Прикладные математические методы в статистической радиотехнике. Федосов В.П. - 36 стр.

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f
1
(l/s
1
)|(2Snm
21
)
-1/2
exp[-(l-nm
11
)
2
/(2nm
21
)],
ɝɞɟ m
10
=m
1
[lnl(x)/s
0
], m
20
=m
2
[lnl(x)/s
0
]; m
11
=m
1
[lnl(x)/s
1
], m
21
= =m
2
[lnl(x)/s
1
]. ȼ
ɷɬɨɦ ɫɥɭɱɚɟ ɢɦɟɸɬ ɦɟɫɬɨ ɫɥɟɞɭɸɳɢɟ ɚɫɢɦɩɬɨɬɢɱɟɫɤɢɟ ɫɨɨɬɧɨɲɟɧɢɹ:
D=1-F [(lnC-nm
10
)/(2Snm
20
)
1/2
] ɢ E=1-F [(lnC-nm
11
)/(2Snm
20
)
1/2
],
ɝɞɟ
F(x)=(2S)
-1/2
³
f
x
exp(-t
2
/2)dt - ɬɚɛɥɢɱɧɵɣ ɢɧɬɟɝɪɚɥ Ʌɚɩɥɚɫɚ.
ɉɨɫɥɟɞɨɜɚɬɟɥɶɧɵɣ ɚɧɚɥɢɡ (ɤɪɢɬɟɪɢɣ Ⱥ.ȼɚɥɶɞɚ)
Ɉɬɥɢɱɢɬɟɥɶɧɨɣ ɨɫɨɛɟɧɧɨɫɬɶɸ ɜɫɟɯ ɢɡɭɱɟɧɧɵɯ ɤɪɢɬɟɪɢɟɜ ɩɪɢ ɩɪɨɜɟɪɤɟ
ɩɪɨɫɬɨɣ ɝɢɩɨɬɟɡɵ ɹɜɥɹɟɬɫɹ ɧɟɢɡɦɟɧɧɨɫɬɶ ɪɚɡɦɟɪɚ
n ɜɵɛɨɪɤɢ X. ɂɦɟɟɬɫɹ ɞɪɭɝɨɣ
ɩɨɞɯɨɞ ɤ ɨɩɪɟɞɟɥɟɧɢɸ ɩɪɚɜɢɥɚ ɜɵɛɨɪɚ ɪɟɲɟɧɢɹ. ɉɪɢ ɷɬɨɦ ɪɚɡɦɟɪ ɜɵɛɨɪɤɢ ɨɝ-
ɪɚɧɢɱɢɜɚɸɬ ɜ ɩɪɨɰɟɫɫɟ ɷɤɫɩɟɪɢɦɟɧɬɚ ɜ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɪɟɡɭɥɶɬɚɬɨɜ ɧɚɛɥɸɞɟɧɢɣ.
ȼɧɚɱɚɥɟ ɧɚɛɥɸɞɚɸɬ ɨɞɧɨ ɟɞɢɧɫɬɜɟɧɧɨɟ ɢɡɦɟɪɟɧɢɟ
x
1
(ɪɚɡɦɟɪ ɜɵɛɨɪɤɢ n=1) ɢ
ɩɪɢɧɢɦɚɸɬ ɨɞɧɨ ɢɡ ɬɪɟɯ ɜɨɡɦɨɠɧɵɯ ɪɟɲɟɧɢɣ: ɩɪɢɧɹɬɶ ɝɢɩɨɬɟɡɭ
H
0
; ɨɬɜɟɪɝɧɭɬɶ
ɝɢɩɨɬɟɡɭ
H
0
ɢɥɢ ɩɪɢɧɹɬɶ ɚɥɶɬɟɪɧɚɬɢɜɭ H
1
; ɩɪɨɞɨɥɠɢɬɶ ɧɚɛɥɸɞɟɧɢɹ, ɬ.ɟ. ɧɟ ɩɪɢ-
ɧɢɦɚɬɶ ɧɢ ɨɞɧɨɝɨ ɢɡ ɞɜɭɯ ɩɪɟɞɵɞɭɳɢɯ ɪɟɲɟɧɢɣ. ȿɫɥɢ ɜ ɩɪɨɰɟɫɫɟ ɷɤɫɩɟɪɢɦɟɧɬɚ
ɩɪɢɧɢɦɚɸɬ ɨɞɧɨ ɢɡ ɞɜɭɯ ɩɟɪɜɵɯ ɜ ɫɩɢɫɤɟ ɪɟɲɟɧɢɣ, ɬɨ ɷɤɫɩɟɪɢɦɟɧɬ ɡɚɤɚɧɱɢɜɚɟɬ-
ɫɹ. ȿɫɥɢ ɩɪɢɧɢɦɚɟɬɫɹ ɬɪɟɬɶɟ ɪɟɲɟɧɢɟ, ɬɨ ɩɨɥɭɱɚɸɬ ɫɥɟɞɭɸɳɟɟ ɟɞɢɧɢɱɧɨɟ ɢɡɦɟ-
ɪɟɧɢɟ,
n ɫɬɚɧɨɜɢɬɫɹ ɪɚɜɧɵɦ ɞɜɭɦ ɢ ɫɧɨɜɚ ɩɪɨɰɟɞɭɪɚ ɩɨɜɬɨɪɹɟɬɫɹ. ȼɷɬɨɦɫɥɭɱɚɟ
n ɬɚɤɠɟ ɹɜɥɹɟɬɫɹ ɋȼ. Ɉɛɥɚɫɬɶ ɩɪɢɧɹɬɢɹ ɪɟɲɟɧɢɣ (ɨɛɥɚɫɬɶ ɜɵɛɨɪɨɤ ɪɚɡɛɢɜɚɟɬɫɹ
ɭɠɟ ɧɚ ɬɪɢ ɨɛɥɚɫɬɢ: ɞɨɩɭɫɬɢɦɭɸ, ɤɪɢɬɢɱɟɫɤɭɸ ɢ ɩɪɨɦɟɠɭɬɨɱɧɭɸ). ȼ ɤɚɱɟɫɬɜɟ
ɤɪɢɬɟɪɢɹ ɩɪɢɧɢɦɚɸɬ
ɦɢɧɢɦɭɦ ɫɪɟɞɧɟɝɨ ɪɚɡɦɟɪɚ ɜɵɛɨɪɤɢ ɩɪɢ ɭɫɥɨɜɢɢ, ɱɬɨ ɭɪɨ-
ɜɟɧɶ ɡɧɚɱɢɦɨɫɬɢ ɧɟ ɩɪɟɜɵɲɚɟɬ
D, ɚ ɦɨɳɧɨɫɬɶ ɤɪɢɬɟɪɢɹ ɧɟ ɦɟɧɶɲɟ 1-E. Ʉɚɤ ɩɨ-
ɤɚɡɚɥ Ⱥ.ȼɚɥɶɞ, ɧɚɢɥɭɱɲɢɦ ɪɟɲɚɸɳɢɦ ɩɪɚɜɢɥɨɦ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɹɜɥɹɟɬɫɹ ɩɨɫɥɟ-
ɞɨɜɚɬɟɥɶɧɨɟ ɩɪɚɜɢɥɨ ɜɵɛɨɪɚ ɪɟɲɟɧɢɹ. Ɉɬɧɨɲɟɧɢɟ ɩɪɚɜɞɨɩɨɞɨɛɢɹ ɫɪɚɜɧɢɜɚɟɬɫɹ
ɫ ɩɨɪɨɝɨɜɵɦɢ ɭɪɨɜɧɹɦɢ
C
0
ɢ C
1
.
ɇɚɢɥɭɱɲɟɟ ɪɚɡɛɢɟɧɢɟ ɩɪɨɫɬɪɚɧɫɬɜɚ ɜɵɛɨɪɨɤ ɨɛɟɫɩɟɱɢɜɚɟɬɫɹ ɫɥɟɞɭɸɳɢɦɢ
ɧɟɪɚɜɟɧɫɬɜɚɦɢ:
- ɞɨɩɭɫɬɢɦɚɹ ɨɛɥɚɫɬɶ
C
0
< l(x
1
,..., x
k
) < C
1
, k=1,..., n-1;
l(x
1
,..., x
k
) < C
0
;
- ɤɪɢɬɢɱɟɫɤɚɹ ɨɛɥɚɫɬɶ
C
0
< l(x
1
,..., x
k
) < C
1
, k=1,..., n-1;
l(x
1
,..., x
k
) > C
0
;
- ɤɪɢɬɢɱɟɫɤɚɹ ɨɛɥɚɫɬɶ
C
0
< l(x
1
,..., x
k
) < C
1
, k=1,..., n.
ȼɵɱɢɫɥɟɧɢɟ ɩɨɪɨɝɨɜɵɯ ɭɪɨɜɧɟɣ
C
0
ɢ C
1
ɜɵɡɵɜɚɟɬ ɡɧɚɱɢɬɟɥɶɧɵɟ ɦɚɬɟɦɚɬɢ-
ɱɟɫɤɢɟ ɬɪɭɞɧɨɫɬɢ, ɨɞɧɚɤɨ ɟɫɥɢ ɭɫɥɨɜɧɵɟ ɜɟɪɨɹɬɧɨɫɬɢ ɨɲɢɛɨɤ ɧɟ ɩɪɟɜɵɲɚɸɬ
ɡɧɚɱɟɧɢɹ 0,5, ɬɨ ɩɨɪɨɝɨɜɵɟ ɡɧɚɱɟɧɢɹ ɨɩɪɟɞɟɥɹɸɬ ɢɡ ɫɥɟɞɭɸɳɢɯ ɧɟɪɚɜɟɧɫɬɜ:
C
1
d
(1-E)/D ɢ C
0
tE/(1-D).
ɉɪɢ ɢɫɩɨɥɶɡɨɜɚɧɢɢ ɪɚɫɫɦɨɬɪɟɧɧɨɝɨ ɤɪɢɬɟɪɢɹ ɞɨɫɬɢɝɚɟɬɫɹ ɷɤɨɧɨɦɢɹ ɪɚɡɦɟ-
ɪɚ ɜɵɛɨɪɤɢ
ɜ ɫɪɟɞɧɟɦ, ɤɨɝɞɚ ɟɟ ɪɚɡɦɟɪ ɢɦɟɟɬ ɫɭɳɟɫɬɜɟɧɧɨɟ ɡɧɚɱɟɧɢɟ, ɧɚɩɪɢɦɟɪ,
ɜ ɥɨɤɚɰɢɢ ɩɪɢ ɨɛɡɨɪɟ ɩɪɨɫɬɪɚɧɫɬɜɚ ɦɨɠɧɨ ɡɧɚɱɢɬɟɥɶɧɨ ɜɵɢɝɪɚɬɶ ɜɪɟɦɹ ɨɛɡɨɪɚ,

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