Прикладные математические методы в статистической радиотехнике. Федосов В.П. - 16 стр.

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4.4.ɋɬɚɬɢɫɬɢɱɟɫɤɚɹ ɜɡɚɢɦɨɡɚɜɢɫɢɦɨɫɬɶ ɢ ɧɟɡɚɜɢɫɢɦɨɫɬɶ
ɋȼ X ɧɚɡɵɜɚɟɬɫɹ ɧɟɡɚɜɢɫɢɦɨɣ ɨɬ ɋȼ Y, ɟɫɥɢ ɡɚɤɨɧ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɜɟɥɢɱɢɧɵ
X ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɬɨɝɨ, ɤɚɤɨɟ ɡɧɚɱɟɧɢɟ ɩɪɢɧɹɥɚ ɋȼ Y. ȼɷɬɨɦɫɥɭɱɚɟf(x/y) = f(x)
ɩɪɢ ɥɸɛɨɦ x. ɇɟɨɛɯɨɞɢɦɨ ɡɚɦɟɬɢɬɶ, ɱɬɨ ɟɫɥɢ ɋȼ X ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɋȼ Y, ɬɨ ɢ ɋȼ Y
ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɋȼ X. Ⱦɥɹ ɧɟɡɚɜɢɫɢɦɵɯ ɋȼ ɬɟɨɪɟɦɚ ɭɦɧɨɠɟɧɢɹ ɡɚɤɨɧɨɜ ɪɚɫɩɪɟɞɟ-
ɥɟɧɢɹ ɢɦɟɟɬ ɜɢɞ:
f(x,y)=f(x)f(y).
ɗɬɨ ɭɫɥɨɜɢɟ ɪɚɫɫɦɚɬɪɢɜɚɟɬɫɹ ɤɚɤ ɧɟɨɛɯɨɞɢɦɨɟ ɢ ɞɨɫɬɚɬɨɱɧɨɟ ɭɫɥɨɜɢɟ ɧɟ-
ɡɚɜɢɫɢɦɨɫɬɢ ɋȼ. Ɋɚɡɥɢɱɚɸɬ ɩɨɧɹɬɢɹ ɮɭɧɤɰɢɨɧɚɥɶɧɨɣ ɢ ɫɬɚɬɢɫɬɢɱɟɫɤɨɣ ɡɚɜɢɫɢ-
ɦɨɫɬɟɣ. ɉɪɢ ɫɬɚɬɢɫɬɢɱɟɫɤɨɣ ɡɚɜɢɫɢɦɨɫɬɢ ɧɟɥɶɡɹ ɭɤɚɡɚɬɶ ɬɨɱɧɨ ɡɧɚɱɟɧɢɟ, ɤɨɬɨ-
ɪɨɟ ɩɪɢɧɢɦɚɟɬ ɨɞɧɚ ɢɡ ɋȼ, ɟɫɥɢ ɢɡɜɟɫɬɧɨ ɡɧɚɱɟɧɢɟ ɞɪɭɝɨɣ, ɦɨɠɧɨ ɥɢɲɶ ɨɩɪɟɞɟ-
ɥɢɬɶ ɜɥɢɹɧɢɟ ɜ ɫɪɟɞɧɟɦ. ɇɨ ɩɨ ɦɟɪɟ ɭɜɟɥɢɱɟɧɢɹ ɜɡɚɢɦɨɡɚɜɢɫɢɦɨɫɬɢ ɫɬɚɬɢɫɬɢɱɟ-
ɫɤɚɹ ɡɚɜɢɫɢɦɨɫɬɶ ɩɪɟɜɪɚɳɚɟɬɫɹ ɜ ɮɭɧɤɰɢɨɧɚɥɶɧɭɸ.
4.5. ɑɢɫɥɨɜɵɟ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɫɢɫɬɟɦɵ ɞɜɭɯ ɋȼ.
Ʉɨɪɪɟɥɢɪɨɜɚɧɧɨɫɬɶ
Ʉɚɤ ɢ ɞɥɹ ɨɞɧɨɣ ɋȼ, ɞɥɹ ɫɢɫɬɟɦɵ ɞɜɭɯ ɋȼ ɦɨɠɧɨ ɢɫɩɨɥɶɡɨɜɚɬɶ ɧɚɱɚɥɶɧɵɟ
ɢ ɰɟɧɬɪɚɥɶɧɵɟ ɦɨɦɟɧɬɵ.
ɇɚɱɚɥɶɧɵɦ ɦɨɦɟɧɬɨɦ ɩɨɪɹɞɤɨɜ k,s ɫɢɫɬɟɦɵ (X,Y) ɧɚɡɵɜɚɟɬɫɹ ɆɈ ɩɪɨɢɡɜɟ-
ɞɟɧɢɹ: X
k
Y
s
;
m
ks
=M[X
k
Y
s
].
ɐɟɧɬɪɚɥɶɧɵɦ ɦɨɦɟɧɬɨɦ ɩɨɪɹɞɤɨɜ k,s ɧɚɡɵɜɚɟɬɫɹ ɆɈ ɩɪɨɢɡɜɟɞɟɧɢɹ k-ɣɢ
s-ɣ ɫɬɟɩɟɧɢ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɰɟɧɬɪɢɪɨɜɚɧɧɵɯ ɜɟɥɢɱɢɧ.
Ⱦɥɹ ɧɟɩɪɟɪɵɜɧɵɯ ɋȼ
m
ks
=
³³
f
f
f
f
x
k
y
s
f(x,y)dxdy, P
ks
=
³³
f
f
f
f
(x-m
x
)
k
(y-m
y
)
s
f(x,y)dxdy.
ɉɟɪɜɵɣ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɟɫɬɶ ɆɈ ɞɥɹ ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɣ ɋȼ X ɢɥɢ Y.
Ⱥɧɚɥɨɝɢɱɧɨ ɢɦɟɸɬɫɹ ɢ ɜɬɨɪɵɟ ɰɟɧɬɪɚɥɶɧɵɟ ɦɨɦɟɧɬɵ ɫɢɫɬɟɦɵ ɋȼ: D
X
ɢ
D
Y
, ɤɨɬɨɪɵɟ ɯɚɪɚɤɬɟɪɢɡɭɸɬ ɫɬɟɩɟɧɶ ɪɚɡɛɪɨɫɚɧɧɨɫɬɢ ɫɥɭɱɚɣɧɨɣ ɬɨɱɤɢ ɜɞɨɥɶ ɨɫɟɣ
x ɢ y ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ.
Ɉɫɨɛɭɸ ɪɨɥɶ ɜ ɫɬɚɬɢɫɬɢɱɟɫɤɨɣ ɪɚɞɢɨɬɟɯɧɢɤɟ ɢɝɪɚɟɬ ɜɬɨɪɨɣ ɫɦɟɲɚɧɧɵɣ
ɰɟɧɬɪɚɥɶɧɵɣ ɦɨɦɟɧɬ P
11
= M[(x-m
X
)(y-m
Y
)] = K
XY
- ɤɨɪɪɟɥɹɰɢɨɧɧɵɣ ɦɨɦɟɧɬ.
Ⱦɥɹ ɧɟɩɪɟɪɵɜɧɵɯ ɋȼ ɤɨɪɪɟɥɹɰɢɨɧɧɵɣ ɦɨɦɟɧɬ ɜɵɪɚɠɚɟɬɫɹ ɮɨɪɦɭɥɨɣ
K
XY
=
³³
f
f
f
f
(x-m
X
)(y-m
Y
)f(x,y)dxdy.
ɗɬɨɬ ɦɨɦɟɧɬ ɤɪɨɦɟ ɪɚɫɫɟɢɜɚɧɢɹ ɋȼ ɯɚɪɚɤɬɟɪɢɡɭɟɬ ɢ ɜɡɚɢɦɨɡɚɜɢɫɢɦɨɫɬɶ ɋȼ
X ɢ Y. ɉɪɢ ɷɬɨɦ, ɟɫɥɢ ɋȼ X ɢ Y ɧɟɡɚɜɢɫɢɦɵ, ɬɨ K
XY
= 0. Ⱦɨɤɚɠɟɦ ɷɬɨ ɩɪɟɞɩɨɥɨ-
ɠɟɧɢɟ: ɟɫɥɢ ɋȼ X ɢ Y ɧɟɡɚɜɢɫɢɦɵ, f(x,y)=f(x)f(y), ɬɨ ɩɨɫɥɟɞɧɢɣ ɢɧɬɟɝɪɚɥ ɪɚɫɩɚɞɚ-

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