. ɉɨɷɬɨɦɭ ɜɟɪɨɹɬɧɨɫɬɢ ɫɭɦɦɢɪɭɸɬɫɹ, ɚ ɮɭɧɤɰɢɹ ɪɚɫɩɪɟ-
ɞɟɥɟɧɢɹ
G(y) =

¦
³
'
i
y
i
f(x)dx. (5.1)
ɉɥɨɬɧɨɫɬɶ ɜɟɪɨɹɬɧɨɫɬɢ ɩɨɥɭɱɚɟɬɫɹ ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɟɦ G(y) ɩɨ y.
ɉɊɂɆȿɊ. ȼɵɪɚɠɟɧɢɟ y = M(x) = ax
2
ɟɫɬɶ ɞɜɭɡɧɚɱɧɚɹ ɮɭɧɤɰɢɹ ɧɚ ɨɫɢ x ɨɬ -f
ɞɨ f. Ɍɨ ɟɫɬɶ ɜ ɮɨɪɦɭɥɟ (5.1) i = 2, ɱɢɫɥɨ ɫɥɚɝɚɟɦɵɯ ɪɚɜɧɨ ɬɚɤɠɟ 2 ɢ ɜɵɪɚɠɟɧɢɟ
ɞɥɹ ɩɥɨɬɧɨɫɬɢ ɜɟɪɨɹɬɧɨɫɬɢ ɦɨɠɧɨ ɧɚɣɬɢ ɬɚɤ: g(y)= =[f(
ay / )+f(- ay / )]
2
ay .
5.3. Ɋɚɫɩɪɟɞɟɥɟɧɢɟ ɮɭɧɤɰɢɢ ɧɟɫɤɨɥɶɤɢɯ ɋȼ
ȿɫɬɶ ɫɢɫɬɟɦɚ, ɧɚɩɪɢɦɟɪ, ɢɡ ɞɜɭɯ ɋȼ X ɢ Y ɫ ɩɥɨɬɧɨɫɬɶɸ ɜɟɪɨɹɬɧɨɫɬɢ f(x, y).
ɇɟɨɛɯɨɞɢɦɨ ɧɚɣɬɢ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɋȼ Z, ɫɜɹɡɚɧɧɨɣ ɫ X ɢ Y ɮɭɧɤɰɢɟɣ Z=M(X,Y).
Ɏɭɧɤɰɢɹ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɋȼ Z ɟɫɬɶ G(z)=P(M(X,Y)<z). Ɋɚɫɫɟɱɟɦ ɩɨɜɟɪɯɧɨɫɬɶ M

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