Прикладные математические методы в статистической радиотехнике. Федосов В.П. - 54 стр.

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ɧɨɫɬɶ ɨɲɢɛɨɤ ɜɬɨɪɨɝɨ ɪɨɞɚ - ɩɪɢɧɹɬɢɟ ɧɟɜɟɪɧɨɣ ɝɢɩɨɬɟɡɵ. ȿɫɥɢ ɞɥɹ ɞɚɧɧɨɣ ɜɵ-
ɛɨɪɤɢ '<'
0
, ɬɨ, ɯɨɬɹ ɩɪɢɧɹɬɨɟ ɩɪɚɜɢɥɨ ɩɪɟɞɩɢɫɵɜɚɟɬ ɭɬɜɟɪɠɞɚɬɶ ɢɫɬɢɧɧɨɫɬɶ
ɜɵɞɜɢɧɭɬɨɣ ɝɢɩɨɬɟɡɵ, ɫɥɟɞɭɟɬ ɤ ɬɚɤɨɦɭ ɭɬɜɟɪɠɞɟɧɢɸ ɨɬɧɟɫɬɢɫɶ ɛɨɥɟɟ ɨɫɬɨɪɨɠ-
ɧɨ, ɬɚɤ ɤɚɤ ɫ ɭɱɟɬɨɦ ɡɚɞɚɧɧɨɝɨ ɤɪɢɬɟɪɢɹ ɫɨɝɥɚɫɢɹ ɧɟɬ ɨɫɧɨɜɚɧɢɣ ɫɱɢɬɚɬɶ, ɱɬɨ
ɜɵɛɨɪɤɚ
X ɧɟ ɫɨɝɥɚɫɭɟɬɫɹ ɫ ɝɢɩɨɬɟɬɢɱɟɫɤɢɦ ɪɚɫɩɪɟɞɟɥɟɧɢɟɦ.
ɇɟɞɨɫɬɚɬɤɨɦ ɤɪɢɬɟɪɢɟɜ ɫɨɝɥɚɫɢɹ ɹɜɥɹɟɬɫɹ ɩɪɨɢɡɜɨɥɶɧɨɫɬɶ ɜɵɛɨɪɚ ɫɚɦɨɝɨ
ɤɪɢɬɟɪɢɹ ɢ ɭɪɨɜɧɹ ɡɧɚɱɢɦɨɫɬɢ D, ɚ ɬɚɤɠɟ ɬɪɭɞɧɨɫɬɢ, ɫɜɹɡɚɧɧɵɟ ɫ ɨɩɪɟɞɟɥɟɧɢɟɦ
ɢɯ ɪɚɫɩɪɟɞɟɥɟɧɢɣ ɞɥɹ ɦɚɥɵɯ
n. Ⱦɪɭɝɨɣ ɩɭɬɶ ɫɨɫɬɨɢɬ ɜ ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɨɦ ɩɨ-
ɫɬɪɨɟɧɢɢ ɧɟɢɡɜɟɫɬɧɨɣ ɮɭɧɤɰɢɢ
F
1
ɩɨ ɜɵɛɨɪɨɱɧɵɦ ɡɧɚɱɟɧɢɹɦ ɛɟɡ ɜɵɞɜɢɠɟɧɢɹ
ɝɢɩɨɬɟɡɵ ɨ ɜɢɞɟ ɷɬɨɣ ɮɭɧɤɰɢɢ.
12.2. Ʉɪɢɬɟɪɢɣ F
2
- ɉɢɪɫɨɧɚ
ɗɬɨɬ ɤɪɢɬɟɪɢɣ ɹɜɥɹɟɬɫɹ ɨɞɧɢɦ ɢɡ ɫɚɦɵɯ ɪɚɫɩɪɨɫɬɪɚɧɟɧɧɵɯ ɜ ɩɪɢɥɨɠɟɧɢɹɯ
ɤ ɬɟɨɪɢɢ ɫɬɚɬɢɫɬɢɱɟɫɤɢɯ ɨɰɟɧɨɤ.
Ɋɚɡɨɛɶɟɦ ɨɛɥɚɫɬɶ, ɝɞɟ ɨɩɪɟɞɟɥɟɧɚ ɝɢɩɨɬɟɬɢɱɟɫɤɚɹ ɮɭɧɤɰɢɹ ɪɚɫɩɪɟɞɟɥɟɧɢɹ
F
1
(x) ɋȼ ɏ, ɧɚ ɤɨɧɟɱɧɨɟ ɱɢɫɥɨ ɧɟɩɟɪɟɤɪɵɜɚɸɳɢɯɫɹ ɢɧɬɟɪɜɚɥɨɜ '
i
, i = 1,..., l.
Ɉɛɨɡɧɚɱɢɦ ɜɟɪɨɹɬɧɨɫɬɶ ɩɨɩɚɞɚɧɢɹ ɟɞɢɧɢɱɧɨɝɨ ɢɡɦɟɪɟɧɢɹ ɜ
i-ɣ ɢɧɬɟɪɜɚɥ p
i
=P(X
'
i
),
¦
l
i 1
p
i
=1. ɉɭɫɬɶ ɜ ɜɵɛɨɪɤɟ X ɱɢɫɥɨ ɜɵɛɨɪɨɱɧɵɯ ɡɧɚɱɟɧɢɣ, ɩɨɩɚɞɚɸɳɢɯ ɜ
ɢɧɬɟɪɜɚɥ '
i
, ɪɚɜɧɨ Q
i
. əɫɧɨ, ɱɬɨ
¦
l
i 1
Q
i
=n.
ɂɫɩɨɥɶɡɭɟɦ ɜ ɤɚɱɟɫɬɜɟ ɤɪɢɬɟɪɢɹ ɫɨɝɥɚɫɢɹ ɜɟɥɢɱɢɧɭ
'=

¦¦
¸
¹
·
¨
©
§
l
i
ii
i
i
i
l
i
i
np
np
p
np
n
1
2
1
1
X
X
.
Ʉɚɤ ɞɨɤɚɡɚɥ ɉɢɪɫɨɧ, ɟɫɥɢ ɩɪɨɜɟɪɹɟɦɚɹ ɝɢɩɨɬɟɡɚ ɨɛ ɢɫɬɢɧɧɨɫɬɢ ɪɚɫɩɪɟɞɟɥɟ-
ɧɢɹ
F
1
(x) ɜɟɪɧɚ, ɬɨ ɩɪɢ nof ɪɚɫɩɪɟɞɟɥɟɧɢɟ ' ɚɫɢɦɩɬɨɬɢɱɟɫɤɢ ɩɪɢɛɥɢɠɚɟɬɫɹ ɤ
2
F
-ɪɚɫɩɪɟɞɟɥɟɧɢɸ ɫ (l 1) ɫɬɟɩɟɧɹɦɢ ɫɜɨɛɨɞɵ, ɧɟ ɡɚɜɢɫɹɳɟɦɭ ɨɬ ɜɢɞɚ ɝɢɩɨ-
ɬɟɬɢɱɟɫɤɨɝɨ ɪɚɫɩɪɟɞɟɥɟɧɢɹ
F
1
(x).
ɂɬɚɤ, ɩɭɫɬɶ
F
2
D
- ɩɪɨɰɟɧɬɧɨɟ ɨɬɤɥɨɧɟɧɢɟ ɋȼ, ɢɦɟɸɳɟɣ F
2
-ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɫ
(
l -1) ɫɬɟɩɟɧɹɦɢ ɫɜɨɛɨɞɵ, ɬ.ɟ. P{F
2
>F
2
D
}=D. ɉɪɢ ɞɨɫɬɚɬɨɱɧɨ ɦɚɥɨɦ D ɢ ɛɨɥɶɲɨɦ
n ɜɟɥɢɱɢɧɚ
', ɜɵɱɢɫɥɟɧɧɚɹ ɫɨɝɥɚɫɧɨ ɤɪɢɬɟɪɢɸ F
2
-ɉɢɪɫɨɧɚ, ɩɪɚɤɬɢɱɟɫɤɢ ɧɢɤɨɝɞɚ
ɧɟ ɛɭɞɟɬ ɩɪɟɜɨɫɯɨɞɢɬɶ ɩɨɪɨɝɨɜɨɟ ɡɧɚɱɟɧɢɟ
F
2
D
, ɟɫɥɢ ɬɨɥɶɤɨ ɝɢɩɨɬɟɡɚ ɨ ɜɢɞɟ
ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɜɟɪɧɚ. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɜɵɛɢɪɚɟɬɫɹ ɫɥɟɞɭɸɳɟɟ ɩɪɚɜɢɥɨ ɩɪɨɜɟɪɤɢ
ɝɢɩɨɬɟɡɵ: ɝɢɩɨɬɟɡɚ ɨɬɤɥɨɧɹɟɬɫɹ, ɟɫɥɢ
'>F
2
D
, ɢ ɩɪɢɧɢɦɚɟɬɫɹ, ɟɫɥɢ '<F
2
D
. ȼɟɪɨ-
ɹɬɧɨɫɬɶ ɨɬɤɥɨɧɢɬɶ ɜɟɪɧɭɸ ɝɢɩɨɬɟɡɭ ɪɚɜɧɚ
D.
Ʉɪɨɦɟ ɨɬɦɟɱɟɧɧɵɯ ɭɠɟ ɨɛɳɢɯ ɧɟɞɨɫɬɚɬɤɨɜ ɤɪɢɬɟɪɢɟɜ ɫɨɝɥɚɫɢɹ ɪɚɫɫɦɨɬ-
ɪɟɧɧɵɣ ɤɪɢɬɟɪɢɣ
F
2
-ɉɢɪɫɨɧɚ ɨɛɥɚɞɚɟɬ ɟɳɟ ɨɞɧɢɦ ɫɭɳɟɫɬɜɟɧɧɵɦ ɧɟɞɨɫɬɚɬɤɨɦ,

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