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1
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A
1
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k
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;
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­ V?VSHVU @QESQE =^=F GQYQZ}RUE HQQ C G=UE=; Ÿ?}^V?RUE A[?VZ ·@QQ?B 9
F[EHVU G=S G=USXXY=F GQYQZ}RUE ERUH HQQ n = C · C = 188496 ;
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P (AB)
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P (AB) + P (A · B)
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=SA=F==@ F=Z==?=FS[U G=U^=Y HVASVV?RUS F=Z==?=YS[U [>VSAY[[A SVEV;
KX?_RYH\E A[EA A , A , . . . , A SV@VE FQQ?QEAQQ F=Z==?=YS[U n [>VSAVY €^=Sb
A=F GQYQZ}HQU G]S]]A [>VSAVY HX@ G[?RUE €^=SA=F Z=S=AY=YXXA P (A ) = p B
;K
==
=
P (A ) = p , . . . , P (A ) = p GQYQS X?_RYH ? G[F [>VSAY[[AB V@^VY YW EW
¤
=
=
=
;
=
=
T €^ SA FS[U G U} GQYEQ VV?F [>VSAY[[ARUE V@?VS [>VSAYRUS F ?S Y>=E
1
2
n
1
2
2
n
1
n
P (A1 ) = q1 , P (A2 ) = q2 , . . . , P (An ) = qn
SV`; ¼V?V^ A EW A [>VSAY[[ARUE (i = 1, n) €A=} EVS EW €^=SA=F [>VSAVY GQY
£ ;«¥
P (A) = 1 − q · q · q · · · q
G=UE=; ¢]?]]? FVYGVYB A , A , . . . , A SV@VE F=Z==?=YS[U [>VSAY[[ARUE €A=}
EVS EW €^=SA=F Z=S=AY=Y HVASVV?RUE V@?VS [>VSAYRUE Z=S=AY=YXXA\E [?}½
^V?RUS EVS}VV@ F=@@=EH=U HVE[[ G=UE=; KXF=UE HQFRQYAQYA A [>VSAY[[A
G[SA R}RY GQYQZ}HQU GQY (P (A ) = p, i = 1, n)) £ ;«¥ HQZ¶·Q
£ ;9¥
P (A) = 1 − q
FVYGV?HVU GQYEQ (q = 1 − p).
#”Ôšš$°¸ <XXAY=S\E H=ZR?TRE G[? G=US QEQF Z=S=AY=Y F=?S=Y>=E
= ; K=ZR?TRE G[? G=U ?XX ž XA== >V?VS
p = 0, 8; p = 0, 7; p = 0, 9 G U^
GXXA=F=A €A=} EVS EW QEQ@QE G=UF [>VSAYRUE Z=S=AY=Y\S QY;
=
:=
i½? H RZ?TRE QEQF [>VSAVYRUS A , (i = 1, 3) B A } EVS EW QEQF [>VSAYRUS½A
SV`; K=ZR?TRE HX@ G[?RUE G=US QEQF Z=S=AY=Y GX@A\EF== QEQFB V@ QEQFQQ@
F=Z==?=FS[U HXY FQQ?QEAQQ F=Z==?=YS[U [>VSAY[[A GQYEQ; P (A ) = q =
;«
1 − 0.8 = 0.2; P (A ) = q = 1 − 0.7 = 0.3; P (A ) = q = 1 − 0.9 = 0.1 XTR? £ ¥
HQZ¶·Q ·@QQ? P (A) = 1−0.2 · 0.3 · 0.1 = 0.994.
i
1
1
2
2
3
n
n
i
i
n
1
2
3
i
1
2
2
1 1 O v s - q t 1 1 r
[>VSAVY
3
1
3
'q \]'
EW G[HVE G[YVS [[@SVF FQ@ FQ@QQ?QQ ERUS[U [>VSAY[[A G=US;
[>VSAY[[ARUE =YW EVS EW €^=SA@=E E]F]YA €^=SAA=S
H1 , H2 , . . . , Hn
A
Hi (i = l, n)
°° M™Õš ‰•Ž•›•›ÃŠ –’‰³´’
¦
SV} @=E=€; ¢]?]]? FVYGVYB A = AH + AH + · · · + AH ; DEV HQFRQYAQYA
== = =
;
== =
H (i = 1, n) [>VSAYRUS H Z SY YXXA SVEV A [>VSAYRUE Z S AY Y EW VASVV?
H==Z=SY=YXXA\E Z=S=AY=Y\S 3 [>VSAYRUE F=?S=Y>=F E]F]YH Z=S=AY=Y==?
[?}[[Y} EVZ@VE ERUYGV?HVU HVE[[ G=UE=;
1
2
n
i
P (A) =
¤VYSV?VES[U GRT^VY2
n
X
P (Hi )P (A/Hi )
£ ;¬¥
i=1
P (A) = P (H1 )P (A/H1 ) + P (H2 )P (A/H2 ) + · · · + P (Hn )P (A/Hn ).
£ ;¬¥ HQZ¶·QS S[UVA Z=S=AY=Y\E HQZW·Q SVEV; £ ;¬¥ EW H==Z=SY=Y HX@ G[?RUE
Z=S=AY=Y\S VEV H==Z=SY=Y\E E]F]Y A=FW A [>VSAYRUE Z=S=AY=Y==? [?b
}[[Y@VE [?}^V?[[ARUE ERUYGV? ~Z;
#”Ôšš$°» 3€Y=ST O FQHQQ@ S=?=F=A QTRF S=>?\E >=Z @=Y@=E XTR?
=YW EVSRUS EW H==ZS==? @QESQE =^=F _==?AY=S=H=U GQY}VV; MQESQ@QE >=Z
HX@ G[? A=FRE FVA FVA @=Y==Y@=E G=U@=E G]S]]A =YW T >=Z==? EW €^@=E QTRF
S=>?== F[?T GQYQF G=UY==; ¼V?^VV =€Y=ST =YW T >=Z==? €^=F GQYQZ}HQU £A]H
>=Z==? €^=F =YG=S[U¥ SV^VY 3 FQHQA QTRF [>VSAYRUE Z=S=AY=Y\S QY; P=Z\E
@F`ZRUS PX?=S ;ž½A [>[[YV^; DEA A=?==F H==Z=SY=YXXA AV^_[[Y} GQYEQ;
== ž½? >=Z\S @QESQE =^=F [>VSAVY
H ½ VFERU XA
O
=
==
H ½ 7½? > Z\S @QESQE ^ F [>VSAVY
3
=
==
1
H ½ ½? > Z\S @QESQE ^ F [>VSAVY
P
K==ZS==? @QESQE =^T GXU XTR? VASVV? >=ZXXA\E
2
=YW EVSRUS @QESQE =^=F GQYQZ} R}RY; ¢]?]]? FVYb
M
N
GVYB
1
=
A
P (H ) = P (H ) = P (H ) = . 3€Y ST A FQHQA QTRF
[>VSAYRUS B ½VV? HVZAVSYV`3; KVS^VY =€Y=ST ž½? >=b
PX?=S ;ž2
Z\S @QESQE =^@=E E]F]YA A FQHQA QTRF [>VSAYRUE
Z=S=AY=Y 1 GQYEQ; ¢]?]]? FVYGVYB P (B/H ) = 1 ,
Ÿ[ETYVE P3(B/H ) = 1 , P (B/H ) = 1 . LUZA £ ;¬3¥ HQZ¶·Q ·@QQ?2
1
2
3
1
2
3
1
2
3
2
4
1 1 1 1 1 1 13
P (B) = P (H1 )P (B/H1 )+P (H2 )P (B/H2 )+P (H3 )P (B/H3 ) = · + · + · = .
3 3 3 2 3 4 36
#”Ôšš$°Æ K=E\ G=?XXE F=Y==@=EA _ 78½H\E Z]ES]B ¡_ ž­½H\E
Z]ES]B >[[E F=Y==@=EA ®_ 78½H\EB _ ž­½H\E Z]ES] G=U@=E SV`; K= G=?XXE
F=Y==@E==@== H==ZS==? ­ Z]ES] =^T >[[E F=Y==@=EA== FRUSVVA A=?== EW >[[E
F=Y==@E==@== H==ZS==? ž Z]ES] =^@=E G=US; K=E\ >[[E F=Y==@E==@== =^@=E
Z]ES] 78½H\E Z]ES] G=UF [>VSAYRUE Z=S=AY=Y\S QY;
DEV HQFRQYAQYA A=?==F H==Z=SY=Y G=U} GQYEQ;
<=?XXE F=Y==@E==@ ž_ 78½H\E B ¡_ ž­½H\E Z]ES] =^=F [>VSAVYB
H½
1
§¨šŽ›”™Š › Œ•‰•—•Œ ±•Š•—© Œ²›“•— –’‰³´’Šµµ
¦¦
½ 7_ 788½H\EB _ ž­
½H\E Z]ES] ==^==F [>VSAVYB;
ž­
½ _ 7 ½H\EB 7_ ½H\E Z]ES] ^ F [>VSAVY
KVS^VY P (H ) = C C = 1 , P (H ) = C C = 4 , P (H ) = C C
C
7
C
7
C
MQER?F@QE [>VSAYVV
A SV^VY 2
LUZA ;
7
8
9
P (A/H ) = , P (A/H ) = , P (A/H ) = .
H2
H3
1
1
1
3
4
4
2
5
7
2
14
P (A) =
3
X
2
3
3
4
3
5
7
3
14
P (Hi )P (A/Hi ) =
i=1
3
3
2
4
5
7
2
= .
7
14
1 7
4 8
2 9
·
+ ·
+ ·
= 0, 54.
7 14 7 14 7 14
1 5 v % u v ' q \ ] '
FQ@ FQ@QQ?QQ ERUS[U H==Z=SY=YXXA G=
½EW G[HVE G[YVS [[@SVFB
==
=
HX?_RYH FRUFRUE ]ZE] VASVV? H Z SY=Y\E Z=S=AY=Y
= ; KX?_RYH FRUSAV} @=E=Z@=?S[U
P (H ), P (H ), . . . , P (H ) ZVAVSAV} G US
K
=
=
;
[>VSAVY A €^ SA@ E SV` VS^VY A [>VSAVY €^=SA@=E\ A=?== =EFE\ H==b
Z=SY=YXXA\E Z=S=AY=Y €=} ]]?TY]SA]F ^V& SV@VE GQAYQSQ =^T [>W`; ¢]?]]?
FVYGVY P (H ), P (A/H ) Z=S=AY=YXXA\E HX@Y=Z}H=US==? P (H /A) Z=S=Ab
Y=YXXA\S QYEQ SV@VE [S;
= ;¬
= ==
P (A · H ) = P (A)P (H /A) = P (H )(A/H ) (i = 1, n) G £ ¥ HQZW·QS EF ?b
=
^Y
P (H )P (A/H )
£ ;ž8¥
P (H /A) =
H1 , H2 , . . . , Hn
1
2
n
i
i
i
i
i
i
i
i
i
n
P
i
P (Hi )P (A/Hi )
£ ;ž8¥ HQZ¶·QS H==Z=SY=YXXA\E H`Q?`Z GX~X <=U`@RUE HQZW·Q SVEV;
#”Ôšš$°ª :Z=? EVS @=YG=?\E ERUH G[HVVSAVF[[ERU 30%½RUS ã [UYAb
^V?B 25%½RUS ãã [UYA^V?B [YA@VE FX^RUS ããã [UYA^V? FRUAVS G=U}VV; ã [UYAb
^V?RUE FRU@VE G[HVVSAF[[ERU 1% B ãã½\E 1.5% B ããã½\E 2% HX@ HX@ SQYQSAQY
G=UA=S G=U^; ¼V?VSYVSTRUE FXA=YA=E =^@=E G[HVVSAVF[[E SQYQSAQY G=U^;
DEVF[[ SQYQSAQY ã [UYA^V?RUE G[HVVSAVF[[E G=UF [>VSAYRUE Z=S=AY=Y\S QY;
=
H ½G[HVVSAVF[[E ã½? [UYA^V?RUEF G UF [>VSAVYB
=
H ½G[HVVSAVF[[E ãã½? [UYA^V?RUEF G UF [>VSAVYB
=
H ½G[HVVSAVF[[E ããã½? [UYA^V?RUEF G UF [>VSAVYB
=
=
=
=
=
A½ FXA YA E ^@ E G[HVVSAF[[E SQYQSAQY G UF [>VSAVY GQY
P (H ) = 0.30, P (H ) = 0.25, P (H ) = 0.45,
P (A/H ) = 0.01, P (A/H ) = 0.015, P (A/H ) = 0.02 SV} ]S]SA@]E HXY
i=1
1
2
3
1
2
1
3
2
3
P (A) = P (H1 )P (A/H1 ) + P (H2 )P (A/H2 ) + P (H3 )P (A/H3 ) =
0.3 · 0.01
0.30 · 0.01 + 0.25 · 0.015 + 0.45 · 0.02 = 0.015 P (H1 /A) =
= 0, 2
0.015
;
B
LUZA ERUH SQYQSAQY G[HVVSAVF[[ERU 78 Q?TRZ FX^RUS ã [UYA^V? [UYA^V?b
YV}VV; ;R_VVEVV@ [>^VY HX?_RYH\E A=?== H H==Z=SY=Y\E Z=S=AY=Y HX?½
_RYH\E ]ZE]F Z=S=AY=Y==@== G=S=@@=E G=UE=;
1
°¸ ¹•™œº”™Š –’‰³´’
¦°
#”Ôšš$°È ¼Q·? H=ZR?TRE GR` GR`V@VV [Y F=Z==?=E G=U ?XX EVS EVS
==
XA GXXA}VV; ã H=ZR?TRE G=US QEQF Z=S=AY=Y 0.8 B ãã H=ZR?TRE G=US QEQF
Z=S=AY=Y 0.4 ; <=U ?XX GXXA@=E\ A=?== FVE EVS EW QEQ}VV; ã GXXA=ST G=US
QEQ@QE G=UF [>VSAYRUE Z=S=AY=Y\S QY;
KX?_RYH €^=SA=F\E ]ZE] A=?==F H==Z=SY=YXXA G=U} GQYEQ2
= =
= =
==
H −ã G ãã H ZR?TE\ FVE EW T QEQFS[UB H −ã G ãã H ZR?TRE FQ·XY QEQFB
;
H −ã EW QEQ}B ãã EW QEQFS[UB H −ã EW QEQFS[UB ãã EW QEQF
K==Z=SY=Y HX@ G[?RUE Z=S=AY=Y2
1
2
3
4
P (H1 ) = 0.2 · 0.6 = 0.12,
P (H3 ) = 0.8 · 0.6 = 048,
A
P (H2 ) = 0.8 · 0.4 = 0.32,
P (H4 ) = 0.2 · 0.4 = 0.08.
<=US ž QEQF [>VSAYRUS ½==? HVZAVSYV}B AVV?F H=Z=SY=Y HX@ G[?RUE [` AVF
== =
A [>VSAYRUE Z S AY Y\S QYGQY
; =
P (A/H ) = 0, P (A/H ) = 0, P (A/H ) = 1, P (A/H ) = 1 ¼ ?RE HX?_RYH
€^=SA@=E\ A=?== £G=US ž QEQ@E\ A=?==¥ H B H ½H==Z=SY=YXXA GQYQZ}S[U
GQYQF G]S]]A H B H H==Z=SY=Y HX@ G[?RUE Z=S=AY=Y EW
1
2
3
4
1
3
P (H3 /A) =
P (H4 /A) =
2
4
0.048 · 1
6
P (H3 )P (A/H3 )
=
= .
P (H3 )P (A/H3 )+P (H4 )P (A/H4 ) 0.48 · 1+0.08 · 1 7
P (H4 )P (A/H4 )
0.08 · 1
1
=
= .
P (H3 )P (A/H3 ) + P (H4 )P (A/H4 )
0.48 · 1 + 0.08 · 1
7
LUEF[[ ã H=ZR?TRE G=US QEQ@QE G=UF [>VSAYRUE Z=S=AY=Y 6 G=UE=;
7
ö ÷ ø ^
ö ü ‡ ý S 5 þ ÿ 1 q / _ u 1 / / r 1 1 % /11 uv 'q \ ]' KX?_RYH\E A[EA A G= A [>VSAYRUE >]^F]E EVS EW €^=SA=F G]S]]A A [>VSAVY p
Z=S=AY=YH=U RYV?AVS SV`; ¢]?]]? FVYGVYB P (A) = p, P (A) = q (p+q = 1) ; ¤VV?F
HX?_RYH\S EVSVE R}RY E]F]YA QYQE A=FRE A=^H=E FRU`; LUZ HX?_RYb
HXXA\E A=?==YY\S [Y F=Z==?=FB @=E=Z@=?S[U HX?_RYH\E A=?==Y=Y GX~X
<`?EXYYRUE @F`Z SVEV; LUZ @=E=Z@=?S[U HX?_RYH\E }R_VV =ZWA?=Y ?=ab
HRaH QYQE H==?=YAA=S; KXF=UYG=YB H`FEQYQSRUE EVS R}RY E]F]YA [UYAb
^V?YV} GXU G[HVVSAVF[[E @H=EA=?H\E _==?AY=S=EA HQFR?QFB V@ HQFR?QF G t
FXS===E\ HX?_RA ?=ARQ RAV^FH GQAR@ >=A?=FB V@ >=A?=F G Z]ES] GQYQE _QQS
QYQE A=FRE F=€F=A @QER?F@QE [>VSAVY €^=SA=FB V@ €^=SA=F SVF ZVH;
KX?_RYH\S n XA== A=^H=E FRUFVA A [>VSAVY €S k XA== €^=SA=F Z=S=AY=b
Y\S QY¶·; MQER?FQ} GXU [>VSAYVV A B H[[ERU Z=S=AY=Y\S P (k) SV`;
=
= =
;
A EW i½Y (i = 1, n) HX?_RYH EA A €^ SA F [>VSAVY GQYQS
KVS^VY A [>VSAYRUS A=?==F G=UAY==? RYV?FRUY} GQYEQ;
n,k
n
i
n,k
An,k = A1 A2 . . . Ak Ak+1 . . . An +. . .+A1 A2 . . . An−k An−(k−1) . . . An
XA== HX?_RYH FRUFVA [>VSAVY XA== €^=SA=F ERUH HQQ C _R?FVS G=UE=;
<[F EVZVSAVF[[E[[A FQ@AFQ@QQ?QQ kERUS[U XTR?
= C p q , GX~X
P (k) = P (A ) = p q
+ ··· + p q
k
n
n
n
n,k
k n−k
|
k n−k
{z
k
Cn
}
k k n−k
n
Pn (k) = Cnk pk q n−k , (k = 0, n)
£¡;ž¥
£¡;ž¥ HQZ¶·QS <`?EXYYRUE HQZ¶·Q SVEV;
#”Ôšš$¸ A, B FQ·? HQSYQST >QQ@ F=€} HQSYQFQQ? GQY}VV; ¼V?V^
>QQ@\S F=€F=A @[YAVV?VV 7 XA== GXX^=Y A HQSYQST FQ}RFQQ?B ¡ XA== F=€F=A
¹œ—Šµ››”™Š ºŒœ‰
¦Æ
XA== @[YA GXX^=Y B HQSYQST FQ}RFQQ? HQFR?TVV; 3YW HQSYQSTRUE FQ}RF
GQYQZ} RY[[ ^V&
PQQ@\S R}RY E]F]YA A=FRE A=^H=E F=€} GXU HXY <`?EXYYRUE @F`Z GQYEQ;
M[YAVV?VV GXXF [>VSAYRUS C SV^VY2 P (C) = P (C) = 1 (p = q = 1 ). A HQSYQSTRUE
2 2 ¡;ž
=
=
=
FQ}RF Z S AY Y £ ¥ HQZ¶·Q ·@QQ? P (2) = C p q = 3 1 · 1 = 3 ,
2
2
8
1
1
1
== =
·
= ,
B HQSYQSTRUE FQ}RF Z S AY Y 2 P (3) = C p q = 4
= = = 2= 2= =; 4
P (2) > P (3) XTR? A HQSYQSTRUE FQ}RF Z S AY Y A ^XX G UE
#”Ôšš$¸¦ wVS FQEQSRUE HX?_ =FRYS==E VE`?SRUE >=?XXY=YH ]S]SA½
@]E EQ?ZQQ@ A=^=FS[U G=UF Z=S=AY=Y p = 0.75 ; ÖU?\E ® FQEQSRUE A]?]^H EW
=FRYS==E VE`?SRUE >=?XXY=YH EQ?ZQQ@QQ A=^=FS[U G=UF [>VSAYRUE Z=S=Ab
Y=Y\S QY;
® FQEQS HXH=ZA =FRYS==E VE`?SR FV^RUE >=?XXY=F [>VSAYRUE Z=S=AY=Y
HQSHZQY p = 0.75 XTR? FQEQS HXH=ZA VE`?SR RY[[ >=?XXY=F [>VSAYRUE Z=b
S=AY=Y q = 1 − 0.75 = 0.25 ; LUZA @QER?F@QE Z=S=AY=Y <`?EXYYRUE HQZ¶·Q
·@QQ?
2
2 2 1
3
3
3
4
3
1
3 3 1
4
4
P6 (4) = C64 p4 q 2 =
6·5
(0.75)4 (0.25)2 = 0, 3.
1·2
q tu v u q t1 1 v ''
5
2 `
HQZ¶·QEA n½RUS GVFVYGVY P (k) EW k½==@ F=Z==?@=E ÅXEa
P (k) = C p q
GQYEQ; ÖAQQ k½RUE €Z=? XHS=EA P (k) ÅXEa Z=a@RZXZ XHS== =^=F ^V&
¢]?]]? FVYGVYB P (0), P (1), . . . , P (n) HQQEXXA\E =YW EW F=ZSRUE RF ^V&
SV@VE =@XXA=Y H=^W€;
KVS^VY np−A QU?FQE G=UF k HQQ EW F=ZSRUE RF Z=S=AY=YH=U SV} F=?XXY}
GQYEQ; KQAQ?FQU FVYGVYB ¼V?V^ np − q G[FVY HQQ GR_ GQY
£¡;7¥
np − q < k < np + p
E]FYRUS F=ES=F k HQQ F=ZSRUE RF Z=S=AY=YH=U HQQ G=UE=;
=
¼V?V^ ÅXEa
np − q EW G[FVY HQQ GQY k = np − q B k = np + p SV@VE FQ·? XHS EA
F=ZSRUE RF XHS== =^E=;
P (k)
#”Ôšš$¸° 3}RYTRE R}RY H]?YRUE ž7 Z=_REA [UYTRYEV; O=_RE ž
=SRUE AQHQ? =}RYTE\ [UYTRYSVV _==?A=F Z=S=AY=Y 1 GQY ž =SRUE AQHQ?
=}RYTE==@ [UYTRYSVV _==?A=F Z=_RE\ F=ZSRUE RF Z=S3=AY=YH=U HQQ G= F=Zb
SRUE RF Z=S=AY=Y\S QY;
O=E=U HQFRQYAQYA n = 12, p = 1 XTR? np − q = 12· 1 − 2 = 3 1 G[FVY GR_;
LZUA £¡;7¥ ·@QQ? F=ZSRUE RF Z=S=3AY=YH=U HQQ k = 34 G=3UE=; 3<`?EXYYRUE
n
k k n−k
n
n
n
n
n
n
0
0
0
01
02
n
0
¸° ѵ•Ù—$a•Ó›•ºÃŠ ›’˜•›Ä© ”Š–œŽ—•› –’‰³´’
¦ª
HQZ¶·Q ·@QQ?
P12 (4) =
4
C12
·p4 ·q 8
4 8
1
2
= 495·
·
= 0, 238.
3
3
5 1 H / Ib . 1 r 1 ' & 1 c u % t 1 ' q \ ] '
<`?EXYYRUE HQZ¶·QS FV?VSYVF €^=A n G= k½RUE F[?VYVVHVU RF XHS=EA P (k)
Z=S=AY=Y\S QYQF _==?AY=S= QYQEH== S=?A=S; KXF=UYG=YB EVS R}RY A`H=Y
[UYA^V?YVAVS [UYA^V?RUE G[HVVSAVF[[E SQYQSAQY G=UF Z=S=AY=Y 0, 02 ; ­88
GVYVE G[HVVSAVF[[ERU AQHQ? ž8 SQYQSAQY G[HVVSAVF[[E G=UF Z=S=AY=Y\S QY
SV@VE GQAYQSQEA XS Z=S=AY=Y P (10) = C (0.02) (0.98) SV} QYAQF G=
= =; %==_RYG=YB 500 G[HVVSAVF[[ERU AQHQ?
C ½HQQS GQAQFQA H[^VSHVU G UE
G=UF SQYQSAY\E HQQ 10½==@ 20½\E AQHQ? G=UF [>VSAYRUE Z=S=AY=Y\S QY¶·
SV^VY P (10 ≤ k ≤ 20) = P (10) + P (11) + · · · + P (20) SV} QYAQEQ; ¢ZE]F
GQAYQSQQ@ FVA A=FRE H]^]SHVU G=UE=; LUZ XT?==@ HX?_RYH\E HQQ n → ∞ [`A
== =
=
= = = == = ==
=
P (k) Z S AY Y\S F€YG ? G]S]]A G S YA H US ? QU?QYQQ GQAQF ?S\S
=ESYRUE Z=H`Z=HRaT OX=^?B Å?=E\E Z=H`Z=HRaT ¾=Y=@ E=? S=?S=}VV;
= ==
= =;
n½F[?VYVVHVU RF [`A A ? F HQZ¶·Q F[TREHVU G UE
1
£¡; ¥
P (k) ≈ √
·ϕ(x)
n
10
500
500
10
500
500
500
10
490
500
n
n
npq
Ÿ[EA 2 x = k√− np , ϕ(x) = √1 ·e
2π
O=
£¡; ¥ HQZ¶·Q EWnpq<`?EXYYRUE HQZ¶·QE\
n → ∞ [`RUE [EVYSVV G]S]]A X ^?½
HQZ¶·Q SV} EV?YVSAVEV;
¾=Y=@\E YQa=YW
=
=
= =;
ϕ(x) ÅXEaRUE ?SXZ`EHRUE V`?VS XHSXXA\E H GYR\S >QFRQ@QE G UA S ϕ(x)
=
ÅXEa x > 0 [`A ZQEQHQE GXX? F G]S]]A ϕ(−x) = ϕ(x) GX~X HVS_ ÅXEa XTR?
@]?]S XHSXXA\E H=GYR EW AVV?F H=GYRH=U =ARY G=UE=; ϕ(5) = 0, 0000015 G=
; LUZA ϕ(x) ÅXEaRUE H=GYR x > 5 [`A V@
x > 5 [`A ϕ(x) ≈ 0 SV} HQQEQ
>QFRQSAQEQ;£K=GYR ڞ½RUS [>¥ ÖAQQ AVV? HQZ¶·QY@QE GQAYQS\S GQA¶·;
¡;
n = 500, p = 0.02, np = 500·0.02 = 10, npq = 10·0.98 = 9, 8 [`A £ ¥ HQZ¶·Q
·@QQ?
−x2 /2
1
1
10 − 10
√
P500 (10) = √ ·ϕ
= √ ·ϕ(0)
9.8
9.8
9.8
ϕ(0) = 0.397
P500 (10) ≈ 0, 124.
n
Pn (k)
K=GYR==@ F=?^=Y
XTR?
P=?RZ HQFRQYAQYA ½RUE F[?VYVVHVU RF XHS=EA
Z=S=AY=Y\S GR_B
=
=
=
==
=
=
=
=
; KVS^VY GRAERU
P (k ≤ k ≤ k ) Z S AY Y\S @QER?FQF _ ?AY S S ?A S
@QER?F@QE [>VSAVY k ½VV@ ]]ES[UB k ½QQ@ QYQES[U XA== €^=SA=F [>VSAYRUE
Z=S=AY=Y OX=^?½¾=Y=@\E REH`S?=Y HQZ¶·QSQQ? [EVYVSAVEV;
=
n½RUE F[?VYVVHVU RF XHS EA
k − np
k − np
£¡;¡¥
P (k ≤ k ≤ k ) ≈ Φ √
−Φ √
n
1
2
1
2
2
n
1
2
1
npq
npq
¹œ—Šµ››”™Š ºŒœ‰
¦È
HQZ¶·Q F[TREHVU G=UE=; Ÿ[EA 2
Φ(x) =
Zx
1
ϕ(t)dt = √
2π
Zx
2 /2
e−t
dt −
¾=Y=@\E ÅXEa;
= == = =
= = =;
Φ(x) ÅXEa A ? F T E ?XXA\S F ES E
 Φ(x)½@QEASQU ÅXEa2 Φ(−x) = −Φ(x)
¦ x EW [0; ∞[ >=^@=?H ]@]F]A Φ(x) ÅXEa [0; 1 [ >=^@=?H ]@E]; ÌXEaRUE
2
S?=ÅRaRUS PX?=S ¡;ž½A [>[[YV^;
0
0
y
y = Φ(x)
1
2
x
0
− 21
PX?=S ¡;ž2
° x = 5 [`A Φ(5) EW 1 ½VV@ 3 · 10 ½VV? €YS=SA=E=; ¢]?]]? FVYGVYB x > 5 [`A
2
1 LUZA
=
= =
= =;
Φ(x) ≈ .
Φ(x) ÅXEaRUE H GYR [0; 4] > ^@ ?H >QFRQSA@QE G UE
£¼=^@?=2YH\E H=GYR Ú7½\S [>¥; ¢ZE]F GQAYQS\E P (10 ≤ k ≤ 20) Z=S=AY=Y\S
QY¶·; £¡;¡¥ HQZ¶·Q ·@QQ?
−8
Pn (10 ≤ k ≤ 20) ≈ Φ
20 − 10
√
9.8
−Φ
10 − 10
√
9.8
≈ Φ(3.19) − Φ(0).
ÅXEaRUE XHS\S H=GYR==@ F=?^=Y Φ(3.19) = 0.49929,
; LUZA P (10 ≤ k ≤ 20) ≈ Φ(3.19) − Φ(0) = 0, 49929.
Á•Š•‰Â2 £¡; ¥ G= £¡;¡¥ HQZ¶·QE\ E=?RU^TY=Y npq [?}^V? ]@]F HXH=Z @=U}?=F
XT?==@ OX=^?½¾=Y=@\E YQa=YW G= REH`S?=Y HQZ¶·QS RFV^TYVE npq ≥ 10
HQFRQYAQYA FV?VSYVEV ; p, q Z=S=AY=Y\E =YW EVS EW ”0”½A FVARU TREVVE
QU?FQE G=U^=Y n½RUS H]ARU TREVVE RFVV? @QESQE =^=F=A F[?EV; LUZA p →
¡; = ¡;¡
;
0 (V@^VY q → 0) [`A £ ¥ G £ ¥ HQZ¶·QS [Y FV?VSYVEV
#”Ôšš$¸¸ P=^QA\E [UYA^V?YV@VE SV?YRUE _RY SQYQSAQY G=UF Z=S=Ab
Y=Y 0.02 ; ž888 SV?YRUE _RYRUS _=YS=F==? @QESQE =^TVV; K[[^V? AVF SQYQSAQY
_RYERU F=?W=ESXU A=^H=Z} EW Z=S=AY=Y==@== 0.01½VV@ G=S==? €YS=SA=F Z=b
S=AY=Y\S [EVY;
K[[^V? AVF SQYQSAQY _RYERU HQQS k SV^VY k − 0.02 < 0.01 G=UF Z=S=Ab
Y=Y\S [EVYEV SV@VE [S; DEV HVEVHSVY GR_ EW1000
11 ≤ k ≤ 29 HVEVHSVY GR_HVU
Φ(x)
Φ(0) = 0
n
¸¸ ص•ºº’ŠÃ –’‰³´’
¦Ê
HVE[[ T=E=?H=U; LUZA P
k
− 0.02 < 0.01 = P1000 (11 ≤ k ≤ 29).
1000
npq = 1000·0.02·0.98 = 19.6 > 10
XTR? ¾=Y=@\E REH`S?=Y HQZ¶·QS FV?VSYV`;
O=E=U HQFRQYAQYA
11 − 1000 · 0.02
29 − 20
x1 = √
≈ −2.03, x2 =
≈ 2, 03. Φ(−2.03) ≈ −0.4788,
4.43
100 · 0.02 · 0.98
Φ(2.03) ≈ 0.4788
P100 (11 ≤ k ≤ 29) ≈ 0.4788 + 0.4788 = 0, 9576.
LUZA £¡;¡¥ HQZ¶·Q ·@QQ?
5 5 d / ' r ' q \ ] '
¢ZE]F >[UYRUE @=E=Z}RA AX?A@=E ·@QQ?B HXF=UE HX?_RYH G[?A A [>VSAYRUE
€^=SA=F Z=S=AY=Y p F=?W=ES[U G=S= [`A £RUZ [>VSAYRUS FQ^Q? [>VSAVY SVEV;¥
OX=^?½¾=Y=@\E HQZ¶·QE\ E=?RU^TY=Y GXX?E=; LUZ XT?==@ VEV HQFRQYAQYA
== = = = = = ¿ =
==
A [>VSAVY k XA €^ SA F Z S AY Y X @@QE\ F€>S ?\E HQZ¶·QSQQ? [EVYVSAV½
;
EV
<`?EXYYRUE @F`ZRUE E]F]YA n → ∞ B p → 0 G]S]]A λ = np FVZ}RSAVF[[E
HQSHZQY £λ = const¥ GQY A=?==F HQZ¶·Q F[TREHVU G=UE=;
Pn (k) ≈
£¡;­¥
λk −λ
e
k!
£¡;­¥ HQZ¶·QE\ E=?RU^TY=Y P (k)− λ e < np HVEVHSVY GR_VV? [EVYVSAVEV;
k! = = =
¿X=@@QE\ HQZ¶·Q £¡;­¥½\E S=UF=Z_RSH
U T E ? EW2 HQAQ?FQU HQQE\ HX?_RY½
H\E Z=S=AY=Y\S QYQF\E HXYA n G= p HQQS ZVAVF _==?AY=S=S[U G]S]]A F=?RE
ERUH HX?_RYH=EA A [>VSAYRUE €^=SA=F AXEA=} HQQ GQYQF λ = n · p HQQS
ZVAVFVA F=ES=YHH=U G=UA=SH Q?_REQ;
¿X=@@QE\ HQZ¶·QE\ P (k) = λ · e ÅXEaRUE H=GYR\S k G= λ½RUE XHSXXb
k!
A=A >QFRQ@QE G=UA=S; £¼=^@?=YH\E
H=GYR Ú ½\S [> ;¥
#”Ôšš$¸» PXX?@=E SX?RY=EA HQAQ?FQU HQQE\ [>VZ FRU} FQYW@E\
A=?== HVE[[ FV@S[[AVA FV?TR} [>VZHVU H=YF FRUEV; wRUH H=YFE\ HQQ N B
G[F [>VZERU HQQ n GQY H==ZS==? @QESQE =^@=E EVS H=YF=EA €S k [>VZ G=UF
Z=S=AY=Y\S [EVY;
=
= = = ==
n _R?FVS [>VZ SX?RY EA FQYRF\SB [>VZ HX@ G[?RUS SX?RY EA F €F > Z ?
HX?_RYH\S n XA== FRU@VE SV} [>V} GQYEQ; <RAERU @QESQE =^@=E H=YF=EA
[>VZ Q?@QE G=UF [>VSAYRUS A SV`; <[F H=YFE\ HQQ N B [>ZVV SX?RY=EA FRUSVVA
FQYW} G=US== XTR? [>VZ G[? EVSVE R}RY Z=S=AY=YH=US==? =YW T H=YF=EA Q?}
GQYEQ; LUZA A [>VSAYRUE Z=S=AY=Y p = 1 B ŸUYA^V?YVF H=YFE\ HQQ N QYQE
XT?==@ p½Z=S=AY=Y G=S= HQQ G=UE=; ¼=?REN λ = np EW n ½HVU HVE[[ G]S]]A ž
N
k
−λ
k
n
−λ
2
¹œ—Šµ››”™Š ºŒœ‰
°Ð
H=YF=EA Q?QF [>VZERU AXEA=} HQQ GQYEQ; MQER?F@QE Z=S=AY=Y P (k) ≈ λ e
HQZ¶·QSQQ? GQAQSAQF G]S]]A λ = 8 HQFRQYAQYA >=?RZ XHSXXA\S QYGQY k!
k
−λ
n
Pn (0) ≈ 0.000 Pn (1) ≈ 0.003 Pn (2) ≈ 0.011
Pn (3) ≈ 0.029 Pn (4) ≈ 0.057 Pn (5) ≈ 0.092
Pn (6) ≈ 0.012 Pn (7) ≈ 0.139 Pn (8) ≈ 0.139
k > 14
Pn (k) < 0.001
N
0.31%
SVF ZVHTRYVE GQAQFQA
[`A
GQYEQ;
DEAVV@ [>^VY ERUH H=YFE\ AXEA}==?
EW ž [>VZB 1.1% EW 7 [>VZB 2.9%
EW [>VZ SVF ZVH HX@ HX@ =SXXY} G=UE=;
#”Ôšš$¸Æ JH=@E\ @H=E ¡88 FV?VSYVSTRA [UYTRYEV; ¼V?VSYVST G[? ž
=SRUE HX?_RA @H=E X?XX XH=@A=F Z=S=AY=Y 0.01 ; =SRUE AQHQ? ­ FV?VSYVST
XS @H=E X?XX XH=@A=F Z=S=AY=Y\S QY;
O=E=U HQFRQYAQYA p = 0.01 n = 400 XTR? ¿X=@@QE\ HQZ¶·QS FV?VSYV}
GQYEQ; λ = 400 · 0.01 = 4. £¡;­¥ HQZ¶·Q ·@QQ? P (5) ≈ 4 e ≈ 0.156293
V?V^ ž =SRUE AQHQ? ¡½]]@ QYQES[U FV?VSYVST XH=@A=F Z5!=S=AY=Y\S QY¶·
¼SV^VY
P (0 ≤ k ≤ 4) = P (0) + P (1) + P (2) + P (3) + P (4) ≈
;
0.018316 + 0.073263 + 0.146525 + 0.195367 + 0.195367 = 0.628838 GQYEQ
5
−4
400
400
400
400
400
400
400
ö ÷ ø e
f ü ÷ ý Sø g øS÷÷ A
÷÷ý üS S‡‡ h
Kþ B q t v - q u t - KX?_RYH\E [? A[EA €Z=? EVS HQQE XHS= =^=F GQYQ^T HV?F[[ [? A[ES X?WATRb
Y=E FVYVF GQYQZ}S[U FVZ}RSAVF[[ERUS @=E=Z@=?S[U SVEV; M=E=Z@=?S[U FVZ½
}RSAVF[[ERUS AR@a?`H G= H=@?=YHS[U SV} FQ·? =ESRYE=; M=E=Z@=?S[U FVZ½
}RSAVF[[ERU =^T GQYQF XHSXXA\S H[[ERU GQYQZ}RH XHSXXA SVEV;
V?V^ @=E=Z@=?S[U FVZ}RSAVF[[ERU GQYQZ}RH XHSXXA EW H]S@S]Y]SB V@^VY
¼
H]S@S]YS[U HQQE A=?==YY\E FVYGV?VV? GRTRSAV} G=U^=Y AR@a?`H @=E=Z@=?S[U
FVZ}RSAVF[[E SVEV; ¼V?V^ @=E=Z@=?S[U FVZ}RSAVF[[ERU GQYQZ}RH XHSXXA
EW HQQE HVEFYVSRUE €Z=? EVS >=^@?==@ G[?AV} G=U^=Y H[[ERUS H=@?=YHS[U
SVEV;
¤R@a?`H @=E=Z@=?S[U FVZ}RSAVF[[E ξ½RUE GQYQZ}RH XHSXXA x , x , . . . , x
=
G VASVV? XHSXXA\S F[YVVE =^=F Z=S=AY=YXXA p , p , . . . , p ½RUE FQQ?QEAQF FQYb
GQQS XS FVZ}RSAVF[[ERU H=?F=YH\E FXXYW SVEV;
KVS^VY HQAQ?FQUYQYH ·@QQ? P (ξ = x ) = p , (i = 1, n) GQYEQ; M=E=Z@=?S[U
FVZ}RSAVF[[ERU H=?F=YH\E FXXYW EW H=GYRB S?=ÅRaB HQZ¶·QSQQ? ]S]SA]F
G= AR@a?`H @=E=Z@=?S[U FVZ}RSAVF[[E SQY H]Y]^ H=GYR==? ]S]SA]E];
1
1
i
ξ:
p:
x1
p1
x2
p2
2
2
n
n
i
...
...
xn
pn
...
...
M=E=Z@=?S[U FVZ}RSAVF[[E−ξ, HX?_RYH\E [? A[EA GQYQZ}RH XHS\EF== =YW
EVSRUS =^=F [>VSAYRUE Z=S=AY=Y EW P (ξ = x )+P (ξ = x )+. . .+P (ξ = x ) GQYQF
G]S]]A VEV EW >=UY_S[U [>VSAVY HXY P P (ξ = x ) = P p = 1 G=UE=;
n
1
2
n
i
i=1
n
i
i=1
Á•Š•‰º•—Ž™ Œš‰Â”ŽšŒŠ© –Š”™ –•—Œ•›–Ê Œµµ›Ä
1 r / / &s uv J °¦
K
2
KQQE HVEFYVS AVV? F (x) SV@VE EVS FX^W@=STRUE ÅXEa HQAQ?FQUYQSA@QE SV}
[>W`; ∀x−RUE FX^WA âξ EW x½VV@ G=S= XHS== =^=F Z=S=AY=Y â½\S RYV?FRUYb
@VE ÅXEaRUS @=E=Z@=?S[U FVZ}RSAVF[[ERU H=?F=YH\E ÅXEa SVEV; ¢]?]]?
FVYGVY2
£­;ž¥
F (x) = P (ξ < x)
== =
= =
;
F (x) ÅXEaRUS ξ ½@ E Z@ ?S[U FVZ}RSAVF[[ERU H ?F YH\E ÅXEa SVEV
K=?F=YH\E REH`S?=Y ÅXEaB REH`S?=Y FXXYW SVF EW GRU;
K=?F=YH\E ÅXEaRUE T=E=?XXA\S =^T [>W`;
 F (x) G[F HQQE HVEFYVS AVV? HQAQ?FQUYQSAQF G= 0 ≤ F (x) ≤ 1 ;
¦ F (x) [Y GXX?E=B ]]?]]? FVYGVY2 x ≤ x [`A F (x ) ≤ F (x ) ;
° ¼V?V^ ξ½RUE GQYQZ}RH XHSXXA ] − ∞, +∞[ >=^@=?H G=U^=Y
;
F (−∞) = lim F (x) = 0, F (+∞) = lim F (x) = 1 GR`YEV
¸ P (x ≤ ξ ≤ x ) = F (x ) − F (x ).
¤R@a?`H @=E=Z@=?S[U FVZ}RSAVF[[ERU H=?F=YH\E ÅXEa EW A=?==F HQZ¶·Qb
SQQ? RYV?FRUYVSAVEV ;
1
x→−∞
1
2
1
2
x→+∞
2
2
1
F (x) =
X
P (ξ = xi )
Ÿ[EAB x < x G=UF G[F i½AXS==?XXA\E FX^WA ERUYGV? =^=SA=E=;
V?V^ AR@a?`H G= H=@?=YHS[U @=E=Z@=?S[U FVZ}RSAVF[[ERU H=?F=YH\E ÅXEa½
¼[[ARUS
S?=ÅRa==? A[?@VYGVY 2
xi ≤x
i
F (x)
F (x)
1
x0 x1 x2 .... xn
1
x
x
a
b
PX?=S ­;72
PX?=S ­;ž2
K=@?=YHS[U @=E=Z@=?S[U FVZ}RSAVF[[ERU H=?F=YH\E ÅXEaRUS @]?]S GR_
XHS=H=UB FV@VS FV@VS H=@?=YHS[U ÅXEa f (x)½RUE HX@Y=Z}H=U A=?==F G=UAb
Y==? RYV?FRUY} GQYEQ;
Z
£­;7¥
F (x) =
f (t)dt
x
= =
= =
=
f (x)½RUS H ?F YH\E E€SH\E ÅXEa GX~X H ?F YH\E ARÅÅ`?`ER Y ÅXEa
SVEV; w€SH\E ÅXEaRUE HQAQ?FQUYQYHQQ@ A=?==F T=E=?XXA Z]?A]E S=?E=;
 f (x) ≥ 0.
−∞
»¦ ɕ—Œ•›–Ê iµŠ˜Õ –Š”™ ±•Š•—
¦ P (x
Rx2
≤ ξ ≤ x2 ) = f (x)dx.
x1
+∞
R
f (x)dx = 1,
ξ ∈ [a, b]
°°
1
°
GQY R f (x)dx = 1
¸ F (x) = f (x).
w€SH\E ÅXEaRUS S?=ÅRa==? A[?@VYGVY2
b
−∞
0
a
f (x)
P (x1 ≤ ξ ≤ x2 )
x
0
x2
x1
PX?=S ­; 2
M=E=Z@=?S[U FVZ}RSAVF[[E E€SH\E ÅXEaVV?VV ]S]SA@]E [`A H[[ERU €Z=?
EVS >=^@?==@ XHS== =^=F Z=S=AY=Y GQYQE H=?F=YH\E ÅXEaRUS G=USXXY} GQYEQ;
#”Ôšš$» M=E=Z@=?S[U FVZ}RSAVF[[E E€SH\E ÅXEaVV?VV ]S]SA]^2

x≤0

 0x
0<x≤4
f (x) =

 8
0
x>4
ÅXEaRUS QY
ÅXEaRUE S?=ÅRa G=USXXY;
1.) P (−1 ≤ ξ ≤ 2) =?
2.) F (x)
3.) F (x), f (x)
w€SH\E ÅXEaRUE T=E=? ·@QQ?
= P (−1 ≤ ξ ≤ 2) = R f (x)dx = R f (x)dx + R f (x)dx =
2
0
2
−1
−1
0
R2 x
x2 2 1
0dx +
= .
dx =
16 0 4
0 8
−1
R0
¦= F (x)½ÅXEaRUS A=?==F ZX} HX@ G[? AVV? G=USXXYG=Y2
R
f (x)dx = 0,
x ≤ 0 [`A
F (x) = P (ξ ≤ x) =
0<x≤4
[`A
x
[`A
ÖAQQ VZFVHSV} GRT^VY
x>4
−∞
;
Rx x
x2
dx =
16
0 8
0
−∞
−∞
4
x
4
0
x
R x
R
R
R
R
0dx + f (x)dx + 0dx =
f (x)dx =
F (x) =
dx = 1.
0 8
4
0
−∞
−∞
F (x)
F (x) =
Rx
f (x)dx =
R0
0dx +
Rx
f (x)dx =
A=?==F FVYGV?HVU GQYEQ;

0
x≤0

 2
x
F (x) =
0<x≤4

 16
1
x>4
°= K=?F=YH\E GQYQE E€SH\E ÅXEaRUE S?=ÅRaRUS PX?=S ­;¡B­;­½A [>[[YV^;
Á•Š•‰º•—Ž™ Œš‰Â”ŽšŒŠ© –Š”™ –•—Œ•›–Ê Œµµ›Ä
°¸
F (x)
f (x)
1
1
0.5
0
4
x
0
4
x
PX?=S ­;¡2
PX?=S ­;­2
K1 j u q 1 / / K[SVVZVY FV?VSYVSAVF >=?RZ H=?F=YH\E FXXYWH=U H=ERY¶€;
ž; ¹”Š’‰ –•—Œ•›– KX?_RYH G[?A A [>VSAVY HQSHZQY p Z=S=AY=YH=U
RYV?AVS SV} [>W`; KVS^VY n A=?==Y@=E HX?_RYH=EA A [>VSAYRUE €^=SA=F HQQ
== =
k ½EW 0, 1, 2 . . . , n SV@VE GQYQZ}RH XHSXXA G[FRU AR@a?`H @ E Z@ ?S[U FVZ}RS½
AVF[[E ξ GQYQF G]S]]A H=?F=YH\E FXXYW EW A=?==F FVYGV?HVU G=UE=;
P (ξ = k) = C p q , [[EA q = 1 − p, k = 0, 1 . . . , n.
; ؕµºº’ŠÃ –•—Œ•›– KX?_RYH\E HQQS F€>S==?S[U RFV@SVFVA np ≈ npq
7GQY
GREQZ H=?F=YH EW ¿X=@@QE\ H=?F=YH GQY} FX^R?=F G]S]]A A=?==F FXXb
=
YR ? RYV?FRUYVSAVEV;
k k n−k
n
[[EA λ = n · p, k = 0, 1, 2 . . .
¢]?]]? FVYGVYB HX?_RYH\E HQQS F[?VYVVHVU RFV@SVFVA A [>VSAYRUE €^=SA=F
Z=S=AY=Y p G=S=@=F GQYQ^T [>VSAVY €^=SA=F AXEA=} HQQ λ½EW HQSHZQY G=UA=S
G=UE=; DEAVV@ [>VFVA ¿X=@@QE\ H=?F=YH λ SV@VE S=EF=E =?=Z`H?VV? G[?VE
HQAQ?FQUYQSAQ} G=UE=; ¿X=@@QE\ H=?F=YH ERUHRUE [UYTRYSVVERU QEQY GQYQE
[UYA^V?YVYRUE H`FEQYQSRUE H=@?=YHS[U =}RYY=S==S >=S^=?TY=F=A ]?S]E FV½
?VSYVSAVEV ;
#”Ôšš$»¦ DV?ZVYRUE [UYA^V?H ž888 VV?[[YVV@ ž ZREXH=A XH=@ H=@=?@=E
HQQSQQ? >QFRQSA@QE
A=P?==F F[@EVSHRUS =^T [>W`;
P
N
m
8 ®88
8m ®8®N
Ÿ[EA N ½H=@?=YH\E HQQ
ž 78 78 8
H=@?=YHH=U VV?[[YRUE HQQ
½
m
−
N
P
ž¡8 «®
= =
7 «8
m − m ½VV?[[YRUE ERUH H @? YH\E HQQ
P
=
=
=
ž8
8 ž
N ½QEQY\E FX^WA N H @? YHH U
¡ 8
8
G=U} GQYQF VV?[[YRUE HQQ
7
P ž888 ¡¬8
ž888
=
=
=
[@EVSHVV@ }RSY F A ž ZREXH=A H=@?=F VV?[[YRUE HQQ EW @=E=Z@=?S[U FVZ½
¼}RSAVF[[E
G]S]]A VV?[[YRUE HQQ RFV@VFVA H=@?=YH\E HQQ G=S=@=} G=US==
EW F=?=SA=} G=UE=; LUZA H=@?=YH\E HQQS ¿X=@@QE\ H=?F=YHH=U SV} [>VF
P (ξ = k) =
i
i
i
e−λ · λk
k!
i
i
i
i
i
i
i
i
»° k•—”‰ –•—Œ•›–µµ
°»
[EAV@HVU ; ¯VFAVV [[ERUS A=?==F G=UAY==? _=YS=€; ž888 VV?[[YRUE AXEA=}
H=@?=YH\E HQQ λ½S QYGQY λ = 490 = 0, 49.
1000 [>VSAYRUE Z=S=AY=Y EW
8 H=@?=YHH=U G=UF £QSH H=@?=YHS[U¥
K
· 0.49
e
= = = =
= 0, 606. VS^VY 0½H @? YHH U G UF ERUH VV?[[YRUE
P (0) =
HQQ EW 1000 · P 0! (0) = 606 ; O]E ž½H=@?=YHH=U G=UF [>VSAYRUE Z=S=AY=b
Y\S QYGQY P (1) = e · 0.49 = 0, 303. :S =ARYF=E==? QEQY\E FX^WA
1!
ž½H=@?=YHH=U G=UF ERUH VV?[[YRUE
HQQ EW 1000 · P (1) = 303 SV} QYAQEQ;
wRUH N H=@?=YHH=U VV?[[YRUE HQQ G= QEQY\E
FX^WA H=@?=YHH=U G=U} GQYQF
=
=
=
==
=
VV?[[YRUE HQQS H @? YH\E HQQEQQ@ F Z ?XXY E S?=ÅRa==? A[?@VY} F=?WXX½
Y=E [>[[YGVY2
−0.49
0
1000
1000
−0.49
1000
1000
i
y(mi )
500
300
100
0
1
2
3
4
x(Ni )
PX?=S ­;®2
LUZA VV?[[YRUE XH=@ H=@?=F HQQS ¿X=@@QE\ H=?F=YHH=U SV} [>V} GQYEQ;
#”Ôšš$»° ¼V@VS FXS===E\ HX?_RA VV?[[YRUE XH=@ H=@?=F HQQ EW
¿X=@@QE\ FXXYR=? H=?F=E=; wVS =}RYTRE ž888 VV?[[YVEA [UYTRYAVS G]S]]A
ž ZREXH=A AXEA}==? ¡ H=@?=YH GQYAQS; KVS^VY ž ZREXH=EA ­ XA== XH=@
H=@?=F [>VSAYRUE Z=S=AY=Y\S QY;
w]F]Y ·@QQ? n = 1000, p = 0.004, k = 5 λ = n · p = 4. LUZA
45 · e−4
P1000 (5) =
= 0, 156.
5!
¤VV?F }R_VVEA ¿X=@@QE\ H=?F=YH\S ]]? FVYGV?VV? =_RSY=} GQYEQ;
KXF=UYG=YB =}RYTRE ž888 VV?[[YVEA [UYTRYAVS G= ž ZREXH=EA AXEA}==? 7½
H=@?=YH\S >=@A=S GQY ž ZREXH=EA H=@?=YH\S =?RYS=F Z=S=AY=Y\S QY¶·;
Ÿ[EA λ = 2, k = 3 ; LUZA P (3) = 2 · e = 0.18
wRUHRUE [UYTRYSVVERU QEQYA =}RSY3!=YH\E HQQ HQSHZQY G=UA=S G]S]]A t
FXS===E\ HX?_ k½XA== [UYTRYSVV FRUAVS SV} [>EV; ¼V?V^ EVS} FXS===EA
FRUSAVF AXEA=} [UYTRYSVVERU HQQ m½ZVAVSAV} G=UE= SV} [>^VY λ = m · t
GQYQF G= t½FXS===E\ HX?_ k XA== [UYTRYSVV FRUSAVF Z=S=AY=Y EW ¿X=@@QE\
FXXYR=? RYV?FRUYVSAVEV;
1
−2
1000
Pk (t) =
(m · t)k · e−mt
k!
Á•Š•‰º•—Ž™ Œš‰Â”ŽšŒŠ© –Š”™ –•—Œ•›–Ê Œµµ›Ä
°Æ
V?V^ k = 0 GQY VEV EW HQEQS H]F]]?]Z} t½FXS===E\ HX?_ H=@?=YHS[U
¼=}RYY
=F Z=S=AY=Y GQYEQ;
P0 (t) = e−mt
O]E H=@?=YHS[U =}RYY=F FXS===E\ H=?F=YH\E ÅXEaRUS A=?==F G=UAY==?
QY} GQYEQ;
F (t) = P (T ≤ t)
Ÿ[ERU HXYA V@?VS [>VSAYRUE Z=S=AY=Y\S QY¶·2
P (T ≥ t) = 1 − P (T ≤ t) = 1 − F (t).
SVASRUS =EF==?^=Y e = 1 − F (t) GQYQF G= VEAVV@
; Λ–ŽšŽ± –•—Œ•›– K=?F=YH\E ÅXEa EW
1 − e , t ≥ 0, [`A
F (x) =
0
t < 0 [`A
HQZ¶·QSQQ? HQAQ?FQUYQSAQF @=E=Z@=?S[U FVZ}RSAVF[[ERUS RYHSVST H=?F=YHb
H=U SVEV; LYHSVST H=?F=YH\E E€SH\E ÅXEa
m·e
t ≥ 0, [`A
f (x) =
0
t < 0 [`A
=
=
= ==
(m > 0) FVYGV?HVU GQYQF G F (t), f (t) ÅXEa[[ARUE S? ÅRaRUS A ? F
=
;
>X? SH A[?@YV^
P (T ≥ t) = e−mt
F (x) = 1 − e−mt .
−mt
−mt
−mt
f (t)
F (t)
1
1
t
t
0
0
PX?=S ­;«2
PX?=S ­;92
¡; #”Ž –•—Œ•›– K=@?=YHS[U @=E=Z@=?S[U FVZ}RSAVF[[E ξ½RUE G[F
GQYQZ}RH XHSXXA EW [a, b] FV?TRZ GQYQS;
¼V?V^ ξ½@=E=Z@=?S[U FVZ}RSAVF[[ERU H=?F=YH\E ÅXEa

[`A

 0x − a x < a
a ≤ x ≤ b [`A
F (x) =

b
−
a

[`A
1
x>b
»° k•—”‰ –•—Œ•›–µµ
°ª
FXXYR=? ]S]SA@]E GQY H[[ERUS }RSA H=?F=YHH=U SVEV; K=?F=YH\E ÅXEaRUS
=_RSY=E E€SH\E ÅXEaRUS HQAQ?FQUYGQY2

[`A
0
x<a

 1
f (x) =
a ≤ x ≤ b [`A

 b−a
[`A
0
x>b
=
F (x), f (x) ÅXEa[[ARUE S? ÅRaRUS A[?@VYGVY 2
F (x)
f (x)
1
1
1
b−a
x
a
x
a
b
b
PX?=S ­;¬2
PX?=S ­;ž82
­; ǚٔ™Š <Š’—‰•›Ä
=
=
= –•—Œ•›– ¼V?V^ H @? YHS[U @=E=Z@=?S[U FVZ}RS½
AVF[[ERU E€SH\E ÅXEa
1
2
2
f (x) = √ e−(x−a) /2σ
σ 2π
FVYGV?HVU GQY H[[ERUS FV^RUE H=?F=YHH=U SVEV; f (x)½ÅXEaRUE S?=ÅRa EW
=
==
x = a GQ@QQ _XYXXE\ FX^WA HVS_ FVZHVUB x = a VS AVV? F ZSRUE RF XHS
=^=F G= x → ±∞ [`A f (x) → 0.
f (x)
√1
2π
x
x=a
0
PX?=S ­;žž2
=
=
=
V^RUE H ?F YHH U FVZ}RSAVF[[ERU H=?F=YH\E ÅXEa A=?==F FVYGV?HVU
¼
G=UE=;
F (x) =
Zx
1
f (x)dx = √
σ 2π
Zx
e
−(x−a)2
2σ 2
dx = 0.5 + Φ
x−a
σ
x1 − a
σ
.
M=E=Z@=?S[U FVZ}RSAVF[[E [x , x ] >=^@?==@ XHS== =^=F Z=S=AY=Y\S
1 R
= =
= =
e
dt SV@VE ¾ Y @\E ÅXEa _RSY E QYGQY 2
Φ(x) = √
−∞
x
2π −∞
1
−∞
2
−t2 /2
P (x1 ≤ ξ ≤ x2 ) = F (x2 ) − F (x1 ) = Φ
x2 − a
σ
−Φ
.
°È
Á•Š•‰º•—Ž™ Œš‰Â”ŽšŒŠ© –Š”™ –•—Œ•›–Ê Œµµ›Ä
DEV [? A[ES =_RSY=E âSX?^=E σ½RUE A[?ZRUS â S=?S=€;
P (|ξ − a| < σ) = P (a−σ < ξ < a + σ) = 2Φ(1) = 0.6826
P (|ξ − a| < 2σ) = P (a−2σ < ξ < a + 2σ) = 2Φ(2) = 0.9544
P (|ξ − a| < 3σ) = P (a−3σ < σ < a + 3σ) = 2Φ(3) = 0.9973
DEAVV@ F=?=F=A FV^RUE H=?F=YHH=U @=E=Z@=?S[U FVZ}RSAVF[[E Z=H`Z=HRa
AXEA}==@== F=>=UF F=>=UYH\E =G@QY~H FVZ}RSAVF[[E 3σ½==@ RF G=UF EW
G=?=S GQYQZ}S[U [>VSAVY G=UE=;
f (x)
S = 0.9973
x
a − 3σ
0
a
a + 3σ
PX?=S ­;ž72
#”Ôšš$»¸ wVSVE G[@ EXHSRUE V?T[[ARUE ]EA]? EW a = 170.2 @Z G=
==
= = == =
σ = 5.6 @Z  ? Z`H?[[AHVU FV^RUE H ?F@ E @ E Z@ ?S[U FVZ}RSAVF[[E GQY
ž®®½ ž«7 @Z ]EA]?HVU F[Z[[@ ERUH V?T[[ARUE FVAVE FX^RUS V>YVF ^V&
<QAYQS\E E]F]Y ·@QQ? x = 172, x = 166 HXY
== =
; OX=^?½¾=Y=@\E REH`S?=Y HQZ¶·QS
P (166 < ξ < 172) Z S AY Y\S QYQF ·@HQU
=_RSY=^=Y2
2
1
172 − 170.2
166 − 170.2
−Φ
=
5.6
5.6
≈ 40%
Φ(0.32)−Φ(−0.75) = 0.3999
40%
P (166 < ξ < 172) = Φ
£GX~X
¥ DEAVV@ ž®®½ ž«7 @Z ]EA]?HVU V?T[[A
==
=
=;
EW AXEA} ? ½RUS V>YV} G UE
#”Ôšš$»» wVSVE G[@ EXHSRUE [EAV@HERU VZVSHVUT[[ARUE VV}ERU
HQU?SRUE X?H EW a = 96.8 @Z G= σ = 4.5 @Z =?=Z`H?[[A G[FRU FV^RUE H=?F@=E
@=E=Z@=?S[U FVZ}RSAVF[[E G=U^; ¼V?^VV Q·AY\E [UYA^V? VZVSHVUT[[ARUE
=YWHQ [UYA^V?YVAVS GQY =YWHQ [UYA^V?YVYRUE ?=>Z`?[[ARUE
F=?W==S HQSHQQ;
KXF=UYG=YB ¡7 ?=>Z`?RUE =YWHQS FVAVE FX^W [UYA^V?YVFRUS QYQF\E HXYA
A=?==F Z=S=AY=Y\S QYQF _==?AY=S=H=U;
¡7½?=>Z`?RUE FX^WA W (42) = P (82 < ξ < 86) B
¡¡½?=>Z`?RUE FX^WA W (44) = P (86 < ξ < 90) S;Z;
86 − 96.8
82 − 96.8
−Φ
=
4.5
4.5
Φ(−2.40) − Φ(−3.29)= 0.008 0.8%
90 − 96.8
86 − 96.8
P (86 < ξ < 90) = Φ
−Φ
= 0.057
4.5
4.5
0.57%.
P (82 < ξ < 86) = Φ
GX~X
£GX~X
¥B
<X@=A G[F ?=>Z`?[[ARUE FX^WA AVV?FRUE =ARY==? QY} [? A[ES
»° k•—”‰ –•—Œ•›–µµ
°Ê
A=?==F F[@EVSHVEA F=?XXY=^;
?=>Z`? [x , x ]
¡7
97B9®m
l
¡¡
9®B¬8m
l
¡®
¬8B¬¡m
l
¡9
¬¡B¬9m
l
­8 l¬9Bž87m
­7 lž87Bž8®m
­¡ lž8®Bžž8m
1
2
t1
;7¬
½ ;¡8
½7ž;­ž
½8;®
½8; «7
7
ž;ž®
7;8¡
t2
;¡8
½7ž;­ž
½8;®
½8; «7
7
ž;ž®
7;8¡
7;¬¡
Φ(t2 )−Φ(t1 )
8;889
8;8­«
8;787
8; ¬
8;7«ž
8;ž87
8;8ž¬
[UYA½
^V?YVF
FX^W £%¥
8;9%
­;«%
78;;¬7%
%
«;ž
7ž8; %
ž;¬7%%
ö ÷ ø n
f ü ÷ ý Sø g øS÷÷ ý
÷ @÷ ÷ ø ÷ ÷ M=E=Z@=?S[U FVZ}RSAVF[[ERU H=?F=YH\E FXXYW EW XS FVZ}RSAVF[[ERUS G[½
?VE HQAQ?FQUYQF GQYQ^T ?=aHRaH @=E=Z@=?S[U FVZ}RSAVF[[ERU XHSXXA G=
HVASVV? XHS== F[YVVE =^=F Z=S=AY=Y\S EVS G[?TYVE ZVAVFVA F[EA?VYHVU H]½
ARUS[UB >=?RZA== GQYQZ}S[U G=UA=S; ¼=?RE @=E=Z@=?S[U FVZ}RSAVF[[ERU
H=?F=YH\E QEYQSRUS ERYVVA HQ^TB `?]EFRU FVYGV?VV? RYV?FRUY@VE HQQEXX½
A\S ZVAVF _==?AY=S= S=?A=S; LUZ HQQEXXA\S @=E=Z@=?S[U FVZ}RSAVF[[ERU
HQQE [>[[YVYH[[A SV} EV?YVEV; DASVV? EW @=E=Z@=?S[U FVZ}RSAVF[[ERU
Z=H`Z=HRa AXEA=}B AR@`?@B €E> G[?RUE V?VZGRUE ZQZ`EHXXAB ZQQAB Z`AR=EB
=@RZZ`H?RUE GQYQE Va@`@@RUE aQVÅÅRR`EHXXA ~Z;
L þ H % q u & / uv J //
¤R@a?`H @=E=Z@=?S[U FVEZ}RSAVF[[E ξ½RUE G[F GQYQZ}RH XHSXXA\S F=?b
S=Y>=F Z=S=AY=Y==? EW [?}[[Y} EVZ@VE ERUYGV?RUS H[[ERU Z=H`Z=HRa AXEb
A=} SVEV; M (ξ), M [ξ], m , E(ξ) SVF ZVHTRYVE HVZAVSYVEV;
ξ
ξi :
pi :
x1
p1
x2
p2
···
···
xn
pn
M (ξ) =
n
X
xi p i
£®;ž¥
i=1
= = =
== =
ξ EW f (x) SV@VE E€SHH U H @? YHS[U @ E Z@ ?S[U FVZ}RSAVF[[E GQY
¼Z=V?V^
=
=
==
H`Z HRa AXEA}RUS A ? F HQZ¶·QSQQ? QYEQ;
Z+∞
M (ξ) =
x · f (x)dx
−∞
£®;7¥
Á•Š•‰º•—Ž™ Œš‰Â”ŽšŒŠ”™ –’’Š ¨›š›–
¸¦
KXF=UE HQFRQYAQYA ξ EW GQYQZ}RH XHSXXA== [a, b] >=^@?==@ =^A=S GQY
M (ξ) =
Zb
£®; ¥
x · f (x)dx
O=H`Z=HRa AXEA}RUE [EA@VE T=E=?XXA\S HVZAVSYV`;
 KQSHZQY FVZ}RSAVF[[ERU Z=H;AXEA=} ]]?HVUS]] HVE[[ 2 M (C) = C.
¦ KQSHZQY HQQS Z=H;AXEA}RUE HVZASRUE ]ZE] S=?S=} GQYEQ2
· ξ) = C · M (ξ).
M=(C
=?S[U FVZ}RSAVF[[E[[ARUE ERUYGV?RUE Z=H;AXEA=} HVASVV?RUE
° M
E=Z@
Z=H;AXEA}XXA\E ERUYGV?HVU HVE[[ 2 M P ξ = P M (ξ )
¸ ¼V?V^ η = P a ξ + b GQY M (η) = P a M (ξ ) + b.
» ¼=Z==?=YS[U @=E=Z@=?S[U FVZ}RSAVF[[E[[ARUE [?}^V?RUE Z=H;AXEA=}
EW HVASVV?RUE Z=H;AXEA}XXA\E [?}^V?HVU HVE[[ 2
a
n
n
i
i
n
i i
i
i=1
i=1
i=1
n
i
i=1
M (ξ1 · ξ2 · · · ξn ) = M (ξ1 ) · M (ξ2 ) · · · M (ξn ).
#”Ôšš$Ɛ ž8 Z A==^XXE\ âXH=@ H=@?=F HQQ â½E\ H=?F=YH\E FXXYW
A=?==F F[@EVSHVV? ]S]SA]^;
8 ž 7
ξ:
8;7 8;¡ 8; 8;ž
p :
DEV @=E=Z@=?S[U FVZ}RSAVF[[ERU Z=H`Z=HRa AXEA}RUS QY;
£®;ž¥ HQZ¶·Q ·@QQ? M (ξ) = 0 · 0.2 + 1 · 0.4 + 2 · 0.3 + 3 · 3 · 0.1 = 1.3
DEV EW ž8 Z A==^XXE\ XH@
=EA AXEA}==? žB H=@?=YH EQQSAQEQ SV@VE [S ~Z;
#”Ôšš$Ʀ K=@?=YHS[U @=E=Z@=?S[U FVZ}RSAVF[[E H=?F=YH\E ÅXEa½
;
VV?VV ]S]SA]^

x≤2

M=E=Z@=?S[U FVZ}RSAVF[[ERU
 0
1
1
F (x) =
(x − 1) −
2<x≤4
Z=H`Z=HRa AXEA}RUS QY;

8
 8
1
>4
DFYVVA E€SH\E
ÅXEaRUS xQY¶·
£f (x) = F (x)¥;
i
2
0
Mξ =
Z4
2

x≤2

 0
1
f (x) =
(x − 1) 2 < x ≤ 4

 4
0
x>4
xf (x)dx =
Z4
2
1
1
x (x − 1)dx =
4
4
x3 x2
−
3
2
4
=
2
19
.
6
Ʀ ҔºÓœ—º© ”ºÓœ—º”™Š ±•Š•—µµ
¸°
L 8 u . % u . % u v J / / 2
M=E=Z@=?S[U FVZ}RSAVF[[E Z=H`Z=HRa AXEA}==@== F=>=UF F=>=UYH\E a^=A?=½
H\E Z=H`Z=HRa AXEA}RUS AR@`?@ SV} EV?YVEV; D(ξ), D[ξ], σ , V (ξ) SVF
ZVHTRYVE HVZAVSYVEV ;
£®;¡¥
D(ξ) = M [ξ − M (ξ)]
== =
[ξ − M (ξ)] @ E Z@ ?S[U FVZ}RSAVF[[ERU GQYQZ}RH XHSXXA EW [x − M (ξ)] ,
== == =
=
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S=Y>=E p , p , . . . , p GQY AR@`?@ EW D(ξ) = P[x − M (ξ)] p SV} GQAQSAQEQ;
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G=UA=S;
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D(ξ) = M (ξ ) − M (ξ)
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2
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2
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Z+∞
D(ξ) = [x−M (ξ)]2 f (x)dx D(ξ) =
x2 f (x)dx−(M ξ)2 
−∞
£®;®¥
−∞
HQZ¶·QSQQ? QYEQ;
M=E=Z@=?S[U FVZ}RSAVF[[ERU AR@`?@VV@ €>SXX? S=?S=@E\S XS @=E=Z@=?S[U
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SV} EV?YVEV;
p
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σ(ξ) = D(ξ)
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 KQSHZQY FVZ}RSAVF[[ERU AR@`?@ HVSHVU HVE[[ 2 D(C) = 0.
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S=?S=E=2
· ξ) = C · D(ξ)
° K(ξ; η) = D(C
M [(ξ − M ξ) · M (η − M η)] = M (ξ · η) − M ξ · M η FVZ}RSAVF[[ERUS
== =
= =
;
ξ, η @ E Z@ ?S[U FVZ}RSAVF[[E[[ARUE aQ??`Y€RUE ZQZ`EH £aQ^ ?R ¥ SVEV
= =;
∀ ξ, η − RUE FX^WA D(ξ +η) = Dξ +Dη+2·K(ξ; η) G UE
¸ ξ , ξ , . . . , ξ −F=Z==?=YS[U GQY
ξ
2
1
2
n
D(ξ1 +ξ2 +. . .+ξn ) = Dξ1 +Dξ2 +. . .+Dξn .
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8 ž 7
ξ:
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DEV HQFRQYAQYA (ξ − M ξ) FVZ}RSAVF[[ERU H=?F=YH\E
FXXYRUS G=USXXY=F _==?AY=S=H=U2
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7
LUZA £®;¡¥ HQZ¶·Q ·@QQ?2
D(ξ) = M [(ξ − M ξ) ] = 1.69·0.2 + 0.09·0.4 + 0.49·0.3 + 2.89·0.1 = 0.81, σ(ξ) = 0, 9.
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2
2
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2
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2
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1
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8;2 8;ž
2 8;0 7 8;¡
ξ2 :
pi
2
2
2
2
LUZA M (ξ ) = 0·0.2 + 1·0.4 + 4·0.3 + 9·0.1 = 2, 5.
£®;­¥ HQZ¶·Q ·@QQ? D(ξ) = M (ξ )−(M ξ) = 2.5 −1.3 = 0.81, σ(ξ) = 0, 9.
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`?@ G= @H=EA=?H F=>=UYH\S GQA¶·2
2
2
D(ξ) =
Zb
2
2
2
x f (x)dx − (M ξ) =
x
21
4
2
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2
a
1
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4
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2
x4 x3
−
4
3
4
2
−
19
6
2
11
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19
6
2
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11
.
6
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ξ =
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SV@VE @=E=Z@=?S[U FVZ}RSAVF[[ERUS =^¶€; KVS^VY
P GQYEQ ; <`?EXYYRUE @F`Z XTR? G[? F =Z==?=YS[U ;
ξ = ξ +ξ +· · ·+ξ =
ξ
ξ
LUZA
i
n
1
2
n
i
i=1
n
n
P
P
P
ξi =
M ξi = (1 · p + 0 · q) = np,
M (ξ) = M
i=1
ni=1 n
P
P
D(ξ) = D
ξi =
D(ξi ), D(ξi ) = M (ξi2 ) − (M ξi )2 =
i=1
i=1
i
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12 ·p+02 ·q−p2 = p−p2 = p(1−p) = pq,
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n
n
P
P
D(ξ)= D(ξi )= pq = npq.
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i=1
i=1
£®;9¥
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Z=H`Z=HRa AXEA=}
M (ξ) = np, D(ξ) = npq,
M (ξ) =
∞
X
k=0
k·
σ(ξ) =
√
npq
∞
X
λk−1
λk −λ
e = e−λ · λ ·
= e−λ λ · eλ = λ.
k!
(k
−
1)!
k=1
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k
=
.
k
k!
k!
k=0
k=0
λ
λe−λ −
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λ + λ2 =
LUZA
k
∞
∞
X
k=0
D(ξ) = λ.
k2
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e = M ξ 2 , Dξ = M ξ 2 −λ2 = λ + λ2 −λ2 = λ
k!
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·e
M (ξ) = λ, D(ξ) = λ, σ(ξ) =
k
t
k!
−λt
√
λ.
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P (ξ = k) = q k−1 ·p = (1−p)k−1 · p, k = 1, 2, . . .
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£®;žž¥
M (ξ) = 1/p , D(ξ) = q/p = (1 − p)/p .
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HQQ S`QZ`H? H=?F=YHH=U G=UE=;
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2
2
m
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n
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== =
= = =
;
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¯R?S`QZ`H? H=?F=YH\E HQQE [>[[YVYH[[A 2
M
M (ξ) = n · ,
N
nm
D(ξ) =
N −1
M
1−
N
n
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.
N
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M (ξ) =
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−∞
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Z+∞
Z+∞
1
−mt
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t · me dt = m ·
x · f (x)dx =
m
0
0
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=
D(ξ) = M (ξ ) − [M (ξ)] =
t2 me−mt dt −
m
2
2
0
=
1
1
2
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2
m
m
m
p
1
σ(ξ) = D(ξ) = .
m
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GQY
M (ξ) = 1/m , D(ξ) = 1/m2 , σ(ξ) = 1/m.
M (ξ) =
Z∞
−∞
x · f (x)dx =
Zb
a
x·
dx
a+b
=
,
b−a
2
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2
Zb Z+∞
a+b
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dx
2
(x−M ξ) f (x)dx =
x−
D(ξ) =
=
,
2
b−a
12
a
−∞
σ(ξ) =
r
√
(b − a)2
3
(b − a)
= √ =
(b − a).
12
6
2 3
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M (ξ) =
(b − a)2
(b − a)
a+b
, D(ξ) =
, σ(ξ) = √ .
2
12
2 3
f (x) =
λr ·xr−1 −λx
·e , x > 0, r = 1, 2, . . .
(r − 1)!
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M (ξ) = r/λ, D(ξ) = r/λ .
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2
2
2
Z+∞
1
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M (ξ) =
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dx = a,
2σ 2
2πσ
−∞
Z+∞
1
(x − a)2
2
(x − a) · √
exp −
dx = σ 2 , σ(ξ) = σ.
D(ξ) =
2
2σ
2πσ
−∞
£®;ž«¥
KXF=UYG=YB
SV@VE H=?F=YH\E E€SH G[FRU
FV^RUE H=?F=YHH=U @=E=Z@=?S[U FVZ}RSAVF[[ERU =?=Z`H?[[A EW
;
M (ξ) = 7, D(ξ) = (0.1) , σ(ξ) = 0.1 GQYEQ
A[?ZRUS FV?VSYV^VY
¼V?V^ VEV @=E=Z@=?S[U FVZ}RSAVF[[ERU FX^WA 3σ½RUE GX~X
== =
P (|ξ − 7| < 3 · 0.1) = P (6, 7 ≤ ξ ≤ 7.3) ≈ 0.9973
ξ @ E Z@ ?S[U
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[>VSAVY GQYEQ;
M (ξ) = a, D(ξ) = λ2 , σ(ξ) = σ.
1
(x − 7)2
√ · exp −
f (x) =
23 · (0.1)2
0.1 · 2π
2
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k
k
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n
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£®;ž¬¥
Z+∞
xk · f (x)dx
νk =
−∞
KXF=UE HQFRQYAQYA ν = M (ξ), ν = M (ξ ) GQYQF HXY
D(ξ) = M (ξ ) − (M ξ) = ν − ν .
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™
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2
2
2
2
2
1
2
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k
k
n
X
µk =
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k = 1, 2, . . .
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µk = [x − M (ξ)]k · f (x)dx
∞
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=
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;
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lim F (x, y) = F (−∞; y) = 0
M (x, y)
x→−∞
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y→−∞
ξ1
0
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lim
F (x, y) = 1,
x → +∞
y → +∞
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=
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¼ZX}RA
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j=1 i=1
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ZZ
£«;9¥
P ((ξ , ξ ) ∈ D) =
ϕ(x, y)dxdy
1
2
== =
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F=ZH\E H=?F=YH\E E€SH SV} EV?YVEV;
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ϕ(x,
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¥
7
ϕ(x, y)dxdy = 1.
¥ ϕ(x, y) = ∂ F (x, y) T=E=?XXA\S F=ES=E=;
∂x∂y
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(D)
2
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2
1
2
RR
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GX~X H=?F=YH\E ÅXEaRUS =_RSY=^=Y2
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1
2
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ϕ(x, y)dxdy
−∞ −∞
(D)
Zx Zy
Rx Ry
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1
1
2
1
F1 (x) = P (ξ1 < x) = P (ξ1 < x, −∞ < ξ2 < +∞) =

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Zx Z∞
=
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DEAVV@ E€SH\E ÅXEa f (x)
−∞ −∞
−∞
−∞
1
£«;ž8¥
Z+∞
ϕ(x, y)dy
f1 (x) =
Ÿ[ETYVE ξ ½RUE FX^WA F (y) =
2
2
−∞
Ry
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+∞
R
−∞
ϕ(x, y)dx dy
Z+∞
f2 (y) =
ϕ(x, y)dx
−∞
GX~X
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ϕ(x, y) = f (x) · f (y)
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== =
;
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P (ξ1 < x, ξ2 < y) = P (ξ1 < x) · P (ξ2 < y)
1
1
2
2
1
2
1
2
M (ξ1 ) =
M (ξ2 ) =
m
n P
P
i=1 j=1
n
m P
P
n
P
xi pij =
yj pij =
xi p i
i=1
m
P
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j=1
j=1 i=1
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M (ξ1 ) =
M (ξ2 ) =
+∞
R
R +∞
−∞ −∞
+∞
R
R +∞
−∞ −∞
x · ϕ(x, y)dxdy =
y · ϕ(x, y)dxdy =
+∞
R
−∞
+∞
R
−∞
x · f1 (x)dx
£«;ž­¥
y · f2 (y)dy
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=
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1
2
D(ξ1 ) =
D(ξ2 ) =
m
n P
P
i=1 j=1
n
m P
P
j=1 i=1
(xi − M ξ1 )2 pij =
(yj − M ξ2 )2 pij =
n
P
(xi − M ξ1 )2 pi
i=1
m
P
j=1
£«;ž®¥
(yj − M ξ2 )2 qj
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D(ξ1 ) =
+∞
R
R +∞
(x−M ξ1 )2 ϕ(x, y)dxdy =
−∞ −∞
+∞
R +∞
R
D(ξ2 ) =
−∞ −∞
+∞
R
(x−M ξ1 )2 f1 (x)dx
−∞
+∞
R
(y−M ξ2 )2 ϕ(x, y)dxdy =
−∞
(y−M ξ2 )2 f2 (y)dy
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1
,
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ž;¥ P ((ξ , ξ ) ∈ D) =?
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2
1
2
2
2
1
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π
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0
1
= 2
π
Z1
0
dy
1 + y2
Z1
1
dx
= 2(
2
1+x
π
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dy
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1
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−ρ
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y
x
y
1
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−
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2σy2
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a
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G
G
Ω
d
2
1
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2
x
a
2
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d
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a
aπ
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2
2
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d
sin αdα =− cos α|π0 = d,
SG =
2
0 2
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b
f (x) =
x−a
b
c
Ÿ[EAB x ≥ a, b > 0, c > 0.
Ë
= = =
= =
=
ξ− `UGXYYRUE H ?F YHH U GQY Z H`Z HRa AXEA } EW 2
1
M (ξ) = a + b · Γ 1 +
.
c
Ÿ[EA 2 Γ(x) = R e · t dt GX~X ¯=ZZ= ÅXEa SV} EV?YVEV;
KXF=UE HQFRQYAQYA Γ(n + 1) = n!, n ∈ N ;
+∞
−t
x−1
0
4 O q q 1 2
f (x) =
1
· xα · e−x/β , (x > 0)
α+1
Γ(α + 1) · β
E€SH G[FRU H=?F=YH\S ¯=ZZ= H=?F=YH SVEV; Ÿ[EAB β > 0, α > −1 E]FYRUS
F=ES=F =?=Z`H?[[A ; ¼V?V^ ξ EW S=ZZ= H=?F=YHH=U GQY H[[ERU Z=H`Z=HRa
AXEA=} G= AR@?`@ A=?==F HQZ¶·QSQQ? HQAQ?FQUYQSAQEQ;
M (ξ) = β(α + 1), D(ξ) = β 2 (α + 1).
ÆÐ
ɍŽšš‰š› Œš—šŽ›šŽšŒ ¨•—”‰ –•—Œ•›–Ê Œµµ›”µ
41 % 1 == =
= =
ξ @ E Z@ ?S[U FVZ}RSAVF[[ERU H ?F YH\E E€SH
f (x) =
Γ(α + β + 2)
· xα · (1 − x)β ,
Γ(α + 1) · Γ(β + 1)
(0 < x < 1)
HQZ¶·QSQQ? HQAQ?FQUYQSAQF GQY H[[ERUS <`H= H=?F=YH SVEV;
Ÿ[EA 2 α > −1, β > −1. O=H;AXEA=} G= AR@`?@ EW2
M (ξ) =
α+1
,
α+β+2
D(ξ) =
(α + 1)(β + 1)
.
(α + β + 2)2 (α + β + 3)
u I& ( 2 , 1 χ
= == =
== =
ξ , ξ , . . . , ξ F Z ? YS[U @ E Z@ ?S[U FVZ}RSAVF[[E[[A G]S]]A ξ (i = 1, n)
=
=
G[? EW a = 0, σ = 1  ? Z`H?[[A G[FRU FV^RUE H=?F=YHH=U G=US;
KVS^VY2 χ (n) = ξ +ξ +· · ·+ξ FVZ}RSAVF[[ERUS n T]Y]]ERU >V?VS G[FRU χ
H=?F=YHH=U @=E=Z@=?S[U FVZ}RSAVF[[E SVEV; χ (n)½ @=E=Z@=?S[U FVZ}RSAV½
F[[ERU H=?F=YH\E E€SH
45
1
2
`
n
i
2
2
2
1
2
2
2
n
f (x) =
1
2n/2 · Γ
2
2
n · x(n/2)−1 · e−x/2 ,
(x ≥ 0)
2
FVYGV?HVU G=UE=; DEV RYV?FRUYVY α = n − 1, β = 2 G=UF [`RUE ¯=ZZ=
2
H=?F=YH\E HXF=UE HQFRQYAQY ~Z;
= =
= =
= =
χ (n) H ?F YH\E Z H`Z HRa AXEA } G AR@`?@ EW 2
2
M (χ2 (n)) = n,
D(χ2 (n)) = 2n.
Ï]Y]]ERU >V?VS n½RUE €E> G[?RUE XHS=EA H=?F=YH\E E€SH\E ÅXEaRUS A=?==F
S?=ÅRaH A[?@YV^; Ï]Y]]ERU >V?VS n G= p = 1−α RHSVF Z=S=AY=Y\E €E> G[?RUE
f (χ2n )
0.5
0.4
n=2
0.3
n=6
0.2
n = 18
0.1
0
χ2
5
10
15
20
25
30
35
40
PX?=S 9;ž2
=
=
=
=
=
XHSXXA A F ?S Y> F χ (n)½\E H GYR\S >QFRQ@QE G=UA=S £F=^@?=YH\E H=GY
Ú¡ ½RUS [>¥;
2
È» a’ŽŠ’—‰•›Ä –•—Œ•›–
Ɛ
Ï]Y]]ERU >V?SRUE RF XHSXXA=A £n > 30¥ χ (n) H=?F=YH N (n, 2n) H=?F=YH
X?XX ERUYEV; LUZA ?=aHRa HQQQQEA n > 30 [`A χ (n) H=?F=YH\S a = n, σ =
==
= = ==
; K=?F=YH\E ÅXEaRUE
2n  ? Z`H?[[A G[FRU FV^RUE H ?F YH ? @QYWAQS
S?=ÅRa AVV? Q?_RF p Q?ARE=HH=U VSRUE =G@R@@ x ½S p½V?VZGRUE a^=EHRYW
SVEV; ¼R½a^=A?=H H=?F=YH\E p½V?VZGRUE a^=EHRYRUS χ (n) SV^VY n > 30 [`A
VASVV? a^=EHRYXXA\S N (0, 1) H=?F=YH\E a^=EHRY==? QY} GQYEQ2
2
2
2
p
2
p
χ2p (n) ≈
√
2
1
Up + 2n − 1
2
Ÿ[EAB U ½EW N (0, 1) H=?F=YH\E a^=EHRYW;
<=S= V?VZGRUE a^=EHRYRXA\E FX^WA RY[[ E=?RU^TY=YH=U A=?==F HQZ¶·QS
FV?VSYVEV ;
p
2
+ Up
χ2p (n) ≈ n 1 −
9n
r
2
9n
!3
.
4K b ' t ' q 1 c 1 V`?VS HQAQ?FQUYQSA@QE @=E=Z@=?S[U FVZ}RSAVF[[E ξ½RUE FX^WA ln ξ½
¼@=V?V^
=
=
E Z@ ?S[U FVZ}RSAVF[[E a, σ =?=Z`H?[[A G[FRU FV^RUE H=?F=YHH=U GQY
H[[ERUS YQSEQ?Z=YW H=?F=YHH=U SVEV;
¢]?]]? FVYGVYB ξ ∼ N (a, σ ) : a = M (ln ξ), σ = D(ln ξ).
¾QSEQ?Z=YW H=?F=YH\E E€SH A=?==F FVYGV?HVU GQYEQ;
2
2
2


h (ln x − a)2 i
1
√
, x≥0
· exp −
f (x) =
2σ 2
 2πσx
0
x<0
¾QSEQ?Z=YW H=?F=YH\E =?=Z`H?[[A 2
M (ξ) = ea+σ
2 /2
4L B c y % u v
,
2
2
D(ξ) = e2a+σ · (eσ − 1).
1 @=E=Z@=?S[U FVZ}RSAVF[[E HX@ G[? a = 0 G= AX?\E σ =?=Z`H½
?[[AHVU FV^RUE H=?F=YHH=U G=US; ξ EW GX@A==@== £ξ ½G[?VV@¥ F=Z==?=YS[U;
=?RE ξ @=E=Z@=?S[U FVZ}RSAVF[[E[[ARUE AQHQ? _XS=Z=E F=Z==?=YS[U n½
¼FVZ}RSAVF[[E
G=UA=S SV`; KVS^VY £n ≤ k ¥
2
ξ, ξ1 , ξ2 , . . . , ξk
i
i
T =r
ξ
1 2
(ξ + ξ22 + · · · + ξk2 )
n 1
Ʀ
ɍŽšš‰š› Œš—šŽ›šŽšŒ ¨•—”‰ –•—Œ•›–Ê Œµµ›”µ
FVZ}RSAVF[[ERUS n½T]Y]]ERU >V?VS G[FRU MHW~A`EHRUE H=?F=YHH=U SVEV;
= =
;M
= =
= ==
t−H ?F YH SV} EV?YVF EW GRU HW~A`EHRUE H ?F YH\E E€SH G  ? Z`H?[[A
1
Sn (x) = √ ·
nπ
Γ
n + 1
Γ
2
n
2
1+
x2 −(n+1)/2
,
n
M [Sn (x)] = 0, D[Sn (x)] =
−∞ < x < +∞
n
n−2
n
p = 1−a
HQZ¶·QSQQ? HQAQ?FQUYQSAQEQ; Ï]Y]]ERU >V?VS G=
Z=S=AY=Y\E
M
=
=
=
=
€E> G[?RUE XHSXXA A HW~A`EHRUE H ?F YH\E ÅXEaRUE H GYR\S >QFRQ@QE
G=UA=S; £Ú« H=GYR\S [>;¥
M
= =
= = =
n → ∞ [`A HW~A`EHRUE H ?F YH EW EQ z?ZTYQSA@QE EQ?Z YW H ?F YH X?XX
M
;
=
=
_XYXXE\ FX^WA HVS_ FVZHVU
ERUYEV HW~A`EHRUE H ?F YH\E E€SH
XTR? VEV H=?F=YH\E a^=EHRYRXA t (n) x= =−t0 (n) E]FY]]? FQYGQSAQEQ; n >
;Ÿ
= =
=
;
30 [`A t (n) ≈ U HQZ¶·QSQQ? QYEQ [EA U EW N (0, 1) H ?F YH\E a^ EHRYW
1−p
p
p
p
p
4N { u _ % u v
1 (|I 1 ,
SV@VE F=Z==?=YS[U @=E=Z@=?S[U FVZ}RSAVF[[ERU FX^WA
FVZ}RSAVF[[ERU H=?F=YH\S n G= n T]Y]]ERU >V?VSHVU
ÌR_`?RUE H=?F=YH SVEV; ÌR_`?RUE H=?F=YH\E E€SH G= HQQE [>[[YVYH[[A 2
χ2 (n1 )
n1
G=
χ2 (n2 )
n2
χ2 (n1 )/n1
F (n1 , n2 ) = 2
χ (n2 )/n2
1
2

0,
x≤0


n + n 

1
2
n n1 /2
Γ
x(n1 /2)−1
1
fF (x) =
2
(n1 +n2 )/2 , x > 0

n1
n2

n
n
1

2
Γ
·
Γ

1+ x
2
2
n2
n2
, (n2 > 2).
n2 − 2
2n22 (n1 + n2 − 2)
D[F (n1 , n2 )] =
, (n2 > 4).
n1 (n2 − 2)2 (n2 − 4)
p
Fp (n1 , n2 )
M [F (n1 , n2 )] =
ÌR_`?RUE H=?F=YH\E ½V?VZGRUE a^=EHRYRUS
SV^VY HV? EW (1 − p)
V?VZGRUE a^=EHRYWH=U A=?==F F=?W==S==? FQYGQSAQEQ;
Fp (n1 , n2 ) =
YV`;
N (0, 1), χ2 (n), Sn (x), F (n1 , n2 )
1
F1−p (n2 , n1 )
H=?F=YHXXA A=?==F FQYGQQHQU GQYQF\S HVZAVS½
Sn2 (x) = F (1, n), F (n, ∞) =
χ2 (n)
, χ2 (1) = N (0; 1).
n
ö ÷ ø }
~ S S‡‡ ‡
<RA [[ERU ]ZE] >]^F]E EVS @=E=Z@=?S[U FVZ}RSAVF[[ERU H=?F=YH\E FXXYW
H[[ERU =?=Z`H?[[ARUE HXF=U @XA=Y@=E GRYVV; KXF=UE EVS @=E=Z@=?S[U
FVZ}RSAVF[[ERU FX^WA TXF=Z €Z=? XHS== =^=F\S FVY} GQYQFS[U XT?==@ QYQE
@=E=Z@=?S[U FVZ}RSAVF[[ERU ERUYGV?RUE FX^WA HQAQ?FQU >[UY FVY} GQYQFS[U
ZVH; ¯V^T QYQE @=E=Z@=?S[U FVZ}RSAVF[[ERU ERUYGV?B EVZVSAVF[[ERU HQQS
QY_?QFQA HQAQ?FQU FXXYW >[U HQSHQYA >=FR?=SA=} R?AVS G=UE=; LUZA RF
HQQE\ FXXYW EWB QYQE HQQE\ HX?_RYH\E AXEA=} [>[[YVYH[[A HQAQ?FQU E]F½
YRUE [`A €Z=? EVS HQQE XHS=EA QU?HQF HXF=U ]S[[Y@VE FVA FVAVE H`Q?`ZXXb
A==@ HQSHQEQ;
 þ € % r _ % r - s - t - 1 u _
Ï`G\_`^\E HVEVHSVY GR_ EW @=E=Z@=?S[U FVZ}RSAVF[[ERU XHS= ]]?RUEF]]
AXEA=} XHS==@ F=>=UF F=>=UYH\E =G@=Y~H FVZ}RSAVF[[E €Z=? EVS V`?VS
HQQEQQ@ FVH?VFS[U G=UF Z=S=AY=Y\S [EVY} GQYQF XX& SV@VE =@XXA=YA F=?RX
]SE];
V?V^ @=E=Z@=?S[U FVZ}RSAVF[[E ξ½RUE AR@`?@ H]S@S]Y]S GQY AX?\E
¼
V`?VS HQQ ε½SRUE FX^WA
D(ξ)
£¬;ž¥
P (|ξ − M (ξ)| ≤ ε) ≥ 1 −
ε
HVEVHSVY GR_ GR`YEV;
2
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2
V?V^ ξ , ξ , . . . , ξ SV@VE FQ@ FQ@QQ?QQ F=Z==?=YS[U @=E=Z@=?S[U FVZ}RSAVF[[½
¼ERU
AR@`?@[[A EW }RSA >==SY=SA@=E (Dξ < C, i = 1, n, C −const) GQY ∀ε > 0
FX^WA
1
2
n
i
lim P
n→∞
ξ1 +ξ2 +. . .+ξn M (ξ1 )+M (ξ2 )+. . .+M (ξn )
−
< ε =1
n
n
Ƹ
Ό –’’ŠÃ Œµµ›”µ
HVEVHSVY F[TREHVU; KQ^TQQ? GRT^VY
!
£¬;7¥
DEAVV@ F=?=F=A @=E=Z@=?S[U FVZ}RSAVF[[E HX@ G[? ]]?@ARUEF]] Z=H`Z=HRa
AXEA}==@== RF €YS=SA=} GQYQF GQYQ^T @=E=Z@=?S[U FVZ}RSAVF[[ERU HQQ F€>b
S==?S[U QY_?QFQA HVASVV?RUE =?RÅZ`HRa AXEA=} EW Z=H`Z=HRa AXEA}XXb
A\E =?RÅZ`HRa AXEA}==@ G=?=S €YS=SA=FS[U G=UE=;
¿?=aHRaH Ï`G\_`^RUE H`Q?`ZRUS A=?==F [EVYSVV FVYGV?VV? =_RSY=E=2
n
lim P
n→∞
n
P
n
1X
1X
ξi −
M (ξi ) < ε
n i=1
n i=1
n
1X
1X
ξi −
M (ξi ) < ε
n i=1
n i=1
KXF=UE HQFRQYAQYA M (ξ ) = a, (i = 1, n) GQY
!
=1
≥1−
C
nε2 .
£¬; ¥
i
M (ξ1 ) + M (ξ2 ) + · · · + M (ξn )
= a.
n
n
1P
ξi − a < ε = 1
lim P
n→∞
n i=1
KVS^VY £¬;7¥ HQZ¶·Q
FVYGV?HVU GQYEQ;
Ÿ[ERUS H=UYG=?Y=^=Y2 :Z=? EVSVE ÅR>Ra FVZ}RSAVF[[ERU }REFVEV XHS= âa â½
S QYQF\E HXYA GR` GR`EVV@VV [Y F=Z==?=F HX?_RYHXXA FRU@VE SV`; ¼VZ}RYH
G[?A HQAQ?FQU FVZ}VVERU =YA== S=?E=; LUZA FVZ}RYH G[?RUE [? A[ES ξ £i½
HX?_RYH\E AXS==?¥ SV@VE @=E=Z@=?S[U FVZ}RSAVF[[E SV} [>VF HXY M (ξ ) =
;D
=
=
= =
;¢
a, (i = 1, n) ~Z EV FVZ}RYH[[A €Z ? EVS E ?RU^TY YH U FRUSAVEV ]?]]?
FVYGVYB D(ξ) = C, (i = 1, n). LUZA RF HQQE\ FXXYRUE E]F]Y ·@QQ? HX?_RYH\E
HQQ F[?VYVVHVU RF G=UF=A FVZ}RYHRUE A[ES[[ARUE =?RÅZ`HRa AXEA=} EW
}REFVEV XHS= âa â½==@ G=?=S [Y €YS=SA=E=;
i
i
 1 % / 1 1 u v % ' % q [Y F=Z==?=F HX?_RYH\E A[EA A½[>VSAVY EVS R}RY p Z=S=AY=YH=Ub
¼S==V?V^
? RYV?AVS GQY H[[ERU €^=SA=F F=?W=ESXU A=^H=Z} G= p Z=S=AY=Y\E €Yb
S=^=?B =G@=Y~H FVZ}RSAVF[[EVV?VV AX?\E V`?VS HQQ ε½SQQ@ G=S= G=UF Z=S=Ab
Y=Y HX?_RYH\E HQQ F[?VYVVHVU ]@]F]A EVSA FRTEVVE T QU?FQE G=U} GQYEQ;
¢]?]]? FVYGVYB
m
£¬;¡¥
−p ≤ε =1
lim P
<`?EXYYRUE H`Q?`ZRUS F€>S==?\E n@QESQAQS XHS==?B G[F n½RUE FX^WA
m
m
= =
;¢
lim
= p G UE SV} QUYSQ} [Y GQYEQ ]?]]? FVYGVYB
− p < ε HVEVHSVY
n >]^_]]?T GQYEQ ;
GR_ n>=?RZ n½RUE XHS=EA =YA=SA=F\S <`?EXYRUE H`Q?`ZA
Ÿ[EHVU FQYGQSAQE Z=S=AY=Y==?== ERUYVF ERUYVYHB G=?=S S=?==S[U ERUYVF
n→∞
n→∞
ʸ ص•ºº’ŠÃ –œ’—œ‰
Æ»
ERUYVYH[[A HQAQ?FQUYQSAAQS GRYVV;
¿?=aHRaH £¬;¡¥ H`Q?`ZRUS A=?==F [EVYSVV FVYGV?VV? =_RSY=E=2
P
£¬;­¥
m
p(1 − p)
−p ≤ε ≥1−
.
n
nε2
 5 d / ' r % ' % q ŸY F=Z==?=F HX?_RYH==? i½Y HX?_RYH G[?A A [>VSAVY €^=SA=F Z=S=AY=Y
p (i = 1, n) GQY
i
lim P
n→∞
m p1 + p2 + · · · + pn
−
≤ε
n
n
£¬;®¥
= 1.
<`?EXYYRUE H`Q?`Z R}RY £HQSHZQY¥ E]FYRUE [`A £HX?_RYH\E HQQ F=Eb
S=YHH=U RF G=UF=A¥ [>VSAYRUE A=^H=Z}RUE HQSH^Q?}RYH\S RYV?FRUY} G=Ub
S== GQY ¿X=@@QE\ H`Q?`Z R}RY GR_ £HQSH^Q?HQU GR_¥ ]]? ]]? E]FYRUE [`A
A=^H=Z}RUE HQSH^Q?}RYH\S RYV?FRUY} G=UE=;
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( [t r t'1 %' %q ,
`
V^RUE H=?F=YHH=UB H]@S]Y]S HQQE\ [Y F=Z==?=F @=E=Z@=?S[U FVZ}RSAVF[[½
¼E[[ARUE
ERUYGV? Z]E FV^RUE H=?F=YHH=U GQYQF\S F€YG=?F=E G=H=Y} GQYEQ;
@=E=Z@=?S[U FVZ}RSAVF[[E HX@ G[? EW FV^RUE H=?F==S[U T HQAQ?b
¼FQUVARUSVV?
E]F]Y GR`YVSAVF [`A HVASVV?RUE ERUYGV? FV^RUE H=?F=YHH=U G=U} GQYEQ;
Ÿ[EA ¾€XEQ^\E RF HQQE\ FXXYW F=?RX ]SE];
V?V^ ξ , ξ , . . . , ξ SV@VE F=Z==?=YS[U @=E=Z@=?S[U FVZ}RSAVF[[ERU
¼
A=?==YY\E FX^WA
ž;¥ M=E=Z@=?S[U FVZ}RSAVF[[E G[? H]S@S]Y]S Z=H;AXEA=}B AR@`?@HVU2
£ (ξ ) = a , D(ξ ) = σ , i = 1, n ¥
;7 ¥ 3M
YW T @=E=Z@=?S[U FVZ}RSAVF[[ERU XHS= GX@=A @=E=Z@=?S[U FVZ}RSAVF[[½
ERU XHSXXA==@ V?@ €YS==H=U GR_ GQY
VASVV? @=E=Z@=?S[U FVZ}RSAVF[[ERU ERUYGV? η = ξ +ξ +. . .+ξ EW n−RUS
F[?VYVVHVU ]@]F]A P a Z=H;AXEA=}B P σ AR@`?@ G[FRU @H=EA=?H FV^RUE
H=?F=YH X?XX ERUYEV; ¢]?]]? FVYGVYB
1
i
2
i
n
2
i
i
n
n
i
i=1

i=1
n
P
n
P

1
2
n
2
i
Zx
 i=1 ξi − i=1 ai
 n→∞
1
2


Fη (x) = P  r n
< x −−−−−→
e−t /2 dt.
P 2
2π
σi
−∞
i=1
£¬;«¥
ÆÆ
Ό –’’ŠÃ Œµµ›”µ
DEV HQFRQYAQYAB @=E=Z@=?S[U FVZ}RSAVF[[ERU A=?==Y=Y ξ , ξ , . . . , ξ −RUS
F€>S==?H== FV^RUE H=?F=YHH=U SVEV;
KXF=UE HQFRQYAQYA ξ , ξ , . . . , ξ EW (M ξ ) = a, D(ξ ) = σ SV@VE Z=H;AXEA=}B
AR@`?@ G[FRU EVSVE R}RY H=?F=YHH=UB [Y F=Z==?=F @=E=Z@=?S[U FVZ}RSAV½
F[[ERU A=?==Y=Y GQY ∀x−RUE FX^WA
1
1

2
n
i
2
n
2
i

n
P
Zx
 i=1 ξi − na
 n→∞
1
2
Fη (x) = P 
< x
e−t /2 dt.
 σ √n
 −−−−−→ 2π
−∞
Ÿ[EAB P M ξ = na, P Dξ = nσ .
==?\E SQY H`Q?`ZRUE Z]?AY]S EW IV G[YSRUE §3½A =^T [>@VE OX=^?½
¼=€>S
¾ Y=@\E YQa=YW G= REH`S?=Y HQZ¶·Q GQYQF\S HVZAVSYV` ;
n
n
i
i=1
i
i=1
2
ö ÷ ø ù
ý ø ÷÷
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ÅXEa ZVHVV? [>V} QYEQ;
KXF=UYG=YB a [EVYVYHRUE H=?F=YH\E FXXYW ZVAVSAV} GXU GQY RHSVF >=^@?\E
FRYRUS A=?==F E]F]Y]]@ QYEQ;
P (a−δ < a < a+δ) = F (δ)−F (−δ) =
Zδ
f (x)dx = p.
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DEA }R_VV GQYSQ} a B σ =?=Z`H? G[FRU FV^RUE H=?F=YHH=U @=E=Z@=?S[U
FVZ}RSAVF[[ERU AR@`?@ £σ ¥ ZVAVSAV} GXU HQFRQYAQYA H[[ERU Z=H`Z=HRa
AXEA=} a½RUE RHSVF >=^@?\S FV?FVE G=USXXY=F\S =^T [>W`;
=
==
n FVZ}VV@H H[[^?RUE XHS x , x , . . . , x HX@ G[? EW a B σ  ? Z`H? G[FRU
FV^RUE H=?F=YHH=U HXYB X = 1 P x FVZ}RSAVF[[E EW a, σ ½=?=Z`H?[[A
n
n
σ
=
=
=
=
=
G[FRU FV^RUE H ?F YHH U G UE (X ∼ N (a, )).
n
δ =
=
=
=
?F
G UASRUS
V^RUE
H
YHH
U
FVZ}RSAVF[[ERU
FX^WA
P (|X−a| < δ) = 2Φ
¼
σ
√
=EF==?^=Y P (|X −a| < δ) = 2Φ δ n E]F]Y GR`YVEV;
σ
Ÿ[EAB Φ(x)½¾=Y=@\E ÅXEa; δ√n = t SV^VY
σ
σt XTR?
σt
GX~X
P (|X − a| < δ) = 2φ(t)
δ=√
P |X − a| < √
= 2Φ(t).
n
n
DEAVV@ P X − √σt < a < X + √σt = 2Φ(t) (∗)
¯VHVY E]S]] H=Y==n@ RHSVF Z=S=AYn=Y p ]S]SA@]E HXY
−δ
2
2
1
2
2
n
n
2
i
i=1
2
σt
σt
P X−√ <a<X+√
= p.
n
n
σt
σt
X−√ ; X+√
n
n
2Φ(t) = p (∗∗)
LUZA Z=H`Z=HRa AXEA}RUE RHSVF >=^@=? EW
Ÿ[EA t½XHS\S
E]F]Y]]@ QYQF G=
σt
δ=√
n
GQYEQ;
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EW [EVYVYHRUE E=?RU^TY=Y ~Z;
#”Ôšš$ ÐÈ σ = 4 AR@`?@HVUB a SV@VE [Y ZVAVSAVF Z=H`Z=HRa AXEA=}½
=
H UB FV^RUE H=?F=YHH=U @=E=Z@=?S[U FVZ}RSAVF[[E AVV? ­ XA== HX?_RYH
¡ ­
ž
FRU@VE A[ES F[@EVSHVA ]STVV; i : ½ ­ 7¡ ½ 8 ž8 ž
7 AXEA}RUE
7 RHSVF7 >=^@?\S
M=E=Z@=?S[U FVZ}RSAVF[[ERU Z=xH`Z: =HRa
p = 0.9
RHSVF Z=S=AY=YH=U G=USXXY;
2
i
ѕ–œ‰•–”˜ º–•–”º–”˜”™Š š›œ‰œŠ–
ÈÆ
K[[^?RUE AXEA}RUS QYGQY X = 1 (−25 + 34 − 20 + 10 + 21) = 4, 2Φ(t) = 0.9
E]F]Y G= ¾=Y=@\E ÅXEaRUE H5=GYR==@ t = 1.65 SV} QYAQEQ;
K
1.65 · 2
σt
= =
= =
δ=√ = √
= 1.47 VS^VY Z H`Z HRa AXEA}RUE RHSVF > ^@ ?
n
5
σt
σt
=
;
X − √ ; X + √ =] − 2.35; 5.47[ SV@VE REH`?^ Y GQYEQ
¤VV? =^Tn [>@VERU n=ARY==?B VF QYQEYQS FV^RUE H=?F=YHH=U E]F]YA H[[ERU
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==?=Z`H?[[ARUE
=
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σ2 −
X
σ2 −
X
σ2
a−
S∗2
σ2
a−
b2
S
σt
σt
X− √ <a<X+ √
n
n
b
b
tS
tS
X− √ <a<X+ √
n
n
nS∗2
nS∗2
2 <
<
σ
χ21−α/2 (n)
χ2α/2 (n)
2
b
b2
(n − 1)S
(n − 1)S
< σ2 < 2
2
χ1−α/2 (n − 1)
χα/2 (n − 1)
σ2
)
n
X ∼ N (a,
X − a√
n ∼ Sn−1 (t)
b
S
σ2 2
χ (n)
S∗2 ∼
n
b2 ∼
S
σ2
χ2 (n − 1)
n−1
=?=Z`H?[[A G[FRU FV^RUE H=?F=YHB
T]Y]]ERU >V?VS G[FRU MHW~A`EHRUE H=?F=YHB
T]Y]]ERU >V?VS G[FRU FR½a^=A?=H H=?F=YH;
¹”Š’‰ –•—Œ•›–Ê p ӕ—•‰œ–—”™Š ¨•Ùº—•Š Šš›š›–
<`?EXYYRUE @F`ZRUE E]F]YA A [>VSAVY €^=SA=F Z=S=AY=Y p = P (A) ZVAVS½
AVFS[U G=US; KX?_RYH\S n XA== A=^H=E FRUFVA A [>VSAVY k XA== €^=SA@=E
GQY p = k EW A [>VSAYRUE F=?W=ESXU A=^H=Z} ~Z; KVS^VY p EW [Y
ZVAVSAVF Zn=S=AY=Y p½RUE VSVE [EVYVYH GQYEQ (p ≈ p) ; ŸY ZVAVSAVF p Z=b
S=AY=Y\E RHSVF >=^@=? (p , p )½S ]S]SA@]E (1 − α) RHSVF Z=S=AY=YH=US==?
QYQF _==?AY=S= ?=aHRaH QYQEH== H==?=YA=E=; ¼V?V^ n > 50, np >
p −p
== =
5, n(1−p ) > 5 E]F]Y[[A GR`YVSAVF GQY Z = p
(q = 1−p)½@ E Z@ ?S[U
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[? A[ES =_RSY=E P (p < p <
= =
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;
p ) ≥ 1−α E]F]YRUS F ES F p , p FRY[[ARUS A ? F HQZ¶·QSQQ? QY} GQYEQ
Ÿ[EAB
σ2
σ2
− a;
N a,
n
n
Sn−1 (t) − (n − 1)
χ2 (n) − n
∗
∗
∗
1
2
∗
∗
∗
1
2
1
p1,2
1
=
1 + t2 /n
2
t2
p∗ +
∓t·
2n
r
p∗ (1 − p∗ )
t2
+ 2
n
4n
Ÿ[EA t½S 2Φ(t) = 1 − α E]F]Y]]@ QYEQ; P=?RZ HQFRQYAQYA
t p ∗
p1,2 ≈ p∗ ∓ √
p (1 − p∗ )
n
!
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HQZ¶·QS =_RSY=F G= GREQZ H=?F=YHH=U @=E=Z@=?S[U FVZ}RSAVF[[ERU FX^WA
H[[^?RUE FVZ}VVS HQAQ?FQUYQFQA RFV^TYVE FV?VSYVEV ;
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EW SQYQSAQY G=U^; KVS^VY ERUH A`H=YXXA\E AQHQ? SQYQSAQY A`H=Y\E V>YVF
FX^RUE RHSVF >=^@?\S 1−α = 0.95 RHSVF Z=S=AY=YH=U G=USXXY;
; = =
=
n > 50, p = 0.1, np = 10, p (1 − p ) = 0.09 ¾ Y @\E ÅXEaRUE H GYR
=_RSY=E 2Φ(t) = 0.95 E]F]Y]]@ t½S QYGQY t = 1.96 ; £ž8;¡«¥ HQZ¶·QSQQ? RHSVF
>=^@?XXA\E FRYRUS QYGQY 0.055 < p < 0.174 GQYEQ;
DEV GQAYQS\E FX^WAB âERUH G[HVVSAVF[[ERU AQHQ?FR SQYQSAQY A`H=YRUE FX^W
£Z=S=AY=Y¥B H[[^V? AVF SQYQSAQY A`H=Y\E F=?W=ES[U A=^H=Z}==@ 0.01½VV@
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FVZ}VV F=ZSRUE G=S=A== FVA G=UF ^V& SVASRUS HQSHQQ·;
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RYV?FRUY} GQYEQ;
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∗
|p∗ − p| < t ·
r
r
p∗ (1 − p∗ )
n
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DEAVV@ H[[^?RUE FVZ}VVS QYGQY n ≥ (0.3 ·n196) = 3457.44 GX~X H[[^?RUE
FVZ}VV F=ZSRUE G=S=A== ¡­9 G=UF=A AVV?F EQHQYSQQS FVY} GQYEQ SV@VE [S;
Á•Š•‰Âo n < 50 ~ZXX V@^VY n(1 − p ) B np HQQEXXA\E =YW EVS EW ­½==@ G=S=
GQY £ž8;¡«¥B £ž8;¡9¥ HQZ¶·QS [Y FV?VSYVEV;
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¿=
==
= = = == =
x EW λ  ? Z`H? G[FRU X @@QE\ H ?F YHH U @ E Z@ ?S[U FVZ}RSAVF[[ERU
=}RSY=SA@=E XHS= G=US; λ =?=Z`H?RUE RHSVF >=^@?\E FRY[[A λ , λ ½\S
QYQFAQQ ¿X=@@QE\ H=?F=YH F€>S==?H== χ ½H=?F=YH X?XX ERUYAVS GQYQF\S
=_RSY=E=;
∗
∗
∗
∗
2
∗
1
2
2
1
λ1 = χ2α/2 (2x)
2
1
λ2 = χ21−α/2 (2x + 2)
2
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¿?=aHRaH λ =?=Z`H?RUE [EVYVYHRUS QYQFAQQ VFYVVA
P FVZ}RSAVF[[ERU
==
=
;
x=
x
nλ ½ ? Z`H?RUE RHSVF > ^@?\E FRY[[ARUS QYEQ
==
=
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nλ  ? Z`H?RUE RHSVF > ^@?\E FRY[[A EW λ < nλ < λ GQY λ  ? Z`H?[[ARUE
RHSVF FRY[[A
λ
λ
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<λ<
n
n
GQYEQ;
n
i
i=1
1
1
2
2
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#”Ôšš$ ÐÐ IREV A`H=Y\S HX?_RF €^=A A A`H=Y\E FX^WA 7 V^A?VYB
=
=
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B G C A`H YXXA\E FX^WA HX@ G[? EVS V^A?VY G[?HSVSA}VV 3YW T A`H Y\E
FX^WA V^A?VY G[?HSVSAVF HQQ EW λ =?=Z`H? G[FRU ¿X=@@QE\ H=?F=YHH=U
SV^VY =?=Z`H?RUE RHSVF >=^@?\S p = 0.9 RHSVF Z=S=AY=YH=US==? G=USXXY;
3}RSY=YH==? HQSHQQ@QE V^A?VYRUE `?]EFRU HQQ x = P x = 4 XTR? £ž8;¡¬¥
HQZW·Q GQYQE χ (n) H=?F=YH\E H=GYR\S =_RSY=^=Y χ (8) = 2.73,
;K
==
χ (10) = 18.3 SV} QYAQEQ VS^VY nλ  ? Z`H?RUE RHSVF FRY[[A
;
=
=
= =
1.365 < 5λ < 9.15 HXY λ  ? Z`H?RUE RHSVF > ^@ ? EW 0.273 < λ < 1.83 GQYEQ
ɍÙ—”™Š Œš‰ÂššŽ ’›’Œ
<RA [[EVV@ ]ZE] H[[^?RUE FVZ}VV n ]S]SA@]E [`A RHSVF >=^@?\S FV?FVE
G=USXXY=F HXF=U =^T [>@VE; ¯VHVY =EFE==@== H[[^?RUE FVZ}VV €Z=? G=UF
[`A _==?AY=S=H=U [? A[ESVV S=?S=E =^T GQYQF ^V& SV@VE =@XXYH S=?E=; LF
HQQE\ FXXYW ·@QQ? F[?VYVVHVU RF G=UF==? @QESQE =^T GQYQF GQYQ^T RESVF EW
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FV^RUE H=?F=YH\E Z=H`Z=HRa AXEA}RUE RHSVF >=^@?\S G=USXXY=F=A QY@QE
£ž8;¡®¥ HQZW·QS =_RSY=E H[[^?RUE FVZ}VVS HQSHQQ} GQYEQ2
σ
£ž8;­ž¥
n=t
(2Φ(t) = p)
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Q?RQE=YB δ XHS=EA X?^XX ?QQ?RQE=Y G=UE=; DEV HQZW·QS >]^F]E VF
QYQEYQSRUE AR@`?@ ZVAVSAV} G=UF=A =_RSY=F EW QUYSQZ}HQU; ¼V?V^ σ [Y
ZVAVSAVF GQY H[[^?RUE FVZ}VVS
Sb
£ž8;­7¥
n=t
δ
HQZ¶·QSQQ? QYEQ; t XHS\S MHW~A`EHRUE H=?F=YH\E H=GYR==@ QYEQ;
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ZVAVSAV} GXU σ G= p Z=S=AY=Y==?B HX?_RYH €^XXY=F==@ ]ZE] H[[^?RUE
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2
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2
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@=?S[U FVZ}RSAVF[[ERU HQQE [>[[YVYH[[ARUE FQQ?QEA\E F=?W== GQYQE H=?½
F=YH\E FXXYRUE FVYGV?RUE HXF=U @H=HR@HRa H==Z=SY=Y AV^_[[YVF _==?Ab
Y=S= €Z=SH S=?A=S; K==Z=SY=Y SVAVS EW V@^VY [EVE V@^VY FXA=Y G=UF €Z=?
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FQUYQYH\S [EAV@YVE @H=HR@HRa H==Z=SY=YXXA\S H=?F=YH\E =?=Z`H?[[ARUE
HXF=U H==Z=SY=YB H=?F=YH\E FXXYRUE HXF=U H==Z=SY=Y SV} `?]EFRUA EW 7
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EW H=?F=YH\E =?=Z`H?RUE HXF=U H==Z=SY=Y GQYEQ;
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HVE[[ SV} [>VF [EAV@ G=US== GQY a = a SV@VE H==Z=SY=Y AV^_[[Y} GQYEQ;
DEV H==Z=SY=Y\S HVS H==Z=SY=Y GX~X =EFE\ H==Z=SY=Y SV} EV?YVVA H SV}
HVZAVSYV`; °=H`Z=HRa HQZ¶·QYYQQ? H : a = a SV} GRTEV; 3EFE\ H==b
Z=SY=YH=U F=?_Y=F ~ZXX H[[ERU V@?VS H==Z=SY=Y\S ]?@]YA]ST H==Z=SY=Y
GX~X =YWH`?E=HR^ H==Z=SY=Y SV} EV?YVVA H SV`; DEV HQFRQYAQYA ]?@]YA]ST
H==Z=SY=Y H EW H : a 6= a , H : a < a , H : a > a SV@VE €E> G=U}
GQYEQ; <QAYQS\E _RE} T=E=?==@ F=Z==?XXY} VASVV?RUE =YW EVSRUS @QESQE
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H[[^?RUE XHSXXA==? _=YS=F=A Q?_REQ; DF QYQEYQSRUE H[[^?RUE @=E=Zb
@=?S[U _RE} T=E=?==@ F=Z==?T @H=HR@HRa _=YS=YH\E A[EA FQ·? HQFRQYAQYA
GX?XX _RUA^V? S=?S=@=E G=U} GQYEQ; ¢]?]]? FVYGVYB A=?==F FQ·? H]?YRUE
=YA== S=?T GQYEQ; Ÿ[EA 2
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== = =
; L = ==
=
H H Z SY Y\S F[YVVE >]^_]]?@]E SV@VE [S ~Z UZ YA S ã H]?YRUE Yb
A== SVEV;
H==Z=SY=Y\S F[YVVE >]^_]]?]F ; ¢]?]]? FVYGVYB H H==Z=SY=Y [EVE
7G=; UF<X?XX
=A H H==Z=SY=Y\S F[YVVE >]^_]]?@]E SV@VE [S; LUZ =YA==S ãã H]?YRUE
=YA== SVEV
;
ã H]?YRUE =YA== S=?=F Z=S=AY=Y\S α SV} HVZAVSYVVA H==Z=SY=Y _=Yb
S=F RHSVF H[^_REB p = 1 − α HQQS @H=HR@HRa E=UA^=? SV} HX@ HX@ EV?b
YVEV; ¿?=aHRaH H`FERaB VARUE >=@SRUE =@XXAYXXA\S _RUA^V?YVFVA RFV^TYVE
=
= == == ==
= ==
α = 0.05, α = 0.01½VV? F ?RE VZEVYVSB E S F XF E\ @XA YS EA α = 0.001
SV} =^A=S;
0
0
0
0
0
0
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1
1
1
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¤V^_[[Y@VE H==Z=SY=Y\S _=YS=F\E HXYA H=?F=YH\E }REFVEV GX~X QU?QYb
QQ XHS= EW X?WATRY=E ZVAVSAV} G=US== HX@S=UY=E @QESQE =^@=EB @=E=Zb
@=?S[U FVZ}RSAVF[[ERUS =_RSY=A=S; DEV FVZ}RSAVF[[ERUS @H=HR@HRa _REb
}[[? SVEV £K¥; ¢]?]]? FVYGVYB H H==Z=SY=Y\S _=YS=F [[?VS S[UVHSVF
@=E=Z@=?S[U FVZ}RSAVF[[ERUS @H=HR@HRa _RE}[[? SVEV; ã G= ãã H]?YRUE
=YA== S=?=F Z=S=AY=YXXA G=S= G=UF==? _RE}[[?RUS @QESQE =^E=; %==_RA
€Z=SH VESRUE H==Z=SY=Y =^T [>VF XT?==@ H H==Z=SY=Y [EVE [`A _RE}[[?RUE
H=?F=YH\S ZVAVSAV} G=US== SV} [>EV; K=?F=YH\E =?=Z`H?RUE HXF=U _REb
}[[?RUS RHSVF _RE}[[?B H=?F=YH\E FXXYRUE HXF=U _RE}[[?RUS >]^_]]?b
YRUE _RE}[[? SV} EV?YVEV;
KQAQ?FQU _RE}[[? @QESQE =^@E\ A=?== H[[ERU GQYQZ}RH G[F XHSXXA\E
QYQEYQSRUS (V ) [Y QSHYQYQF 7 AVA QYQEYQSH FX^==E= £V = D ∪ K, D ∩ K = ∅¥;
3EFE\ H==Z=SY=Y\S F[YVVE >]^_]]?]F XHSXXA\E QYQEYQSRUS _RE}[[?RUE
XHS\E ZX} £D ¥B =EFE\ H==Z=SY=Y\S FV?VS@VFS[U G=UYS=F XHSXXA\E QYQEb
YQSRUS _RE}[[?RUE a?RHRa ZX} £K ¥ SV} EV?YVEV;
LHSVF H[^_RE α½RUE FX^WA a?RHRa ZX} K ½S @QESQE =^=F QYQE €E>\E
GQYQZ}XXA G=UA=S; LHSVF H[^_RE α EW a?RHRa ZX}RUE FVZ}VVS HQAQ?½
FQUYEQ; IRE}[[?RUE XHS\E QYQEYQSH a?RHRa ZX}RUE G=U?Y=Y EW ]?@]YA]ST
H==Z=SY=Y H ½VV@ F=Z==?E=; KXF=UYG=YB =EFE\ H==Z=SY=Y H : a = a G]S]]A
]?@]YA]ST H==Z=SY=Y EW H : a 6= a GQY a?RHRa ZX} K ½S K : |t| > ε E]F]Y
GR`YVSAV} G=UF==? @QESQ} =^E=; DEV ZX}RUS FQ·? H=YH ZX} SVF G= PX?=S
ž8;¬½A A[?@VY@VE HVS_ FVZH ZX} GQYEQ;
0
0
1
0
1
0
0
f (t/H0 )
α/2
K
−ε
p=1−α
α/2
t
ε
D =V \K
K
PX?=S ž8;¬2
N?RHRa ZX}RUE AVV? G=US== >X?==@Y=@=E ZX} HX@ G[?RUE H=YG=U α/2½HQU
HVE[[ G=UE=; ¼V?V^ ]?@]YA]ST H==Z=SY=Y EW H : a > a (H : a < a ) GQY
HVS_ FVZH H=?F=YH\E FX^WA a?RHRa ZX} EW K : t > ε (K : t < ε) E]F]Y]]?
HQAQ?FQUYQSAQEQ; £PX?=S ž8;ž8B PX?=S ž8;žž¥
DASVV? ZX}XXA\S EVS H=YH ZX}XXA SVF G= F=?S=Y>=E G=?XXE ]?]]@S]YB >[[E
]?]]@S]Y a?RHRa ZX} SVEV; PX?==@Y=@=E ZX} HX@ G[?RUE H=YG=U α½H=U HVE[[
G=UE=;
1
0
1
0
ÐªÁ–•–”º–”˜ –••‰•Ž›•› ԕ›Ž•Œ ԔŠÂ—
ʐ
f (t/H0 )
p=1−α
α
t
ε
K
D
H0 : a = a0 H1 : a > a0
PX?=S ž8;ž82
f (t/H0 )
p=1−α
α
K
−ε
t
D
H0 : a = a0
H1 : a < a0
PX?=S ž8;žž2
K=?F=YH EW HVS_ FVZHVU GR_ GQYQ^T V`?VS E€SHH=UB _RE}[[?RUE ]?@]Y½
A]ST H==Z=SY=Y EW H : a 6= a [`A a?RHRa ZX}RUS K : 0 < t < ε B ε < t
E]F]Y GR`YVSAV} G=UF==? @QESQE =^E=; LUZ ZX}RUS a^=>R HVS_ FVZH a?Rb
HRa ZX} SVEV; £PX?=S ž8;ž7¥
DASVV? >X?==@Y=@=E H=YG=UEXXA Z]E
f (t/H )
α ½HQU HVE[[ G=UE=; MH=HR@HRa _RUAb
2
^V?
S=?S=FA== FV?V^B _RE}[[?RUE
=}RSY=SA@=E XHSXXA a?RHRa ZX}RA
p=1−α
F=?W€=Y=SA=} G=U^=Y H==Z=SY=Y\S
FV?VS@VFS[UB
α/2
α/2
F=?RE _RE}[[?RUE =}RSY=SA@=E
ε
t
ε
XHSXXA _RE}[[?RUE XHS\E ZX}RA
K
K
D
=?W€=Y=SA=} G=U^=Y H==Z=SY=Y\S
F
PX?=S ž8;ž72
F[YVVE >]^_]]?E];
ÖAQQ AVV? H=UYG=?Y=@E\S Z=H`Z=HRa HVZAVSYVSVV =_RSY=E EVSHSV`; M=E=Zb
@=?S[U FVZ}RSAVF[[E X AVV? HX?_RYH FRU}B n FVZ}VV@H H[[^?RUE XHSXXA
== = =
;
=
x , x , . . . , x ½RUS QY@QE SV` 3}RSY YH\E A[EA H H Z SY Y AV^_RSA@VE
G]S]]A AVV?F H[[^?RUE XHSXXA==@ F=Z==?@=E T (x , x , . . . , x , H ) SV@VE _REb
}[[? @QESQE =^@=E SV`; LHSVF H[^_RE α ]S]SA@]E G=US; KVS^VY a?RHRa ZX}
;
K EW P (T ∈ K) = α (∗) E]F]Y]]? HQAQ?FQUYQSAQEQ ¼V?V^ _RE}[[?RUE
=}RSY=SA@=E XHS=2 Te = T (x , x , . . . , x , H ) K ZX}RA F=?W€=Y=SA=} G=U^=Y
1
0
1
0
2
1
1
2
n
0
1
H0
1
2
n
0
2
n
0
2
ѕ–œ‰•–”˜ º–•–”º–”˜”™Š š›œ‰œŠ–
ʦ
H==Z=SY=Y\S FV?VS@VFS[U; Te ∈/ K GQY H H==Z=SY=Y\S F[YVVE
=
>]^_]]?]F G VEV HQFRQYAQYA H H==Z=SY=Y\S F=>=UFS[U SVEV;
¢?@]YA]ST H==Z=SY=Y [EVE [`A _RE}[[?RUE XHS= a?RHRa ZX}RA F=?W€=Y=Sb
A=F Z=S=AY=Y £P (T ∈ K)¥½\S _RE}[[?RUE T=A=Y SVEV; ¢]?]]? FVYGVYB
]?@]YA]ST H==Z=SY=Y [EVE G=UF=A HVS H==Z=SY=Y\S FV?VS@VFS[U GQYSQF Z=½
S=AY=Y ~Z; K==Z=SY=Y _=YS=F €^=A RHSVF H[^_RE G= H[[^?RUE FVZ}VV EW
HQSHZQYB F=?RE a?RHRa ZX}== €E> G[?VV? @QESQE =^=F GQYQZ} G=US== XT?==@
_RE}[[?RUE T=A=Y F=ZSRUE RF G=UF==? a?RHRa ZX}RUS @QESQE =^E=; IREb
}[[?RUE T=A=Y FRTEVVE RF G=UF HXH=Z ãã H]?YRUE =YA== S=?=F Z=S=AY=Y
H]ARU TREVVE G=S= G=UE=; DEV XHS==? a?RHRa ZX}RUE QEQ^THQU G=U?Y=Y EW
=YWH`?E=HR^ H==Z=SY=Y H ½RUS €=} @QESQE =^@E==@ F=Z==?A=S GQYQF\S A=FRE
HVZAVSYV`;
(Te ∈ K) H0
0
0
H1
1
#”Ôšš$ Ð OVAVSAV} G=US== σ AR@`?@ G[FRU FV^RUE H=?F=YHb
=
H U @=E=Z@=?S[U FVZ}RSAVF[[ERU Z=H`Z=HRa AXEA}RUE HXF=U2 âO=H`Z=HRa
AXEA=} EW a XHS=H=U HVE[[ SV@VE â H==Z=SY=Y\S p RHSVF Z=S=AY=YH=U _=YS=;
2
0
¢?@]YA]ST H==Z=SY=Y\S H : a 6= a SV`;
ÅXEaRUS @H=HR@HRa _RE}[[?VV? @QESQE =^G=Y H H==Z=SY=Y
[EVE [`A EW EQ?ZTYQSA@QE FV^RUE H=?F=YHH=U G=UE=; ¢S]SA@]E RHSVF
H[^_RE ½RUE FX^WA a?RHRa ZX} EW K : |t| > ε SV@VE HVS_ FVZHVU ZX}
GQYEQ; OX}RUE FRY ε½S P (|t| < ε) = 2Φ(ε) = 1 − α E]F]Y]]@ QYEQ; Φ(t)½
EQz?ZTYQSA@QE FV^RUE H=?F=YH\E ÅXEa;
¼V?V^ Te = Xσ/−√an XHS= K ZX}RA F=?W€=Y=SA=} G=U^=Y H H==Z=SY=Y\S
FV?VS@VFS[U ;
KXF=UE HQFRQYAQYAB σ = 16, n = 9, X = 18 G]S]]A a = 16 [`A H H==b
Z=SY=Y\S α = 0.05 RHSVF H[^_REHVUSVV? _=YS=€ SV^VY2 2Φ(ε) = 0.95 E]Fb
]Y ·@QQ? EQz?ZTYQSA@QE FV^RUE H=?F=YH\E H=GYR==@ ε = 1.96 SV} QYAQEQ;
KVS^VY K : |t| ≥ 1.96 GX~X Te = 18 − 16 = 1.5 < 1.96 XTR? H H==Z=SY=Y
SV@VE [S ~Z;
F=>=UFS[U G= F[YVVE >]^_]]?T GQYEQ4/3
=
=
=
=
H ?F YHH U VF QYQEYQSRUE  ?=Z`H?[[ARUE HXF=U >=?RZ H==Z=SY=Y\S
¼_V^RUE
=YS=F _RE}[[? G= HVASVV?RUE H=?F=YHB a?RHRa ZX}RUS QYQF E]F]Y >V?SRUS
A=?==F F[@EVSHVEA [>[[YV^;
H 0 : a = a0
X − a0
√
T =
σ/ n
T
α
1
0
0
0
0
2
0
0
0
ÐªÁ–•–”º–”˜ –••‰•Ž›•› ԕ›Ž•Œ ԔŠÂ—
ˆ’†‹†
±Œ²”³–
ˆ
’
’
š‹‰
š
š”
µ‹
· ‹‹Ž‹šˆ‹ˆ †’‰“’ˆ
¹º
X ∼ N (a, σ 2 )
H0 : a = a0
ȼ
X ∼ N (a, σ 2 )
H0 : a = a0
¼º
X ∼ N (a, σ 2 )
H0 :
½º
¾º
σ2
=
σ02
X ∼ N (a, σ 2 )
H0 : σ 2 = σ02
X ∼ N (a1 , σx2 )
Y ∼ N (a2 , σy2 )
H0 : a1 = a2
Ž‡”‡š”‡‰
σ2 −
˜ˆ
Ž‡”‡š”‡‰
σ2 −
Ž‡”‡š”‡‰
a−
ˆ
Ž‡”‡š”˜ ‡‰
a−
Ž‡”‡š”‡‰
T = (X − a0 )
T = (X − a0 )
√
n
σ
√
n
b
S
T = nS∗2 /σ02
n
1 P
2
(xi − a)2
S∗ =
n i=1
b2 /σ 2
T = (n − 1)S
0
σx2 , σy2 −
b2
σx2 = S
x
2
b
σy = Sy2
´Œ––´†
Ž±¶–—
‰–ˆ†’–—š‰“’ˆ¸ˆ¸‰
˜”
˜”
˜”
˜”
˜”
˜”
˜”
´
‹––
µ–†¶˜˜Œ
T=s
X −Y
σy2
σx2
+
n1
n2
n1 , n2 −
˜˜œŒ–—†
‰‡Ž¶‡‡
Ê°
ˆ‹ˆ
‹‹Ž‹š
†‡†
˜
˜”
†¶˜˜Œ–—
µ–‹Œ‰‹
ˆ †
H0
H1 : a 6= a0
2Φ(ε) = 1 − α.
H1 : a > a0
2Φ(ε) = 1 − 2α.
H1 : a < a0
2Φ(−ε) = 1 − 2α.
H1 : a 6= a0
Sn−1 (ε) = 1 − α.
H1 : a > a0
Sn−1 (ε) = 1 − 2α.
H1 : a < a0
Sn−1 (−ε) = 1 − 2α.
H1 : σ 2 6= σ02
χ21−α/2; n < χ2 <
< χ2α/2; n , (H0 +)
H1 : σ 2 > σ02
χ2
≥
χ2α; n
˜”
˜”
˜”
T ∼ N (0, 1)
T ∼ Sn−1 (t)
T ∼ χ2n (t)
, (H0 −)
H1 : σ 2 < σ02
χ2 ≤ χ21−α; n , (H0 −)
H1 : σ 2 6= σ02
χ21−α/2; n−1 < χ2 <
< χ2α/2; n−1 , (H0 +)
˜”
˜”
˜”
˜”
H1 : σ 2 > σ02
χ2 ≥ χ2α; n−1 , (H0 −)
H1 : σ 2 < σ02
χ2 ≤ χ21−α; n−1 , (H0 −)
H1 : a1 6= a2
2Φ(ε) = 1 − α.
H1 : a1 > a2
˜”
2Φ(ε) = 1 − 2α.
H1 : a1 < a2
2Φ(−ε) = 1 − 2α.
T ∼ χ2n−1 (t)
T ∼ N (0, 1)
ѕ–œ‰•–”˜ º–•–”º–”˜”™Š š›œ‰œŠ–
ʸ
ÃÄ
Â
ÅÆÁÆÇÁÆ¿È
2
X ∼ N (a1 , σx
)
2
Y ∼ N (a2 , σy
)
T = (X − Y )/
v
u
2
2 u n1 σx
+ n2 σy
1
1
t
+
n1 +n2 − 2
n1
n2
n1 , n2 −
É¿¿ÊËÌÍÎ
ÈÆÅÏÆÆ
2
b2
σx
=S
x
H0 : a1 = a2
2
b2
σy
=S
y
ÐÄ
2
X ∼ N (a1 , σx
)
ÅÆÁÆÇÁÆÈ
T=
2
2
H0 : σx
= σy
ÑÄ
2
S∗
1=
1
n1
1
2
S∗
2=
n2
¿Â
ÅÆÁÆÇÁÆÈ
2
X ∼ N (a1 , σx
)
2
S∗
1
(xi − a2 )
2
2
H1 : σx
> σy
F > Fα (n1 ; n2 ) , (H0 −)
2
2
2
H1 : σx
< σy
i=1
2
2
H0 : σx
= σy
T=
b2
S
x
b2
S
2
2
H1 : σx
> σy
y
a1 = X
a2 = Y
T ∼ F (n1 ; n2 )
F1−α (n1 ; n2 ) = 1/Fα (n2 ; n1 )
F < F1−α (n1 ; n2 ) , (H0 −)
2
2
H1 : σx
6= σy
F1−α/2 (k1 ; k2 ) < F <
< Fα/2 (k1 ; k2 ) , (H0 +)
a1 , a2 −
2
Y ∼ N (a2 , σy
)
k = n1 +n2 −2
H1 : a1 < a2
Sk (−ε) = 1 − 2α.
2
2
H1 : σx
6= σy
F1−α/2 (n1 ; n2 ) < F <
< Fα/2 (n1 ; n2 ) , (H0 +)
2
S∗
2
n
P1
(xi − a1 )2
i=1
n
P2
T ∼ Sk (t)
Sk (ε) = 1 − 2α.
a1 , a2 −
2
Y ∼ N (a2 , σy
)
¿ÀÁ
¿ÀÁ
¿ÀÁ
¿ÀÁ
¿ÀÁ
¿ÀÁ
¿ÀÁ
¿ÀÁ
¿ÀÁ
H1 : a1 6= a2
Sk (ε) = 1 − α.
H1 : a1 > a2
2
2
σx
, σy
−
T ∼ F (k1 ; k2 )
F > Fα (k1 ; k2 ) , (H0 −)
2
2
H1 : σx
< σy
F1−α (k1 ; k2 ) = 1/Fα (k2 ; k1 )
F < F1−α (k1 ; k2 ) , (H0 −)
k1 = n1 −1
k2 = n2 −1.
Ÿ[EAB (H +) EW H H==Z=SY=Y\S F[YVVE >]^_]]?E]B (H −) EW H H==Z=SY=Y\S
E€==E= SV@VE HQ^TRY@QE HVZAVSYVSVVB F (n , n ) EW n G= n T]Y]]ERU >V?S[[A
G[FRU ÌR_`?RUE H=?F=YH £F ½H=?F=YH¥; ÌR_`?RUE H=?F=YH\E F[@EVSHRUS
F=^@?=YH==@ [> £K=GYR Ú9¥;
MH=HR@HRa H==Z=SY=Y _=YS=F €^=A €Z=? EVS H H==Z=SY=Y\S F[YVVE
>]^_]]?T G=US== EW H[[ERUS €S G=H=Y@=E SV@VE [S GR_B F=?RE H[[ERUS [S[U@b
SVF G[?VE [EAV@ G=UFS[U SV@VE [S ~Z; ¢]?]]? FVYGVYB H H==Z=SY=Y @XA=Yb
S==E\ [? A[EA F=?_Y=FS[U SVASRUS Y EQHQYYQQ SV@VE [S; Ÿ[ETYVE H H==b
Z=SY=Y\S FV?VS@VFS[U GQYSQ} G=US== EW H[[ERUS F[YVVE >]^_]]?]F G[?VE
[EAV@YVY G=UFS[U SV@VE [S ~Z;
DEV G[SAVV@ [EAV@YVE VESRUE H==Z=SY=Y\S F[YVVE >]^_]]?@ERU A=?== EVZVYH
@XA=YS== _==?AY=S=H=U GQYQF\S HVZAVSYV`;
Á•Š•‰Âo K=?F=YH\E =?=Z`H?[[ARUE HXF=U AVV?F _RE}[[?[[AVAB H :
== = =
a = a H Z SY Y\S α½RHSVF H[^_REHVUSVV? F[YVVE >]^_]]?]F ZX} EW a
=
=
 ? Z`H?RUE 1 − α RHSVF Z=S=AY=Y G[FRU RHSVF >=^@=?H=U A=^F=E=; ¢?]]@b
S]Y ZX} G[FRU RHSVF _RE}[[?H ]?]]@S]Y RHSVF >=^@=?B FQ·? H=YH ZX} G[FRU
RHSVF _RE}[[?H FQ·? H=YH RHSVF >=^@=? HX@ HX@ F=?S=Y>=E=; DEV G[FVE
EW AV^_[[Y@VE H H==Z=SY=Y\S a =?=Z`H?RUE RHSVF REH`?^=Y==? _=YS=F
GQYQZ}RUS QYSQ} GXU ~Z;
==
= = = =
= ==
= = =
a  ? Z`H?RUE a XHS F ?S Y> F RHSVF REH`?^ Y ? FXTRSA } G U^ Y
¼V?V^
== = =
;D
== = = = =
H H Z SY Y\S F[YVVE >]^_]]?E] @?VS HQFRQYAQYA H H Z SY Y F > U½
E=; H : a = a SV@VE H==Z=SY=Y _=YS=} G=US== GQY a − a €YS=^?\E RHSVF
REH`?^=Y\S G=USXXYE=; <=USXXY@=E REH`?^=Y a − a €YS=^?\E HVS XHS\S
FXTR} G=U^=Y H H==Z=SY=Y\S F[YVVE >]^_]]?E]; ¼=?RE AR@`?@[[ARUE
HVEYRUE HXF=U H : σ = σ H==Z=SY=Y\E FX^WA RHSVF REH`?^=Y EW AR@`?b
@[[ARUE F=?W==E\ FX^WA G=USXXY=SA=F XT?==@ G=USXXY@=E RHSVF REH`?^=Y
=?=Z`H?[[ARUE EVSHVU HVE[[ XHS\S FXTR} G=U^=Y H H==Z=SY=Y\S F[YVVE
0
0
0
1
2
1
0
2
0
0
0
0
0
0
0
0
0
0
1
2
1
1
2
0
0
2
x
2
y
0
2
ÐªÁ–•–”º–”˜ –••‰•Ž›•› ԕ›Ž•Œ ԔŠÂ—
Ê»
>]^_]]?E];
Ÿ ¡¢¡Ò £¤¥ ß ¤¦§¨ Ý ß ­­¦«¬ Ý §­ ß ¤¬ ® « Ý ¯ àà ¥
MXAY=} GXU @=E=Z@=?S[U FVZ}RSAVF[[ERU H=?F=YH\E FXXYW ZVAVSAVFS[U
HQFRQYAQYA @H=HR@HRa X^==E\ HX@Y=Z}H=US==? SR@HQS?=ZZ G=USXXY}B @=E=Zb
@=?S[U FVZ}RSAVF[[E QEQY\E H=?F=YH\E €Z=? FXXYWH=U G=U} GQYQF HXF=U
H==Z=SY=Y AV^_[[YEV; ¢]?]]? FVYGVYB QEQY\E H=?F=YH\E ÅXEa EW @H=HR@b
HRa _RE}[[? _==?A=F @H=HR@HRa H==Z=SY=Y G=UA=S; DEV H==Z=SY=Y\S _=Yb
S=FA== QEQY\E H=?F=YH\E E€SH\E ÅXEa G= SR@HQS?=ZZ\S F=?WXXY} [>VF
S?=ÅRaRUE =?S\S FV?VSYV} GQYEQ; ;R_VVYGVYB FVZ}RYHRUE [? A[ESVV?
>QFRQ@QE REH`?^=Y\E ^=?R=\E X^== =^T [>W`;
x : [−4; −3[ [−3; −2[ [−2; −1[ [−1; 0[ [0; 1[ [1; 2[ [2; 3[ [3; 4]
8;8žž 8;8­ž 8;ž¡8 8;7«8 8;7¡¡ 8;ž«7 8;8¬8 8;877
p :
¯R@HQS?=ZZ\S G=USXXY}B S?=ÅRaH=U F=?WXXYG=Y2 â@=E=Z@=?S[U FVZ}RSAV½
F[[E FV^RUE H=?F=YHH=U â SV@VE H==Z=SY=Y\S AV^_[[Y} GQYQFQQ? G=UE=; ÖAQQ
H=?F=YH\E =?=Z`H?[[ARUS QY¶·; Ÿ[ERU HXYA H[[^?RUE VY`Z`EH[[ARUS FV?T½
ZRUE AXEA}XXA==? @QESQE =^T H[[^?RUE AXEA=} G= AR@`?@RUS GQA¶·2
∗
8
8
n
k
X
X
1X
1X
2
∗
2
x2i p∗i − X ≈ 2.18675
pi xi ≈ 0.171, S =
xi =
ni x i =
X=
n i=1
n i=1
i=1
i=1
FX^W G[FRU =?S√= ·@QQ? FV^RUE H=?F=YH\E =?=Z`H?[[A
¼=ZSRUE RF [EVERU
;K
= =
=
=
a = X, σ = S G UA S XTR? S = 2.18675 ≈ 1.479 VS^VY QEQY\E H ?F Yb
H\E E€SH\E ÅXEa
2
2
F (x) =
(x−0.171)2
1
√ e− 2·2.186752
1.479 2π
GQYEQ;
=
=
y = f (x) ÅXEaRUE XHSXXA\S FV?TZRUE > F\E VS[[A AVV? QY} H GYR >QFRQ½
^QY
¡ ½
¡
ž 8
ž
½
½
½
7
7
x:
8;88¡ 8;87­ 8;8¬8 8;ž¬¬ 8;7«¡ 8;7 ¡ 8;ž7¡ 8;8¡7 8;889
f (x) :
DEV H=GYR==@ f (x) ÅXEa\E S?=ÅRaRUS G=USXXY} ]ZE]F G=USXXY@=E SR@b
HQS?=ZZH=U F=?WXXY=F >=Z==? A[SEVYH S=?S=} GQYEQ;
ÖAQQ FV^RUE H=?F=YH\E FXXYRUS _=YS=F FVA FVAVE ?=aHRa _RE}[[?[[½
ARUS =^T [>W`;
 ÒµŠ•Â •“º’›Ó– Œ•¨•™›– •Ô”Ž›•Œ ԔŠÂ—
K[[^?RUE FVZ}VV RF GR_ £n < 120¥ [`AB âFV?V^ H[[^?RUE H=?F=YH EW FV^RUE
H=?F=YHH=U QU?QYQQ H=?F=YHH=U GQY
X
0.4
− 0.7979 < √
σ
n
(∗∗)
ѕ–œ‰•–”˜ º–•–”º–”˜”™Š š›œ‰œŠ–
ÊÆ
E]F]Y GR`YVSAVEV â SV@VE VESRUE >]^Y]Z}]]? _=YS=} GQYEQ;
Ÿ[EAB X = 1 P |x − X| AXEA=} =G@QY~H F=>=UYHB
n
= = =
;
σ = S £>]^F]E FV^RUE H ?F YHH U E]F]YA VEV HVZAVSYVSVVS FV?VSYVEV¥
;R_VVYGVYB EVS =ESRUE Q~XHEXXA\E ]EA?RUE FVZ}VVSVV? >QFRQSA@QE @H=HR@b
HRa X^== =^T [>W`;
Ú [X ; x ] n p £A=^H=Z}¥ x £REH;AXEA=}¥
ž lž®­Gž®¬l ­
8;89¬
ž®«
8;7 7
ž«ž
Gž« l ž
7 lž®¬
ž­
8;
®9
ž«­
ž«
ž««
G l
7
l
¡ lž««Gž9žl ž¡
8;7­8
ž«¬
­ ž9žGž9­ ­
8;89¬
ž9
® llž9­Gž9¬ml ¡
8;8«7
ž9«
K[[^?RUE =?=Z`H?[[ARUS QY¶·;
n
i
i=1
i−1
X=
6
X
i=1
i
∗
i
i
i
v
u 6
uX
2
p∗i xi ≈ 175.93, S = t
p∗i x21 − X ≈ 5.55,
i=1
56
1 X
247
= 4.41
X=
|xi − X| ≈
56 i=1
56
DASVV? FQYGQSAYXXA==? (∗∗) E]F]YRUS _=YS=^=Y 4.41 − 0.7979 < √0.4 GX~X
5.55
56
;
0.0032 < 0.0535 GQYEQ
LUZA H[[^?RUE H=?F=YH EW FV^RUE H=?F=YHH=U SV@VE H==Z=SY=Y\S F[YVVE
>]^_]]?T GQYEQ;
¦ Ί–œ—Ù•›ÃŠ •›•™Õ •Ô”Ž›•Œ ԔŠÂ—
K[[^?RUE FVZ}VV 3 < n < 1000 [`A FV^RUE H=?F=YH\E FXXYRUS REH`?^=Y\E
A=Y=U =_RSY=E _=YS=} GQYEQ; LEH`?^=Y\E A=Y=U R = x − x ½RUS
QY@E\ A=?== R/σ F=?W==S GQAQ} F=^@?=YH\E H=GYR==@ £H=GY;Ú®¥ VEV F=?Wb
==E\ F=?S=Y>=F AVVAB AQQA XHSXXA\S QY} }R_EV; ¼V?V^ R/σ F=?W== =Yb
A==E\ 0.1 Z=S=AY=YH=US==? AVVA G= AQQA F€>S==?\E FQQ?QEA Q?_R} G=U^=Y
FV^RUE H=?F=YH\E HXF=U H==Z=SY=Y\S VEV _RE}VV? F[YVVE >]^_]]?T GQYEQ;
KXF=UYG=YB ]ZE]F }R_VVEA R = 189 − 165 = 24. R/σ = 24 ≈ 4.324, n =
5.55
= == = = =
=
­; =
56, p = 0.1 [`A H GYR @ F ?S Y> F AVVA G AQQA XHSXXA\S QYGQY 7 G
¡;8 GQYEQ; O=E=U HQFRQYAQYA 4.03 < 4.324 < 5.23 XT?==@ H[[^V? FV^RUE
H=?F=YHH=U SV} [>V} GQYEQ;
° ɍÙ—”™Š š˜ºÕœºº “• •º”‰‰œ–—”™Š ˜’šii”Õ”œŠ– •Ô”Ž›•Œ
ԔŠÂ—
max
min
ÐªÁ–•–”º–”˜ –••‰•Ž›•› ԕ›Ž•Œ ԔŠÂ—
ʪ
K[[^?RUE FVZ}VV G=S= £n < 20¥ [`A FV^RUE H=?F=YH\E FXXYRUS Va@`@@ G=
=@RZZ`H?RUE aQVÅÅRR`EHXXA\E HX@Y=Z}H=US==? _=YS=} GQYEQ; K[[^?RUE
VASVV? aQVÅÅRR`EHXXA\S A=?==F HQZ¶·QSQQ? QYAQS GQYQF\S GRA ZVAVF GRYVV;
n
1 X
A=
(xi − X)3 ,
3
nS i=1
n
1 X
E=
(xi − X)4 − 3
4
nS i=1
Ÿ[EA S H[[^?RUE AR@`?@; DASVV? @=E=Z@=?S[U FVZ}RSAVF[[E HX@ G[?RUE
AR@`?@[[A EW F=?S=Y>=E
6(n − 2)
£ž8;­ ¥
D(A) =
2
(n + 1)(n + 3)
D(E) =
HQZ¶·QSQQ? GQAQSAQEQ; ¼V?V^
24n(n − 2)(n − 3)
(n + 1)2 (n + 3)(n + 5)
£ž8;­¡¥
£ž8;­­¥
q
|A| ≤ 3 D(A)
£ž8;­®¥
E]F]Y[[A GR`YVSAV} G=U^=Y XS H[[^?RUS FV^RUE H=?F=YHH=U SV} [>V} GQYEQ;
¢ZE] =^T [>@VE }R_VVERU FX^WA A = 0.262, E = −0.861, D(A) = 0.098,
=
D(E) = 0.329 G |E| ≤ 2.867, |A| ≤ 0.939 XTR? ]S]SA@]E H[[^V? FV^RUE
=
=
=
=
H ?F YHH U G UE=;
¸ ؔ—º’ŠÃ Œ”$˜Ù•—•– (χ ) ԔŠÂ—
M=E=Z@=?S[U FVZ}RSAVF[[ERU H=?F=YH\E FXXYRUE HXF=U H==Z=SY=Y\S _=Yb
S=F=A ?=aHRaH ]?S]E FV?VSYVSAAVSB HQFR?QZ}HQU G]S]]A `?]EFRU =?S= EW
N;¿R?@QE\ FR½a^=A?=H _RE}[[? ~Z; DEV _RE}[[?B HX?_RYH==? HQSHQQ@QE
H=?F=YH\E FXXYW EW X?WATRY=E ]S]SA@]E ~ZXX V@^VY X?WATRY=E H]@]]Y} GXU
H=?F=YH\E FXXYWH=U HQFR?QF V@VF HXF=U H==Z=SY=Y\S _=YS=E=;
DFYVVA n FVZ}VV@H H[[^?VV? >QFRQSA@QE @=E=Z@=?S[U FV}RSAVF[[ERU ]]?T½
Y]SA]F ZX}RUS k REH`?^=YA FX^==E=; LEH`?^=Y\E HQQS 8 < k < 20 G=U½
F==? =^G=Y HQFR?QZ}HQU G]S]]A ?=aHRaH RFV^TYVE k = [1 + 3.2 ln n] HQZ¶·Qb
SQQ? QYAQS; Ÿ[EA [x] EW x HQQE\ G[FVY FV@VS; i½? REH`?^=YA F=?W€=Y=Sb
A=F H[[^?RUE VY`Z`EHRUE HQQS n SV`; Ÿ[ERU A=?== H[[^?RUE HX@Y=Z}H=U
SR@HQS?=ZZ\S G=USXXY=EB @=E=Z@=?S[U FVZ}RSAVF[[ERU H=?F=YH\E FXXYRUE
FVYGV?RUE HXF=U H==Z=SY=Y AV^_[[Y}B H=?F=YH\E [Y ZVAVSAVF =?=Z`H?[[½
ARUS QYEQ; ¿=?=Z`H?[[ARUS QYQFAQQ F=ZSRUE RF [EVERU FX^W G[FRU =?S\S
FV?VSYV} GQYEQ;
MQESQE =^@=E H=?F=YH\E FXXYRUE HX@Y=Z}H=US==? @=E=Z@=?S[U FVZ}RSAVF[[E
= == == =^=F Z=S=AY=Y p ½S QYEQ; K[[^?VV? >QFRQSA@QE H=?F=YH G=
i½Y > ^@? @ XHS
q
|E| ≤ 5 D(E)
2
i
i
ѕ–œ‰•–”˜ º–•–”º–”˜”™Š š›œ‰œŠ–
ÊÈ
@QESQE =^@=E H=?F=YH\E FQQ?QEAQF F=>=UYH\S HQAQ?FQUYQF\E HXYA χ SV}
EV?YVSAVF A=?==F _RE}[[?RUS =^T [>AVS;
2
2
χ =
k
X
(ni − npi )2
npi
i=1
£ž8;­«¥
k
X
n ∗
=
(pi − pi )2
p
i=1 i
O=H`Z=HRa @H=HR@HRaH AVV?F FVZ}RSAVF[[ERUS l = k −r −1 T]Y]]ERU >V?VS
G[FRU χ ½H=?F=YHH=U SV} G=H=YA=S; Ÿ[EAB k½ REH`?^=Y\E HQQB n½H[[^?RUE
FVZ}VVB r½@QESQE =^@=E H=?F=YH\E FXXYRUE =?=Z`H?RUE HQQ;
¢S]SA@]E RHSVF H[^_RE α G= χ H=?F=YH\E H=GYR==@ AVV?F FVZ}RSAVF[[ERU
QEQY\E XHS= χ ½S QYEQ;
V?V^ χ ≤ χ E]F]Y GR`YVSAV} G=U^=Y @QESQE =^@=E H=?F=YH\E FXXYRUS
¼F[YVVE
>]^_]]?E]; D@?VS HQFRQYAQY EW XS H[[^V? VEV H=?F=YH\E FXXYWA
ERUVFS[US F=?XXYE=;
=
χ ½ _RE}[[?RUS H[[^?RUE FVZ}VV n ≥ 50÷150 G n ≥ 5÷8 [`A FV?VSYV^VY
;
=
=
=
HQFR?QZ}HQU ¼V?V^ YW EVS i½? REH`?^ YA F ?W€=Y=SA=F VY`Z`EH[[ARUE
HQQ n < 5 G=U^=Y F]?_ >V?SVYAVV Q?_RF FQ·? GX~X FVA FVAVE REH`?^=YXX½
A\S EVSHSVEV; DF QYQEYQS FV^RUE H=?F=YHH=U HQFRQYAQYA r = 2 G]S]]A X B
= ==
==
= =
σ  ? Z`H?[[A EW H[[^?VV? HQAQ?FQUYQSAQF\E S AE p ½S QYQFAQQ A ? F
HQZ¶·QS =_RSY=E=;
2
2
2
2
l,α
2
1,α
2
i
i
2
i
pi = P (xi ≤ X ≤ xi+1 ) = Φ
xi+1 − X
σ
−Φ
xi − X
σ
#”Ôšš$ Ð¦ ¼=UYXXY} GXU S=ESRUE T=E=?\S XAR?A=F\E HXYA H[[ERU
Z`F=ERa T=E=?\E H=?F=YH\E FXXYRUS ZVAVF _==?AY=S=H=U GQYAQS G=UE=;
­¯M Z=?aRUE S=ES @XA=Y} n = 374 XA== =}RSY=YH FRU@ERU A[EA S=ESRUE
Z`F=ERa T=E=?XXA\E EVS GQYQF âX?@=F F€>S==? FV^RUE H=?F=YHH=U â SV@VE
H==Z=SY=Y AV^_[[Y}VV; KX?_RYH==? X = 42.37aSÔZZ B σ = 0.94aSÔZZ SV}
HQSHQQ@QE G]S]]A REH`?^=Y\E ^=?R=\E X^==S A=?==F F[@EVSHVEA ]S]^;
K=?F=YH\E FXXYRUE HXF=U
Ú [x ; x ] n
ž [40; 41[ 78
VEVF[[ H==Z=SY=Y\S α = 0.01
H[^_REHVUSVV? _=Yb
7 [41; 42[ žž
7 RHSVF
L
ž­¡
=
;
S € EH`?^=Y\E ^=?R=\E
[42; 43[
¡ [43; 44[ «
X^==E==@ F=?^=Y ®½? REH`?b
­ [44; 45[ ž
^=Y\E =}RSY=YH\E XHS= ­½
® [45; 46] 7
==@ G=S= XT?==@ ­B ® AXS==?
P
«¡ REH`?^=YXXA\S EVSHSV`;
½
2
i
i+1
2
i
[41; 42[ [42; 143[ [43; 44[ [44; 46]
8
žž7
ž­¡
«
ž­
7
¢S]SA@]E @=E=Z@=?S[U FVZ}RSAVF[[ERU XHS= FX^==YH\E REH`?^=Y HX@ G[?A
F=?W€=Y=SA=F Z=S=AY=Y p ½S QY¶·;
[xi ; xi+1 ]
ni
[40; 41[
i
ÐªÁ–•–”º–”˜ –••‰•Ž›•› ԕ›Ž•Œ ԔŠÂ—
ÊÊ
41−42.37 40−42.37 p1 = P (40 < X < 41) = Φ
−Φ
= 0.4940−0.4280 = 0, 066.
0.94
0.94
p2 = 0.2762, p3 = 0.3971, p4 = 0.2128, p5 = 0, 0417.
2
χ
Ÿ[ETYVE
_RE}[[?RUE XHS\S QYQF\E HXYA A=?==F H=GYR\S >QFRQF EW HQFR?QZ}b
HQU G=UA=S;
(n − np )
Ú [x ÷ x ] n
p
np
|n − np | (n − np )
np
ž [40; 41[ 78 8;8®® 7¡;«® ¡;«®
ž;­®
8;9«ž
7
8;7«®7 ž8 ;«8 9; 8
®9;9¬
8;®®¡
7
7 [41; 42[ žž
ž­¡ 8; ¬«ž ž¡9;­7 ­;¡9
8;ž7«
¬;8
7
[42; 43[
¡ [43; 44[ « 8;7ž79 «¬;­¬ ®;­¬
¡ ;¡
8;­¡®
­ [44; 46] ž­ 8;8¡ž« ž­;®8 8;®8
8; ®
8;87
P
«¡ 8;¬¬ 9 «7;ž«
½
½
½
= 2.231
O=E=U HQFRQYAQYA
=U SV}
REH`?^=Y\E HQQ k = 5 B VF QYQEYQS FV^RUE H=?Fχ =YHH
L
;
=
[>V} GXU HXY r = 2 UZA T]Y]]ERU >V?VS l = 5 − 2 − 1 = 2. χ ½H ?F=YH\E
H=GYR==@ α = 0.01, l = 2 G=UF [`RUE XHS\S QYGQY χ = 9.2, 2.231 < 9.2
XTR? ­¯M ½Z=?aRUE S=ESRUE âX?@=F F€>S==? FV^RUE H=?F=YHH=U â SV@VE H==b
Z=SY=Y\S F[YVVE >]^_]]?T GQYEQ;
» ‘’›‰’Ž’—’ÙÊ ԔŠÂ—
==
= == =
= =
== =
n XA SRUE [Y F Z ? F HX?_RYH\E [? A[ESVV?B H @? YHS[U @ E Z@ ?S[U
=
=
=
FVZ}RSAVF[[ERU HX?_RYH\E H ?F YH\E ÅXEa F (x)½RUS G USXXY@=E SV`;
KVS^VY HX?_RYH\E HQQ F[?VYVVHVU RF [`A F (x) ÅXEa VF QYQEYQSRUE H=?½
F=YH\E ÅXEa F (x) ÕYÕÕ ERUYAVS GQYQF\S GRA ZVAVF GRYVV; Ö?Q@\E Z=H`½
Z=HRaT 3;w;NQYZQSQ?Q^ n → ∞ [`A D = max|F (x)−F (x)|√n FVZ}RSAVF[[E
= == =
F (x) ÅXEaRUE FVYGV?VV@ F Z ? FS[USVV?
i
i+1
i
i
i
i
i
i
i
i
2
i
2
i
2
2
2
2;0.01
n
n
n
K(λ) =
+∞
X
n
(−1)k e−2k
£ž8;­9¥
2 λ2
k=−∞
F€>S==?H=U G=UE= SV} G=H=Y}VV; £ž8;­9¥ ÅXEaRUS NQYZQSQ?Q^\E ÅXEa SVEV;
M=E=Z@=?S[U FVZ}RSAVF[[ERU H=?F=YH\E ÅXEa G= NQYZQSQ?Q^\E ÅXEaRUE
HQAQ?FQUYQYHQQ@B F[?VYVVHVU RF n G= AX?\E λ > 0 HQQE\ FX^WA
P (Dn < λ) ≈ K(λ) =
+∞
X
(−1)k e−2k
£ž8;­¬¥
2 λ2
k=−∞
P (Dn ≥ λ) ≈ 1 − K(λ) = 1 −
+∞
X
(−1)k e−2k
2 λ2
£ž8;®8¥
E]F]Y[[A Z]?AE];
£ž8;®8¥ ÅXEaRUE H=GYR\S λ½RUE XHSXXA=A >QFRQ@QE G=UA=S; £H=GY;Ú­¥
ÖAQQ QEQY\E G= HX?_RYH\E H=?F=YH\E ÅXEa[[A\E FQQ?QEA\E F=>=UYH\S
NQYZQSQ?Q^\E _RE}[[?VV? _=YS=F A=?==YY\S =^T [>W`; DFYVVA HX?_RYH\E
k=−∞
ѕ–œ‰•–”˜ º–•–”º–”˜”™Š š›œ‰œŠ–
ÐÐ
H=?F=YH\E ÅXEaRUE XHS= QEQY\E ÅXEaVV@ F=>=UF F=ZSRUE RF =G@QY~H F=½
>=UYH\S A=?==F HQZ¶·QSQQ? QYEQ;
£ž8;®ž¥
D = max|F (x) − F (x)|
¤=?== EW
√
£ž8;®7¥
λ= n·D
E]F]Y]]@ λ½FVZ}RSAVF[[ERUS QYEQ; ¢S]SA@]E RHSVF H[^_RE α½A F=?S=Y>=F
= == = ==
;
λ½RUE QEQY\E XHS\S A ? F H GYR @ QYEQ
8;8ž 8;87 8;8­ 8;ž 8;ž­ 8;78 8;7­ 8;
α
ž;® ž;­7 ž; ® ž;77 ž;ž¡ ž;8« ž;87 8;¬«
λ
8;¡ 8;­ 8;® 8;« 8;9 8;¬ 8;¬¬
α
8;9¬
8;9 8;«« 8;«ž 8;®¡ 8;­« 8;¡¡
λ
V?V^ λ < λ E]F]Y GR`YVSAV} G=U^=Y @=E=Z@=?S[U FVZ}RSAVF[[ERU Q½
¼EQY\E
H=?F=YH\E ÅXEa F (x) FXXYWH=U G=UE=; NQYZQSQ?Q^\E _RE}[[?RUE
A=^XX H=Y EW K(λ) Z=S=AY=Y X?WATRY=E @QESQE =^@=E H=?F=YH\E FXXYR=@
F=Z==?A=SS[UA Q?_REQ;
#”Ôšš$ Ð° M=E=Z@=?S[U FVZ}RSAVF[[ERU HX?_RYH\E [? A[ESVV?
>QFRQ@QE @H=HR@HRa X^== =^T [>W`;
n
1−α
1−α
1−α
Ú [x ; x ] x = x + x
2
ž [−20; −15[
½ ž«;­
½ ž«;­
7 [−15; −10[
7;­
½ ;­
−5[
¡ [−10;
½7;­
0[
­ [−5;
5[
7«;­
® [5;[0;10[
«
ž7;­
15[
9 [10;
ž«;­
[15; 20[
¬ [20; 25[
;­
77
ž8 [25; 30]
7«;­
K[[^?RUE =?=Z`H?[[ARUS QYGQY
i
i
X=
10
X
i=1
i+1
p∗i xi = 4.30,
i+1
i
ni
«
žž
ž­
7¡
¡¬
¡ž
7ž«®
«
p∗i
8;8 ­
8;8­­
8;8«­
8;ž78
8;7¡­
8;78­
8;ž 8
8;89­
8;8 ­
8;8ž­
v
u 10
uX
σ=t
p∗i (xi − X)2 = 9.71
i=1
M=E=Z@=?S[U FVZ}RSAVF[[E FV^RUE H=?F=YHH=U G=UF HXF=U H==Z=SY=Y\S χ
G= NQYZQSQ?Q^\E _RE}[[?VV? A=^F=? _=YS=€; Ÿ[ERU HXYA A=?==F H=GYR\S
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2
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Ú [x ; x ] n np nF (x) nF (x) n|F (x)−F (x)| (n − np )
np
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ž;ž
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8;77
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9;
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7
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8;žž
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8;8
8;8®
ž8 [25; +∞[
88 788
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8;88
7
88 ½
½
½
½
nD = 8.3
χ = 6.642
7
DEA p Z=S=AY=YXXA\S χ _RE}[[?H =^T [>@VE HQZ¶·QSQQ?
QY@QE
GQYEQ;
KXF=UYG=YB
i
i
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i
i
n
i
2
n
i
2
2
i
−10 − 4.3
−15 − 4.3
−Φ
9.71
9.71
2
10
P (ni − npi )
χ2 =
= 6, 642.
npi
i=1
6.642 < 14.1
p2 = Φ
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Ï]Y]]ERU >V?VS l = 10 − 2 − 1 = 7 ; χ = 14, 1 ;
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nD
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0
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Ÿ
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= == =
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i
N
i=1
i=1
m
1X 2
1
Q1 =
Ti −
n i=1
N
m
X
Ti
i=1
!2
£žž;ž¬¥
Ÿ[EA B N = mn, P = y ,
T =
y , (i = 1, m)
LESV} F€YG=?TRY@=E\ A[EA AVV?F G[F ERUYGV?[[ARUS F[@EVSH =_RSY=E S=?==?
GQAQ} GQYEQ £;R_VVS F=?¥;
¿?=aHRaH HQFRQYAQF ]]? EVS F[EA?VY EW F[TRE >[UYRUE H[^_RE G[? R}RY½
FVE HQQE\ VY`Z`EHHVU G=UF =YG=S[U G=UA=SH Q?_REQ; DEV HQFRQYAQYA AVV?
=^T [>@VE HQZW·QEXXA FV?FVE ]]?TY]SA]FRUS HVZAVSYV`; i½Y H[^_ERU VY`b
Z`EHRUE HQQ n £i = 1, m¥ G= H[[^?RUE FVZ}VV N = P n GQYQF\S =EF==?=E
£žž;ž¬¥ HQZ¶·QS GRT^VY2
i
n
P
j=1
2
ij
i
n
P
ij
j=1
m
i
i
i=1
m
X
1
1
Q1 =
Ti −
ni
N
i=1
GQYQF G= P , T ERUYGV?[[A P
K[[^?RUE AR@`?@[[A
i
i
i
S2 =
=
ni
P
j=1
yij2 ,
m
X
i=1
Ti =
Q
Q1
; S12 =
;
N
m−1
Ti
ni
P
!2
yij
£žž;78¥
FVYGV?HVU GQYEQ;
j=1
S22 =
Q2
N −m
£žž;7ž¥
GQYEQ;
<X@=A HQZW·QEXXA FV^VV? F=AS=Y=SA=E=; £žž;ž9¥½ £žž;78¥ HQZW·QS FV?FVE =_RS½
Y=F\S A=?==F }R_VVSVV? [>[[YW`;
#”Ôšš$  ¢S]SA@]E H[[^?RUE XHSXXA==? âF[TRE >[UYRUE AXEA=}
XHSXXA HVE[[ â SV@VE H==Z=SY=Y\S RHSVF H[^_RE α = 0.05 [`A _=YS=;
KX?_RYH\E ¼[TRE >[UYRUE H[^_RE
¡
ž 7
HQQ
ž
−19 −31 −35 −31
7
−28 −33 −32 −27
−39 −35 −26 −28
¡
−36 −25 −35 −35
­
−44 −28 −30 −40
®
−39 −31 −17 −31
° ǒ´— Œ±”Š ¨™›– ”ºÓœ—º”™Š ԔŠÂ”›Žšš
ÐÊ
¢S]SA@]E H[[^?RUE FX^WA N = 24, n = 6, m = 4 ; ¼[TRE >[UYRUE H[^_RE
G[?RUE AXEA}RUS GQAQ} £x = −34.2, x = −30.5, x = −29.2, x = −32.0¥B
A=?== EW H[^_RE FQQ?QEA\E £G[YVS FQQ?QEA\E¥ AXEA}RUS GQA^QY H[[^?RUE
`?]EFRU AXEA=} x = −31.5 GQYEQ; C = −31 SV} @QESQE =^T A=?==F F[@b
EVSHRUS >QFRQ`;
[TRE >[UYRUE H[^_RE
P
¡
ž ¼ 7
i
j
y
y
y
y
y
ž
ž7 ž¡¡
8 y 8 −4
ž® y 8 y 8
¬ −2 ¡ −1 ž ¡ ž®
7
®¡
ž® ­ 7­
¬
−8
−4
¡
­ ® ® −4 16 −4 ž®
−5
7
­
ž®¬
¬ ž ž −9 9ž
−13
®
®¡
8
8 ž¡ ž®¬
8
−8
žž
T
−19
−6
−11
®8;
ž;­
8;
®
9«;¬
T /n
7 ¡«­
7 7 ­
®­
ž
¬ž«
P
77
77
£žž;ž9¥B £žž;ž¬¥ HQZ¶·QS =_RSY=E ERUYGV?[[ARUS QYGQY2
1
2
2
j1
j1
j2
2
j2
3
j3
2
j3
4
2
j4
j4
i
2
i
i
Q = 917 −
121
= 912.0,
24
Q1 = 87.9 −
121
= 82.9, Q2 = 893.1
24
ÖAQQ £žž;ž8¥½ £žž;žž¥ HQZ¶·QSQQ? AR@`?@[[ARUE [EVYVYHRUS QY¶·;
S12 =
82.9
829.1
= 27.63; S22 =
= 41.46
4−1
24 − 4
£žž;ž ¥ ·@QQ? _RE}[[?RUE XHS= EW F = 27.63 = 0.67 GQYEQ; O=E=U HQFb
RQYAQYA α = 0.05, k = 3, k = 20 XTR?41.46
H=GYR==@ F
= 3.10 SV}
;
QYAQEQ
==
=
= == =
F < 3.10 XT? @ _RE}[[?RUE XHS a?RHRa ZX}RA F ?W€ Y SA FS[U GQY}
L
=EFE\ H==Z=SY=Y\S [S[U@SVF [EAV@S[U GQYEQ; ESVFYVV? =^T [>V} GXU F[TRE
>[UYRUE E]Y]]S HQQQFS[U G=U} GQYEQ; KX?_RYH\E [? A[ES[[A R}RY AR@b
`?@ G[FRU FV^RUE H=?F=YH=U G=UF ·@HQU XT?==@ =EF==?=YH=U F=EA=} HQAQ?b
FQU _=YS=YH EQHQYSQQE AVV? [EAV@YVE V@RUE A[SEVYHVV S=?S=F _==?AY=b
S=H=U GQYQF\S HVZAVSYV`;
1
þþ1
'] Ju
`t - -
2
v 1 u . % u v
0.05,3,20
_u
u19
V?V^ HX?_RYH\E [? A[EA FVA FVAVE F[TRE >[UY >V?VS E]Y]]Y} G=U^=Y QYQE
¼F[TRE
>[UYH AR@`?@RUE _RE}RYSVV FRUF _==?AY=S=H=U G]S]]A VEV HQFRQY½
AQYA F[TRE >[UY FQQ?QEA\E F=?RY=E [UYTYVYRUS HQQQF XT?==@ EVS F[TRE
>[UYH _RE}YVYVV@ QEYQS €YS==H=U;
ҔºÓœ—º”™Š ԔŠÂ”›Žšš
Ð
wVSVE >V?VS [UYTYVF FQ·? F[TRE >[UYRUE E]Y]]S HQQQF GQAYQSQ =^T [>W`;
Ÿ[ERU HXYA R}RY H]?YRUE FVA FVAVE @XX?W Z=_REB EVS H]?YRUE FVA FVAVE
FV@VS H[[FRU VA G=UA=S SV`; DASVV? @XX?W Z=_RE GQYQE H[[FRU VARUE FV@b
S[[A [UYA^V?YVE S=?S=} GXU G[HVVSAVF[[ERU T=E=?H E]Y]]Y} GXU V@VF HXF=U
GQAYQSQ EW FQ·? F[TRE >[UYH AR@`?@RUE _RE}YVYRUE ERUHYVS GQAYQS\E
EVS GQYEQ; ¤R@`?@RUE _RE}YVY FRUF [EA@VE _==?AY=SXXA GR`YVSA@VE SV}
@=E=€; MXX?W Z=_REXXA==@ [>[[YVF E]Y]]S A F[TRE >[UYB H[[FRU VARUE
E]Y]]S B F[TRE >[UY GQYSQE @QESQ} =^W€; r HQQE\ @XX?W Z=_REB v H]?b
YRUE H[[FRU VA G=UA=S SV^VY A F[TRE >[UY r H[^_REHVUB B F[TRE >[UY v
H[^_REHVU GQYEQ;
3}RSY=YH\E Z=H?R\S A=?==F F[@EVSHVV? [>[[YV^; (i, j)½Y E[AVEA A G= B
F[TRE >[UY[[ARUE F=?S=Y>=E i G= j ½? H[^_ERU =}RSY=SA@=E XHS= x ½S G=U?b
YXXY@=E GQYEQ;
=
B Å aHQ?\E H[^_RE
=
A Å aHQ?\E
B
B
···B ···
B
x
H[^_RE
ij
1
2
A1
x11
x12
A2
x21
Ai
j
v
i∗
· · · x1j · · ·
x1v
x1∗
x22
· · · x2j · · ·
x2v
x2∗
xi1
xi2
· · · xij · · ·
xiv
xi∗
Ar
xr1
xr2
· · · xrj · · ·
xrv
xr∗
x∗j
x∗1
x∗2
· · · x∗j · · ·
x∗v
X
=F >Q?RYSQQ?B VFYVVA £i, j ¥½Y E[AVEA =}RSY=YH\E >]^F]E S=E x
¼XHS€YG= G==?TY
=
UA S SV`; K[[ETYVE A G= B F[TRE >[UY[[A FQQ?QEAQQ F=?RY=E [UYTYVY
[>[[YAVSS[U £F=Z==?=YS[U¥ SV`; A F[TRE >[UYRUE i½? H[^_ERU HX?_RYH\E
AXEA}RUS β B B F[TRE >[UYRUE j ½Y H[^_ERU HX?_RYH\E AXEA}RUS β
SV^VY €Z=?T HX?_RYH\S A=?==F FVYGV?HVU A[?@VY} GQYEQ;
£žž;77¥
x =µ+γ +g +e
Ÿ[EAB µ−`?]EFRU AXEA=}B
==
γ −A F[TRE >[UYRUE i½? H[^_ERU E]Y]]S]]? [[@VF F > UYHB
==
g −B−F[TRE >[UYRUE j ½Y H[^_ERU E]Y]]S]]? [[@VF F > UYHB
= =
= ==
=
= =
=
e − x XHS\E }RSY YH\E YA £>]^F]E S E x }RSY YH\E [`A Yb
A== HVSHVU HVE[[¥; ¼[TRE >[UYRUE F=?RY=E [UYTYVYRUS HQQQQS[U £žž;77¥
FVYGV?RUE >=S^=?\S _XS=Z=E >=S^=? SVEV;
µ, β , β AXEA}XXA\E [EVYVYHRUS GQA^QY 2
1 XX
£žž;7 ¥
x
X=
ij
jA
jB
ij
i
j
ij
i
j
ij
iA
ij
ij
iB
r
rv
v
ij
i=1 j=1
° ǒ´— Œ±”Š ¨™›– ”ºÓœ—º”™Š ԔŠÂ”›Žšš

£žž;7¡¥
v
1X
xi∗ =
xij , i = 1, r
v j=1
£žž;7­¥
F[TRE >[UYH AR@`?@RUE _RE}RYSVVEAB H[[^?RUE `?]EFRU AR@`?@RUS
¼Q·?
=
=
G
A
B F[TRE >[UY HX@ G[?RUE E]Y]]HVU X€YA@ E FV@S[[AB Z]E HQQQSAQQS[U
=
=
=
=
[YA@VE GX@ A @ E Z@ ?S[U F[TRE >[UYRUE E]Y]]HVU X€YA@=E FV@VS SV@VE SX?b
^=E EVZVSAVF[[EA >=A=YA=S; ¢]?]]? FVYGVY £žž;®¥ HQZ¶·Q A=?==F FVYGV?HVU
GQYEQ2
£žž;7®¥
Q=Q +Q +Q
Ÿ[EA 2
X
£žž;7«¥
Q =v
(x − X)
=
=
= =
Q EW Z]?RUE AXEA}XXA G `?]EFRU AXEA}RUE €YS ^?\E a^ A? HXXA\E ERUYb
GV? G]S]]A A F[TRE >[UYRUE ]]?TY]YH]]? HQAQ?FQUYQSAQEQ;
r
x∗j =
1X
xij , j = 1, v
r i=1
1
2
3
r
1
i∗
2
i=1
1
Q2 = r
v
X
j=1
£žž;79¥
(x∗j − X)2
EW G=S=E\ AXEA}XXA G= `?]EFRU AXEA}RUE €YS=^?\E a^=A?=HXXA\E ERUYGV?
G]S]]A B F[TRE >[UYRUE ]]?TY]YH]]? HQAQ?FQUYQSAQEQ;
Q3 =
r X
v
X
i=1 j=1
(xij − xi∗ − x∗j + X)2
£žž;7¬¥
EW a^=A?=HXXA\E [YAVSAVY ERUYGV? SV} EV?YVSAVF G= HQQQSAQQS[U [YA@VE
@=E=Z@=?S[U F[TRE >[UY[[ARUE ]]?TY]YH]]? HQAQ?FQUYQSAQEQ; Q½S `?]EFRU
ERUYGV? SVEV;
¼=?S=Y>=F AR@`?@[[ARUE [EVYVYHRUS GQA^QY2
1 XX
Q
£žž; 8¥
S =
(x − X) =
Q3
r
v
2
rv − 1
ij
i=1 j=1
2
rv − 1
r
S12
v X
Q1
(xi∗ − X)2 =
=
r − 1 i=1
r−1
v
S22 =
r
S32
r X
Q2
(x∗j − X)2 =
v − 1 j=1
v−1
v
XX
1
Q
(xij − xi∗ − x∗j + X)2 =
=
(r − 1)(v − 1) i=1 j=1
(r − 1)(v − 1)
£žž; ž¥
£žž; 7¥
£žž; ¥
ҔºÓœ—º”™Š ԔŠÂ”›Žšš
¦
Q·? F[TRE >[UYH AR@`?@RUE _RE}RYSVVEA A G= B F[TRE >[UYRUE E]Y]]S
¼
HQQQFAQQ F[TRE >[UY HX@ G[?RUE AR@`?@RUS [YAVSAVY AR@`?@H F=?WXXYE=;
KXF=UYG=Y A F[TRE >[UYRUE E]Y]]S HQQQFQA F = S SV@VE _RE}[[?RUS
S
=_RSY=F G= VEV @=E=Z@=?S[U FVZ}RSAVF[[E r−1 G= (r−1)(v−1)
T]Y]]ERU >V?VS
G[FRU ÌR_`?RUE H=?F=YHH=U G=UE=; B F[TRE >[UYRUE E]Y]]S HQQQFAQQ v−1
G= (r−1)(v−1) T]Y]]ERU >V?VS G[FRU F = S SV@VE _RE}[[?RUS =_RSY=E=;
S
Ì=aHQ?XXA\E E]Y]]ERU HXF=U V@RUE _RUA^V?
S=?S=F =?S= EW EVS F[TRE
¤
=
=
=;
>[UYH AR@`?@RUE _RE}RYSVVHVU ARY G UE VV? =^T [>@VEB H[^_RE G[?
AVV?VV >]^F]E S=E HX?_RYHH=U FQ·? F[TRE >[UYH AR@`?@RUE _RE}YVYRUE
[EA@VE [? A[ES A=?==F F[@EVSHVEA EVSHSVE F=?XXY=^;
A
B
Q1 = v ·
Q2 = r ·
v
r P
P
r
P
¤R6`Y6[[ARUE
[EVYVYH
(xi∗ − X)2
r−1
S12 =
(x∗j − X)2
v−1
i=1
v
P
j=1
(xij −xi∗ −x∗j +X)2
i=1 j=1
r P
v
P
Q=
(xij − X)2
i=1 j=1
Q3 =
2
2
2
3
Ï]Y]]ERU
>V?VS
¤R@`?@ N^=A?=HXXA\E ERUYGV?
A
F[TRE
>[UYRUE
B
F[TRE
>[UYRUE
ŸYAVSAVY
q?]EFRU
2
1
2
3
Q1
r−1
Q2
v−1
Q3
(r−1)(v−1) S32 =
(r−1)(v−1)
Q
rv − 1
S2 =
rv − 1
S22 =
#”Ôšš$ ¦ <RAVEA A=?==F =}RSY=YH\E Z=H?R ]S]SA@]E SV`;
=
B Å aHQ?\E H[^_RE
=
A Å aHQ?\E
B
B
B
x
H[^_RE
1
A1
A2
x∗j
1
5
3
2
2
6
4
3
i∗
3
10
6.5
2
7
4.5
O=E=U HQFRQYAQYA r = 2, v = 3, n = rv = 6 GQYEQ; ¼[@EVSHRUE @[[YRUE
G=S=E= G= Z]?RUE VY`Z`EH[[ARUS GQAQ} QY@QE GQYEQ; KXF=UYG=Y2 A Å=aHQb
?\E A H[^_ERU FX^WA £Z]?RUE AXEA=}¥ i = 2, j = 1, 3 HXY
5 + 6 + 10
; q?]EFRU AXEA=} X = 4.5 £žž;7«¥÷ £žž;7¬¥
= 7 SVF ZVH
x =
HQZ¶·QSQQ? 3Q , Q , Q ERUYGV?[[ARUS QYGQY2 Q = 37.5, Q = 3, Q = 53.5
¤R@`?@[[ARUE [EVYVYHRUS GQA¶·;
2
2∗
1
S12 =
37.5
= 37.5;
1
2
3
S22 =
1
13
= 6.5,
2
S32 =
3
= 1.5
1·2
2
3
° ǒ´— Œ±”Š ¨™›– ”ºÓœ—º”™Š ԔŠÂ”›Žšš
¯=?S=} =^@=E [? A[ES[[ARUS A=?==F F[@EVSHVA EVSHSV^VY2
¤R6`Y6[[ARUE
¤R@`?@ N^=A?=HXXA\E Ï]Y]]ERU
>V?VS
ERUYGV?
[EVYVYH
F[TRE
«;­
ž
«;­
A
>[UYRUE
ž ;8
®;­
B F[TRE
7
>[UYRUE
ŸYAVSAVY
;8
ž;­
7
AR@`?@
q?]EFRU
­ ;­
­
ž8;«
AR@`?@
°
ÖAQQ F B F _RE}[[?RUE XHSXXA\S GQAW·;
A
B
FA =
S22
S12
37.5
6.5
=
25,
F
=
= 4, 3.
=
=
B
2
2
S3
1.5
S3
1.5
LHSVF H[^_RE α = 0.05 G= k = 1, k = 2 T]Y]]ERU >V?SRUE FX^WA ÌR_`?RUE
H=GYR\E XHSXXA\S QYGQY F = 18.5;
; LUEF[[ F > F ; F < F F=?W== GR`Y} G=US==
F
= 19.0 GQYEQ
XT?==@ A F[TRE >[UY @XA=Y} GXU FVZ}RSAVF[[EA ZVAVSAVF[U E]Y]]HVUB B
F[TRE >[UYRUE E]Y]]S HQQQFS[U G=U} GQYEQ SV@VE A[SEVYHRUS S=?S=} GQYEQ;
<RA FQ·? F[TRE >[UYH AR@`?@RUE _RE}RYSVVERU HXF=UE HQFRQYAY\S
=^T [>YVV; <[? `?]EFRU HQFRQYAQYA A F[TRE >[UYRUE i½? H[^_REB B F[TRE
>[UYRUE j ½Y H[^_RE AVV? FRU@VE HX?_RYH\E HQQ ££x , x ¥FQ@ XHS\E HQQ¥ EVS
GR_ FVA FVA G=U} GQYQF\E S=AE= VASVV? HQQEXXA ]]? ]]? G=U} GQYEQ; LUZ
HX?_RYH\S A=^H=YHH=U HX?_RYH GX~X >V?VSVV HX?_RYH G[FRU =}RSY=YH
£HX?_RYH¥ SVEV; Ÿ[EVV@ S=AE= A G= B F[TRE >[UY[[A FQQ?QEAQQ F=?RY=E
[UYTYVYHVU G=U} GQYEQ;
q?]EFRU HQFRQYAQYA FQ·? F[TRE >[UYH _RE}RYSVVERU =}RSY=YH\E XHS\S
=
A ?==F FVYGV?HVU GRTR} GQYEQ;
£žž; ¡¥
x =µ+γ +g +ν +e
Ÿ[EAB µ−`?]EFRU AXEA=}B
= ==
γ −A F[TRE >[UYRUE i½? H[^_ERU E]Y]]HVU X€YA@ E F > UYHB
= ==
g −B F[TRE >[UYRUE j ½Y H[^_ERU E]Y]]HVU X€YA@ E F > UYHB
=
=
=
==
ν −A G B F[TRE >[UYRUE F ?RY E [UYTYVYVV? [[@@VE F > UYHB
=
= ==;
e −i G j H[^_ERU FQ@ AVV?F >V?VSVV HX?_RYHXXA\E YA
¡
žž;
=
=
=
=
=
=
;
£ ¥ > S^ ?\S _XS Z E GR_ > S^ ? SVEV
£i, j ¥ H[^_ERU FQ@ G[? AVV? EVSVE R}RY n HQQE\ >V?VSVV HX?_RYH FRU@VE
SV^VY
FQ·? F[TRE >[UYH AR@`?@RUE _RE}YVYRUE [EA@VE =ARYHS=Y
£žž; ­¥
Q=Q +Q +Q +Q
1
2
α,A
α,B
A
α,A
B
α,B
i
ijk
i
j
ij
ijk
i
j
ij
ijk
1
2
3
4
j
ҔºÓœ—º”™Š ԔŠÂ”›Žšš
¸
FVYGV?HVU GQYEQ;
Ÿ[EA 2
£žž; ®¥
n
v X
r X
X
(xijk − X)2
Q=
i=1 j=1 k=1
£žž; «¥
r
X
(xi∗∗ − X)2
Q1 = nv
Q2 = nr
i=1
n
X
j=1
Q3 = n
v
r X
X
i=1 j=1
£žž; ¬¥
(xij∗ − xi∗∗ − x∗j∗ + X)2
n
v X
r X
X
Q4 =
£žž; 9¥
(x∗j∗ − X)2
£žž;¡8¥
(xijk − xij∗ )2
=
= =
= =
Q EW A G B F[TRE >[UYRUE F ?RY E [UYTYVYRUS [EVYVF a^ A? HXXA\E
ERUYGV?
=
= =
;
Q EW GX@ A F[TRE >[UY[[ARUE E]Y]]S [EVYVF a^ A? HXXA\E ERUYGV? ~Z

1 P

H[^_ERU
FQ@
AVV?F

x =
x − (i, j)


n


>V?VSVV HX?_RYH\E AXEA=} 





1P
=
Z]?RUE
AXEA
}

x =
x −
£žž;¡ž¥
v


1P
==
=


x =
x − G S E\ AXEA }


r

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1
2
3
4
2
1
2
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GQYEQ;
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k1 = 1, k2 = 24
α = 0.05
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ρ
£ž7;ž9¥
M (z) = ln
+
SV@VE Z=H`Z=HRa AXEA=}H=UB
2
1−ρ
σz2 =
2(n − 1)
£ž7;ž¬¥
3
n−3
SV@VE AR@`?@HVU FV^RUE H=?F=YH X?XX ERUYEV;
KVS^VY VF QYQEYQSRUE ρ aQVÅÅRR`EHRUE RHSVF >=^@=?
= =; th(z ), th(z ) XHSXXA\S H=GYR==@
th(z ) < ρ < th(z ) FVYGV?HVU G UE
£K=GYR Ú¬¥ z , z ½\E A=?==F FQYGQSAYXXA=A QYEQ;
1
2
1
1
2
2
1 1 + r
1
− tα √
z1 = ln
2
1−r
n−3
1 1 + r
1
z2 = ln
+ tα √
2
1−r
n−3
£ž7;78¥
Ÿ[EAB Φ(t ) = 1 − α, Φ(x)½¾=Y=@\E ÅXEa;
K[[^?RUE aQ??`Y€RUE aQVÅÅRR`EH RHSVYHVU GR_ HQFRQYAQYA VF QYQEYQSRUE
==
=
=
= = =
=
;
ρ  ? Z`H?RUE RHSVF > ^@?\S G USXXY F HXF U @XXAY\S ^T [>VFS[U
#”Ôšš$ ¦° ¢ZE]F AVA G[YVSH =^T [>@VE }R_VVERU X, Y −@=E=Z@=?S[U
FVZ}RSAVF[[E FQQ?QEA\EB H[[^?RUE aQ??`Y€RUE aQVÅÅRR`EH RHSVYHVU V@VFb
RUS _=YS=}B QEQY\E aQVÅÅRR`EH ρ−RUE RHSVF >=^@?\S G=USXXY; LHSVF
H[^_RE α = 0, 05 ;
£r = 0.74 SV} QY@QE GRYVV;¥ £ž7;ž®¥ ·@QQ? MHW~A`EHRUE _RE}[[?RUS
GRT^VY
√
α
MHW~A`EHRUE H=GYR==@
t0.05;48 = 2.01
0.74 50 − 2
t= √
= 7.62
1 − 0.742
α = 0.05, k = 50 − 2 = 48
[`A QEQY\E XHS\S QYGQY
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KX?_RYH\E XHS= G= QEQY\E XHS\E FX^WA t > t
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HXY r aQVÅÅRR`EH RHSVYHVU aQVÅÅRR`EH GQYEQ AQQ ρ aQVÅÅRR`EHRUE
RHSVF >=^@?\S G=USXXYW€; Ÿ[ERU HXYA £ž7;ž«¥ HQZ¶·QSQQ? ÌR_`?RUE z FX^R?½
S=YH\E XHS\S GQA¶·;
0.05;48
z=
E]FY]]@ t
0.05
= 1.96
1 1 + 0.74
ln
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2 1 − 0.74
Φ(tα ) = 0.95
HXY £ž7;78¥ HQZ¶·Q ·@QQ?
1
= 0.6646,
50 − 3
1
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z2 = 0.9505 − 1.96 · √
50 − 3
z1 = 0.9505 − 1.96 · √
¯R`?GQY H=ES`E@ th (z) ÅXEaRUE H=GYR ~ZXX V@^VY th(z) = e − 1
e +
1
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>=^@=? 0.51 ≤ ρ ≤ 0.844 GQYEQ;
2z
2z
' %1 [s u v
cs u %&
þ K
2
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<RAERU AVV? =^T [>@VE aQ??`Y€RUE aQVÅÅRR`EH EW >]^F]E F=ZH\E FV^RUE
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?`Y€ F=Z==?=YH=U HQFRQYAQYA Y F=Z==?Y\E F[TRUS RYV?FRUYVF [>[[YVYH
GQY} T=AE=; ¯VHVY ?=aHRaH @XA=Y} GXU FVZ}RSAVF[[E[[A ]]? FQQ?QEAQQ
@H=HR@HRa F=Z==?=YH=U GQYQ^T =YW EVS EW @=E=Z@=?S[U GR_ T ~ZXX V@^VY
F=ZHA== FV^RUE H=?F=YHH=U GR_ SVF ZVHTRYVE aQ??`Y€RUE _RE}RYSVVERU
X?WATRY@=E E]FYRUS F=ES==S[U G=U} GQYAQS; DEV HQFRQYAQYA FVZ}RSAV½
F[[E[[ARUE F=?RY=E F=Z==?Y\S RYV?FRUYVF `?]EFRU [>[[YVYH EW aQ??`Y€b
RUE F=?W== ~Z;
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G=UAY==? ]]?TY]SA]F GR_ >]^F]E ]]? FQQ?QEA\E X€YA== G=UAY==? RY?VF F=b
Z==?Y\S F=?RY=E F=Z==?=Y SV} QUYSQEQ; DEV [>[[YVYHRUS S=?S=F\E HXYA
== =
=
;
η @ E Z@ ?S[U FVZ}RSAVF[[ERU H[[^?RUE `?]EFRU AR@`?@RUS ^T [>W`
s
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=
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n j=1
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V?V^ η @=E=Z@=?S[U FVZ}RSAVF[[ERU HX?_RYH\E XHSXXA FQQ?QEAQQ [Y QSHb
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FVA FVAVE G[YSVV@ HQSHQ} G=U^=Y H[[ERU `?]EFRU AR@`?@RUS A=?==F
FQ·? ERUYGV?H H=^W} GQYEQ;
2
2
Sη2 = M M (η|ξ = x) − M η + M η − M (η|ξ = x)
‘’——œ›²Õ”™Š ԔŠÂ”›Žšš
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HQ^TQQ?
£ž7;77¥
δ ½EW G[YVS FQQ?QEA\E AR@`?@ G]S]]A ξ F[TRE >[UY η ½A E]Y]]Y]F E]Y]]Yb
YRUS RYV?FRUYEV;
1X
£ž7;7 ¥
δ =
(y − Y ) n
n
;Ÿ
=
σ ½[YAVSAVY AR@`?@ YAVSAVY AR@`?@ EW G[YSRUE AR@`?@[[ARUE AXEA }
G]S]]A HQQQSAQQS[U GX@=A F[TRE >[UY[[ARUE E]Y]]YYRUS RYV?FRUYEV;
2
Sη2 = δ η|ξ + σ 2η|ξ
2
η|ξ
k
2
η|ξ
2
i
i
i=1
2
η|ξ
k
1P
2
S iη ni
n i=1
s
1 P
2
(yj − y i )2 nij , (i = 1, k)
S iη =
ni j=1
v
u 2
uδ
η|ξ
τηξ = t 2
Sη
£ž7;7¡¥
σ 2η|ξ =
£ž7;7­¥
FVZ}RSAVF[[ERUS HX?_RYH\E GX~X H[[^?RUE aQ??`Y€RUE F=?W== £η½RUE ξ
AVV?F¥ SV} EV?YVVAB ξ G= η FVZ}RSAVF[[ERU F=?RY=E F=Z==?Y\E E€SH?=Y\S
RYV?FRUYVF [>[[YVYH GQYSQE =^A=S;
= =
τ FVZ}RSAVF[[ERU Q?QEA τ FVZ}RSAVF[[ERUS FV?VSYV} > E_@ E G]S]]A
=
;
H[[ERUS A`H`?ZRE RUE aQVÅÅRR`EH SVEV
M=E=Z@=?S[U FVZ}RSAVF[[E[[ARUE F=?RY=E F=Z==?=Y FRTEVVE F[THVU G=UF
HXH=Z τ F=?W== RF G=UE=; DEV EW ξ @=E=Z@=?S[U FVZ}RSAVF[[ERU ]]?TY]YHB
HQQQSAQQS[U [YA@VE @=E=Z@=?S[U F[TRE >[UY[[ARUS GQA^QY RY[[ RFVV? η
@=E=Z@=?S[U FVZ}RSAVF[[ERU ]]?TY]YH]EA E]Y]]Y]FRUS RYV?FRUYEV ;
Ÿ[ETYVEB
q
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τ = δ /S
SV@VE aQ??`Y€RUE F=?W==S =^T [>EV; NQ??`Y€RUE τ , τ F=?W==E\ T=b
E=?XXA\S AX?A^=Y2
 NQ??`Y€ F=?W== @]?]S GR_ G=UE=; £0 ≤ τ ≤ 1¥
¦ τ = 0 G=UF >=UY_S[U G]S]]A F[?VYVVHVU
E]F]Y EW y = Y = const G=UF
€^A=Y ~Z;£PX?=S ž7;¬¥
¢]?]]? FVYGVYB η½RUE ξ AVV?F ?`S?`@b
η
@RUE _XS=Z H=?F=YH\E H]^RUS A=U?b
@=E FV^HVV _XS=Z G=UE= SV@VE [S; DEV
τ =0
HQFRQYAQYA ξ, η FVZ}RSAVF[[E[[A
aQ??`Y€ F=Z==?=YS[U G=UE=; τ ½
aQVÅÅRR`EHRUE FX^WA Z]E RUZ T=b
0
ξ
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ηξ
ηξ
ηξ
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° τ = 1 G=UF >=UY_S[U G]S]]A F[?VYVVHVU E]F]Y EW σ = 0 G=UF
€^A=Y ~Z; ¢]?]]? FVYGVYB @=E=Z@=?S[U F[TRE >[UY[[ARUE E]Y]] G=UFS[U
G]S]]A G[F XHSXXA EW ?`S?`@@RUE ZX?XUE Q?TRZA H]^Y]?@]E G=UE= SV@VE [S;
2
η|ξ
ηξ
η
η
τηξ = 1
0
0 ≤ τηξ ≤ 1
0
ξ
ξ
PX?=S ž7;ž82
PX?=S ž7;žž2
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~Z; £aQ??`Y€RUE aQVÅÅRR`EHRUE FX^WA r = r = r ;¥ ¢]?]]? FVYGVYB
aQ??`Y€RUE F=?W==S GQAQFQA €Z=? FX^W@=STRUS [Y F=Z==?=F FX^W@=STB €Zb
?\S EW F=Z==?=E FX^W@=ST GQYSQE @QESQE =^=F==@ F=Z==?E= SV@VE [S;
KX?_RYH\E aQ??`Y€RUE F=?W== τ EW y XHSXXA\S FQYGQ@QE ?`S?`@@RUE
_XS=Z==@ £FXS=?F=U _XS=Z G=UE=¥ aQ??`Y€RUE Q?E\ VS[[ARUE @=?ERF @=?ERb
Y\S [>[[Y@VE [>[[YVYH ~Z; ¯V^T @=E=Z@=?S[U F[TRE >[UYRUE E]Y]]YY]]@
GQY} y XHSXXA\E ]]?TY]YHRUE FXXYW EW >]?TRSA]F XT?==@ τ F=?W== RFV@b
AVS; Ÿ[EHVU X€YA=E QEQY\E aQ??`Y€RUE F=?W== GX~X η½RUE ξ AVV?F aQ?b
?`Y€RUE REA`a@ R SV@VE [>[[YVYHRUS =^T [>AVS;
NQ??`Y€RUE REA`a@ EW aQ??`Y€RUE Q?E\ VS[[A }REFVEV ?`S?`@@RUE
_XS=Z y ½VV@ @=?ERF @R=?ERY\S HQAQ?FQUYEQ2
ηξ
ξη
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ηξ
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ηξ
x
s
=
s
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Ÿ[EAB δ G= σ AR@`?@[[A EW £ž7;7 ¥B £ž7;7¡¥ HQZW·QSQQ? GQAQSAQF GQYQ^T
G[YSRUE AXEA}XXA y ½S ?`S?`@@RUE HVS_RHSVYVV? QY@QE E]F]YH Z=H`Z=HRa
AXEA=} y ½VV? @QYREQ; Ÿ[EHVU H]@HVUSVV?
Rηξ =
2
η|ξ
2
δη|ξ
Sη2
1−
2
ση|ξ
Sη2
2
η|ξ
i
x
Rξη =
s
2
δξ|η
Sξ2
1−
2
σξ|η
Sξ2
£ž7;79¥
SV@VE aQ??`Y€RUE REA`a@RUS =^T [>EV;
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HVU F=?W=ESXUS==? ]EA]? S=?=F GQYQ^T H[[ERUS QYQFQA ?`S?`@@RUE HVS_RHb
SVYRUS ZVAVF _==?AY=S=S[U XT?==@ A=^XX H=YH=U ~Z;
=
= == = == ==
;
r G τ, R [>[[YVYH[[A A ? F F ?W S ? FQYGQSAQEQ
£ž7;7¬¥
0 ≤ |r| ≤ R ≤ τ ≤ 1
‘’——œ›²Õ”™Š ԔŠÂ”›Žšš
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X^W@=STA\E F=?RY=E F=Z==?=Y _XS=Z=E [`A R = R = |r| G=UE=; NQ?½
¼?`Y€RUE
F=?== τ ½ RUE RHSVYHVU V@VFRUS _=YS=F=A A=?==F _RE}[[?RUS
=_RSY=E=;
τ (n − m)
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F =
(1 − τ )(m − 1)
Ÿ[EA 2 m½@=E=Z@=?S[U FVZ}RSAVF[[ERU
XHSXXA\S G[YVSYV@VE REH`?^=Y\E HQQ;
=
Ì
F EW k = m − 1 G k = n − m T]Y]]ERU >V?S[[A G[FRU R_`?RUE F ½
=
=
=
=
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GVYB τ ½S H[[^?RUE aQ??`Y€RUE F=?W==S RYV?FRUYVF RHSVYHVU aQVÅÅRR`EH
SV} [>EV; F ½H=GYR==@ QY@QE QEQY\E XHS=;
Q·? FX^W@=STRUE aQ??`Y€RUE REA`a@ R½RUE RHSVYHVU T=E=?\S A=?==F _REb
¼}[[?RUS
=_RSY=E _=YS=E=;
R (n − 2)
£ž7; ž¥
F =
1−R
T]Y]]ERU
>V?S[[A G[FRU ÌR_`?RUE F ½
FVZ}RSAVF[[E
B
F ½
k = 1 k = n−2
O
=
=
=
=
=;
H ?F YHH U G UE ]E F > F
E]F]Y GR`YVSA^VY R½RUS RHSVYHVU SV}
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=
= =;
τ G aQ??`Y€RUE REA`a@ R ½S QY} HVASVV?RUE RHSVYHVU V@VFRUS _ YS
DFYVVA ;R_VV½ ž7;7½A QY@QE [? A[ES[[AVV? Y , S ½S HX@ HX@ QY¶·;
ξη
ηξ
2
τ
τ
1
2
2
α,k1 ,k2
α,k1 ,k2
2
R
R
1
2
2
R
α,k1 ,k2
η,ξ
ηξ
2
η
H
<[YSRUE AXEA=} G= REH`?^=Y\E A=^H=Z}XXA\S Q?QYXXY=E HQQQQS FRUFAVV
A=?==F F[@EVSHRUS =_RSY=F EW HQFR?QZ}HQU;
Y =
846
15226
= 16.92( ), Sη2 =
− 16.922 ≈ 18, 23.
50
50
77«;­;­
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7«;­
¡7;­
xi
y
ž8;
ž;
ž«;9
78;;8
7
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ni
ž
7žžž
7
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£ž7;7 ¥ HQZW·Q ·@QQ?2 δ
y
(y − Y ) n
ž8;¡
ž «;­
ž ;9 ž77®;­
ž«;7
ž;®
78;®;¬ ž¡¬;8
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7
­87;8
½
517.8
ž;­
= ==
= 10.36 HXY £ 7 7 ¥ HQZW·QS _RSY ^ Y
=
i
2
η
ž ž;­
ž«8;¡
ž®;
ž7­;«
« ;¬
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(y i − Y )2 ni
50
τηξ =
r
x
10.36
= 0.754
18.23
x
2
i
NQ??`Y€RUE F=?W== τ ½RUE XHS= EW ]ZE]F >[UYA QY@QE r = 0.74½HVU QU?QYb
QQ S=?T G=UE=; DEV EW @=E=Z@=?S[U FVZ}RSAVF[[E[[A _XS=Z=E aQ??`Y€
ηξ
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F=Z==?=YH=U G=UF HXF=U AV^_[[Y@VE H==Z=SY=Y\S EQHQY} GXU ~Z;
D
R ½REA`a@RUS QYQF\E HXYA 2 y = 0.6762x − 4.79 £ EVF[[ ?`S?`@@RUE HVS_RHb
SVYRUS ž ;7½A S=?S=} [>[[YEV¥ SV@VE ?`S?`@@RUE HVS_RHSVYVV? y XHSXXA\S
QYEQ; DASVV? XHSXXA\S ]ZE]F F[@EVSHRUE @[[YVV@VV 7 A=FW G=S=E=EA
G=U?YXXY@=E;
ηξ
x
xi
2
δη|ξ
502.0
=
= 10.04,
50
Rηξ =
r
10.04
= 0.742
18.23
P=^@?\E [? A[ES[[ARUS GQAQFQA S=?@=E G[A[[^TYVYRUE =YA==S V@ HQQ^QY R
G= r EW G=?=S R}RY S=?T G=UE=; LUZA FX^W@=STXXA _XS=Z=E aQ??`Y€ F=Zb
==?=YH=U [`A R REA`a@RUS GQAQF _==?AY=S=S[UB >]^F]E r aQVÅÅRR`EHRUS
GQAQFQA F=ES=YHH=U GQYQF EW F=?=SA=} G=UE=; ¤`H`?ZRE=RUE aQVÅÅRR`EH
= ==
R = 0.551 G US EWB FQEQSH [UYA^V?YVF G[HVVSAVF[[E £Y ¥½RUE ]]?TY]YHRUE
= = = =
55.1% EW [UYA^V?YVYRUE [EA@VE ÅQEA £X ¥½RUE ]]?TY]YH]]? H UYG ?Y SA F\S
F=?XXY} G=UE=; <[YVSYV@VE REH`?^=Y\E HQQ m = 5 GQYQF\S =EF==?=E τ
aQVÅÅRR`EHRUE RHSVYHVU V@VFRUS _=YS=€;
ηξ
ηξ
2
ηξ
ηξ
Fτ =
0.7542 (50 − 5)
= 14.82
(1 − 0.754)2 (5 − 1)
LHSVF H[^_RE α = 0.05 G= k = 4, k = 45 [`A ÅR_`?RUE F ½H=?F=YH\E
;
H=GYR==@ F
= 2.57 SV} QYAQEQ
XTR? τ ½S RHSVYHVU SV} HQQEQ; Ÿ[ETYVE R aQVÅÅRR`EHRUS
F >F
_=YS=€;
1
2
0.05;4;45
τ
0.05;4;45
ηξ
FR =
ηξ
0.7422 (50 − 2)
= 58.8,
(1 − 0.742)2
HXY aQ??`Y€RUE REA`a@ R
ηξ
= 0.742
FR > F0.05,1,48 = 4.04
EW RHSVYHVUA HQQQSAQEQ;
þ L ú 1 ' - q - - & ' % 1 [ s u v _ u u 1 t - - 2
' %1 [ s u v % 7 u v / v &'-/ / u 9
ê
su% 3ZWA?=Y ?=aHRaRUE [>VSAY[[A RFV^TYVE QYQE F[TRE >[UYH >=S^=?==? A[?@½
YVSAAVS; LUZ XT?==@ ]ZE] =^T [>@VE FQ·? FVZ}VV@H aQ??`Y€RUE >=S^=?\S
QYQE FVZ}VV@H HQFRQYAQYA ]?S]HS]F _==?AY=S=H=U; ÖYQE FVZ}VV@H aQ?b
?`Y€RUE _RE}RYSVVERU >Q?RYSQ EW X , X , . . . , X SV@VE £n > 2¥ FX^W@=F
FVZ}RSAVF[[E FQQ?QEA\E F=?RY=E F=Z==?Y\S @XA=Y}B VASVV?VV@ F=Z==?@=E
£Y ¥½FVZ}RSAVF[[EA F=ZSRUE RFVV? E]Y]]Y]F X FX^W@=STA\S @QESQE =^=F=A
Q?_REQ;
= = = == =
X , X , . . . , X , . . . , X ½SV@VE QYQE FVZ}VV@H FV^RUE H ?F YHH U @ E Z@ ?S[U
FVZ}RSAVF[[E[[A ]S]SA@]E G=US; DASVV? @=E=Z@=?S[U FVZ}RSAVF[[ERU F=½
?RY=E F=Z==?Y\S FQ@ aQ??`Y€RUE aQVÅÅRR`EH[[AVV? >QFRQSA@QEB A=?==F
1
2
n
i
1
2
j
m
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...
...
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ρ2m 

... 
1
...
...
...
...

r1m
r2m 

... 
1
1
ρ12
 ρ21 1
Qm = 
 ... ...
ρm1 ρm2
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rm = 
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rm1 rm2
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n
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FVZ}RSAVF[[EVV@ F=Z==?=F F=?RY=E F=Z==?Y\S HQSHQQF ;
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xk
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Ri.12....m =
1−
|rm |
qii
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k
m
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i
s
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i.12...m
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1
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r −r −r
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ij
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ij
ij.k
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j
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=ARY==? −1 ≤ r
= = =; K[[ETYVE n FVZ}VVH H[[^?RUE XHS==?
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RUE HXF=UE aQVÅÅRR`EHRUE RHSVYHVU T=E=? EW HX?_RYH\E HQQ n − m + 2
G=UF [`RUE FQ@ aQ??`Y€RUE aQVÅÅRR`EHRUE =ARY==? [EVYVSAVEV;
#”Ôšš$ ¦» ¼]A]YZ]?RUE G[HVVZ} £X ¥B =}RYTA\E E=@E\ FVZ}VV
£X ¥B [UYA^V?YVYA =}RYY=@=E }RYRUE £X ¥ F=?RY=E F=Z==?Y\S @XAY=F\E
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FQ@ aQ??`Y€RUE aQVÅÅRR`EH[[A r = 0.20, r = 0.41, r = 0.82½\S
QY} RHSVYHVU aQVÅÅRR`EHXXA SV} HQSHQQ}VV; KVS^VY aQ??`Y€RUE `?]EFRU
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£ž7; ®¥ HQZ¶·QS =_RSY=^=Y2
rst
ij.12...m
1
2
3
12
13
23
1.23
R1.23 =
s
0.282 + 0.412 − 2 · 0.20 · 0.41 · 0.82
= 0.47
1 − 0.822
Ÿ[EVV@ [>^VYB EVS H=Y==@ F]A]YZ]?RUE G[HVVZ}B E]S]] H=Y==@ =}RYTA\E
E=@E\ FVZ}VV G= =}RYY=@=E }RY FQ·?\E FQQ?QEA ZVAVSAVF[U aQ??`Y€ F=b
Z==?=Y G=US== EW RYV?T G=UE=;
¤`H`?ZRE=RUE `?]EFRU aQVÅÅRR`EH R = 0.225 G=US== EW =}RYTA\E
F]A]YZ]?RUE G[HVVZ}RUE ]]?TY]YHRUE 22.5% EW HVAERU E=@ G= =}RYY=@=E
}RYHVU FQYGQQHQUSQQ? H=UYG=?Y=SA=E= SV@VE [S; ÖAQQ R ½RUE RHSVYHVU
V@VFRUS _=YS=€;
2
1.23
1.23
0.472 (100 − 3)
= 14.1
(1 − 0.472 )(3 − 1)
= 3.09
F > F0.05;2;97
F =
K=GYR==@ F
SV} QYAQF G=
ZVAVSAVF[U €YS=SA=E=;
ÖAQQ aQ??`Y€RUE HXF=UE aQVÅÅRR`EHXXA\S GQAW·;
0.05;2;97
r12.3 = p
r12 − r13 · r23
2
2
(1−r13
)(1−r23
)
= 0.44, r23.1 = 0, 83.
=p
0.20−0.41 · 0.82
(1−0.412 )(1−0.822 )
Ÿ[ETYVE r
= =;
r ½aQVÅÅRR`EH RHSVYHVU V@VFRUS _ YS €
13.3
12.3
XTR?
R1.23
= −0.26
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n − m + 2 = 100 − 3 + 2 = 99 XTR? £ 7 ¥ _RE}[[? ·@QQ?
√
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t= √
1 − 0.262
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;
XTR? aQ?b
= 1.99 SV} QYAQEQ |t| > t
=
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HXY Z]E r , r aQVÅÅRR`EH[[A RHSVYHVU; DEV G[FEVV@ [>^VYB
r
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£r = 0.20; r = −0.26¥;
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=YRUS EW T _=YHS==E GQYSQE =^T GQYEQ; KVS^VYB F[ERU GR`RUE }RE G= ]EA?RUE
F=Z==?Y\E FX^WA =YRUS EW T _=YHS==E GQYSQE @QESQE =^T GQYQFS[U; P=?RZ
HQFRQYAQYA FX^W@=ST FQQ?QEA\E HQQE F=?W==E AVV? HXYSXX?Y=@=E V^V? ÅQ?b
Z=YW F=Z==?=Y G=US== GQYQ^T GQARH XHS==?== €=} T H=UYG=?Y=SA=FS[U FXX?½
Z=S aQ??`Y€ G=UA=S; wVS [SVV? FVYGVYB HQQE [EVYVYHVV? FVZ}RSAVFS[U F=b
Z==?=Y Q?_RE G=UE=; ¼V@VS @X?=STA\E Z=H`Z=HRa @VHSVYSVV G= F]S}ZRUE
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@=STRA EW QYQE FVZ}VV@H FV^RUE H=?F=YHH=U V@VFRUS _=YS=F `?]EFRU _REb
}[[? ?=aHRaH G=UFS[U; LUZA QEQY\E >=S^=?\EF== _RE} T=E=?B FX^W@=STb
A\E EVS FVZ}VV@H H=?F=YHXXA FV^RUE H=?F=YH=EA F=?_Y=FS[U G=UF T=E=?B
FQ·? FVZ}VV@H aQ??`Y€RUE Q?E\ VS[[A EW Z]E FQ·? FVZ}VV@H FV^RUE H=?½
0.05;97
0.05;97
12.3
12.3
12
0
13.2
12.3
12.3
23.1
13.2
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F=YHH=U G=U} GQYQF HXF=U _=YS=@=E H==Z=SY=YXXA AVV?VV HXYSXX?Y=E QYQE
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KXF=UYG=YB =^T [>V} GXU FX^W@=F FVZ}RSAVF[[ERU =YW EVS EW @=E=Z@=?S[U
GR_ V@^VYB ?`S?`@@RUE ZX?XU _XS=Z=E GR_ SVF ZVH G=U} GQYEQ; DASVV?
HQFRQYA\E [`A HQQE ]S]SAY[[AVA F=ZSRUE @=UE A]F]F ZX?XUS £S=A=?SXXS¥
G=USXXY=F _==?AY=S=H=U G]S]]A RUZ =?SXXA\S ?`S?`@@RUE _RE}YVYA =^T
[>EV; C`S?`@@RUE _RE}RYSVVERU [EA@VE GQAYQSQ EW2
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¼ÅXEaRUE
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@=E E]S]Y GR`YVSA@VE E]F]YA ?`S?`@@RUE _RE}YVY FRUEV;
 ¼=Z==?=E FX^W@=ST ý½EW HQSHZQY AR@`?@HVUB F=Z==?=YS[U @=E=Z@=?S[U
FVZ}RSAVF[[E[[A G=UF ;
¦ ŸY F=Z==?=F FX^W@=ST ü EW @=E=Z@=?S[U GR_ FVZ}RSAVF[[E[[A G=UF ;
° w]F]YH Z=H`Z=HRa AXEA=} M ( ý/ ü = þ )½RUS M ( ý/ ü = þ ) = ϕ(þ )
FVYGV?HVU GRTR} GQYAQS G=UF ;
DEV G[YVSH >]^F]E _XS=Z=E ?`S?`@@RUE _RE}RYSVVS =^T [>EV; ¢]?]]? FVYb
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EW _XS=Z=E GR_VV? Q?QYQ} GQYQF ϕ( þ , β) FVYGV?RUE F=Z==?=Y\S =^T [>EV;
KXF=UYG=Y2
M (Y /X = x) = β0 + β1 x, M (Y /X = x) = β0 + β1 x + β2 x2 ,
M (Y /X = x) = β0 +
β1
, M (Y /X = x) = β0 + β1 x + β2 x2 + β3 x3
x
לŽ—œºº”™Š ԔŠÂ”›Žšš
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RYV?FRUY} GQYAQS; ££ž ;¬¥½HQZ¶·QS [>¥
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x
yx
y
yx
לŽ—œºº”™Š ԔŠÂ”›Žšš
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M=E=Z@=?S[U FVZ}RSAVF[[ERU =}RSY=YH\E XHSXXA £x , y ¥½VV? b
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i
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£ž ;ž¡¥
s
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xi yj nij − X · Y
=
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£ž ;ž ¥
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n j=1
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; ž ;¬¥B £ž ;ž8¥ HQZW·QEQQ@ r = b · b GX~X
S , S −H[[^?RUE AR@`?@[[A £
p
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V?V^ aQ??`Y€RUE aQVÅÅRR`EH r½RHSVYHVU GQY b , b [EVYVYH[[A RHb
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£ž ;ž¬¥
£ž ;78¥
MHW~A`EHRUE H=?F=YH\E H=GYR\E XHS=;
#”Ôšš$ ° ;R_VV½ ž7;7½A =^T [>@VEB [UYA^V?YVYRUE [EA@VE ÅQEA½X B
FQEQSH [UYA^V?YVF G[HVVSAVF[[ERU FVZ}VV Y ½RUE ]S]SAY]]? HVASVV?RUE ?`S½
?`@@RUE HVS_RHSVY G= ?`S?`@@RUE aQVÅÅRR`EHXXA\S QY} [EVYVYHRUS G=U½
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XT?==@
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byx =
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50
14.768
14.768
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= 0.8099.
21.84
18.2336
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ž @=€ H]S?]S]]? EVZVSA[[YGVY FQEQSH [UYA^V?YVF G[HVVSAVF[[ERU FVZ}VV
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x
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p
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yx
q
√
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£ž ;ž¬¥B £ž ;78¥ HQZ¶·Q ·@QQ?B RHSVF >=^@?XXA
GQYEQ;
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£ž ;7ž¥
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α = 0.05, k = n = 48
t0.05;48 = 2.01
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ϕ(xi )
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k
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£ž ;7«¥
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P
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yi −
i=1
KXF=UE HQFRQYAQYA
k=0
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GQY EQ?Z=YW @R@H`Z A=?==F FVYGV?HVU GQYEQ;
n
P
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n
n
n
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Ÿ[EA 2 S − y XHS\E @H=EA=?H =YA==;
#”Ôšš$ °¸ ;R_VV½ ž7;7½\E ]S]SAY]]? [UYA^V?YVYRUE [EA@VE ÅQEA EW
® £@=€ H]S?]S¥
G=UF [UYA^V?[[ARUE FQEQSH [UYA^V?YVF G[HVVSAVF[[ERU AXEb
A=} FVZ}VV GQYQE HX@S=UY=E @QESQE =^@=E S=E==?TRY@=E XHS\E RHSVF >=^@½
?\S HX@ HX@ G=USXXY;
<RA [[EVV@ ]ZE]F }R_VVE[[AVA
x
0
0
0
0
2
y0
2
0
0
y0
X = 32.1,
2
x
y0
x
2
0
2
2
y0
α;k
∗
0
x
y0
α;k
0
Sx2 = 21.84, y x = 0.6762 · x − 4.79,
n
P
e2i =
n
P
(y xi − yi )2 = 502.0
SV@VE [? A[ES[[ARUS EVSVEH S=?S=E =^@=E GRYVV; ÖAQQ [YAVSAVY AR@`?@RUS
QY¶·; S˜ˆ” = 502.0 = 10.46. £ž ;¡®¥ HQZ¶·Q ·@QQ? E]F]YH AXEA}RUE AR@b
`?@RUS QYGQY502 − 2
i=1
i=1
2
Sy20
1
(36 − 32)2
= 10.46
= 0.355, Syx0 = 0.596.
+
50
50 · 21.84
לŽ—œºº”™Š ԔŠÂ”›Žšš
»Ð
C`S?`@@RUE HVS_RHSVY ·@QQ? y = 0.6762 − 36 − 4.79 = 19.55.
£ž ;¡9¥ HQZ¶·QSQQ? E]F]YH AXEA}RUE RHSVF >=^@?\S QYGQY2
;
18.35 ≤ M (Y /X = 36) ≤ 20.75, £ t
= 2.01 ¥ GQYEQ
¢]?]]? FVYGVYB [UYA^V?YVYRUE [EA@VE ÅQEA EW ® £@=€ H]S?]S¥ G=UF [UYAb
^V?[[ARUE FQEQSH [UYA^V?YVF G[HVVSAVF[[ERU AXEA=} FVZ}VV 13.35÷20.75(H)
G=UF\S 8;¬­ SV@VE RHSVF Z=S=AY=YH=US==? FVY} GQYEQ; ÖAQQ HX@S=UY=E @QESQb
@QE >]^F]E EVS [UYA^V?RUE FX^WA AVV?F >=^@?\S G=USXXY¶€; (y )
x0
0.05;48
∗
36
(36 − 32)2
1
∗ = 3.29.
+
= 10.46 1 +
= 10.81, Sy36
50
50 · 21.84
∗
≤ 26.16( )
12.94 ≤ y36
Sy236
∗
LUZA
H
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X
X
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£ž ;­ž¥
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;
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P
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=
;
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P
== =
Q =
(y − y ) EW HQQQSAQQS[U [YA@VE @ E Z@ ?S[U F[TRE >[UY[[ARUE
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Q (n − 2)
£ž ;­7¥
F =
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0
1
1
2
1
n
n
i
i=1
xi
n
1
i=1
n
2
i=1
2
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i
xi
2
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2
xi
i
2
2
n
xi
2
0
1
Q2
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»
_RE}[[?RUS =_RSY=E=; DEV _RE}[[? k = 1, k = n − 2 T]Y]]ERU >V?S[[A
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E]F]Y GR`YVSA^VY
==
=
=
= =
H H Z SY Y\S [S[U@SV} ?`S?`@@RUE HVS_RHSVYRUE RYV?FRUYVF T A^ ?\S
[EVZ_RYHVUA HQQEQ; F ½ÌR_`?RUE H=?F=YH\E H=GYR\E XHS=;
¼V?V^ ?`S?`@@RUE HVS_RHSVY k =?=Z`H?RUS =SXXY@=E G=U^=Y2
Q (n − k)
£ž ;­ ¥
F =
Q (k − 1)
=
=
=;
D
_RE}[[?RUS _RSY E EV _RE}[[? £ž ;­7¥½\E =ARY H=?F=YHH=U G=UE=;
#”Ôšš$ °» ;R_VV½ ž7;7½H =^T [>@VE FQEQSH [UYA^V?YVF G[HVVSAVF[[ERU
FVZ}VV £Y ¥ [UYA^V?YVYRUE [EA@VE ÅQEA £X ¥½QQ@ F=Z==?=F ?`S?`@@RUE HVS_RHb
SVYRUE RYV?FRUYVF T=A^=?\S _=YS=;
1
2
α;k1 ;k2
0
α;k1 ;k2
1
2
n
P
Q1 =
i=1
(yxi − Y )2 = 502.0
n 2
P
1
yi = 911.68
Q = (yi − Y ) =
− ·
n
i=1
i=1
i=1
Q2 = Q − Q1 = 911.68 − 502.0 = 409.68
n
P
2
n
P
yi2
£ž ;­7¥ _RE}[[?RUE XHS\S GQA^QY F = 502(50 − 2) = 58.8 LHSVF H[^_RE
409.68XTR? ;R_VV ž ;ž A S=?S=E
½ ½
α = 0.05½\E FX^WA F
= 4.04 Gè F > F
=^@=E ?`S?`@@RUE HVS_RHSVY
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0.05;48
0.05;1;48
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£ž ;­¡¥
y = β x + β x + β x + ··· + β x + ε
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Q?XXY¶€2
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ü
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... 
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n · (q + 1)
...
...
...
...
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ü −[Y F=Z==?=F FX^W@=STA\E
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£ž ;­­¥
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£ž ;­®¥
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
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 e2 
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

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£ž ;®8¥
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£ž ;® ¥
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1
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£ž ;«­¥
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==
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F =
£ž ;««¥
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Q2 · q
EW £ž ;­ž¥ HQZ¶·QSQQ? HQAQ?FQUYQSAQF G= F > F
(k = q, k =
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=
;
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=ERa}RYH\E H[^_RE½X £%¥G VVY}RUE FXS===EA EVS =}RYTRE\ QYGQ?YQF
E[[?@ERU FVZ}VV½Y £H¥;
Q1 , Q2
n−q−1)
α;k1 ;k2
1
2
1
2
¡¬
97
9­
9
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ž88
­
97
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®­
žž x9¡ x9¡ y­;­
ž7 ¬¡ ®¡ ®;ž
ž žž 9ž 9;«
ž¡ ž¡ž ž88 žž;®
ž­ 99 « ¡;­
ž® ¬9 97 ¬;®
ž« 97 ¡¬ ­;­
ž9 9 « ®;9
ž¬ žž­ ®« «;9
78 ž8­ 9¡ ž8;­
=¥ Y = Y (X , X ) F=Z==?Y\E =E=YRHRa RYV?FRUYYRUS QY;
G¥ C`S?`@@RUE aQVÅÅRR`EHXXA\E RHSVYHVU T=E=? G= ?`S?`@@RUE HVS_RHb
SVYRUE RYV?FRUYVF T=A^=?\S _=YS=;
^¥ ¤=^F=?S\E >X>==E EW ž888 ZB Z`F=ERa}RYH\E H[^_RE EW ®8% G=UF XX?F=U½
EXXA\E FX^WA ž =}RYTE\ VVY}REAVV QYGQ?YQF E[[?@ERU FVZ}VVERU AXEA=}
G= S=E==?TRY@=E XHS\E ¬­%½RUE RHSVF >=^@?\S HX@ HX@ G=USXXY;
¤=?==F Z=H?R=E HVZAVSYVSVVS Q?XXY¶€;
1
i
ž
7
¡
­
®
«
9
¬
ž8
xi1
«®
ž89
žž®
¬9
¬ž
ž­
9­7
žž
«®
ž8ž
xi2
1
1
1
...
1
1
76
108
116
...
115
105
49
82
85
...
67
84
­;8
ž8;
¬;«
«;­
®;­
¬;«
®;7
ž8;¬
­;7
«;8
yi
i
i1
i2
i
2










5.0
10.3
9.7






 , Y =






 7.8
10.5
;;





.



KVS^VY2
לŽ—œºº”™Š ԔŠÂ”›Žšš
»Æ

1 1
1 ...
X T · X =  76 108 116 . . .
49 82 85 . . .


 

1
1

115 105  · 


67 84

1
1
1
...
1
1

20
1992
1380
 1992 204130 142480  ,
1380 142480 103232

5.0
10.3

 

1 1
1 ...
1
1
 9.7

X T · Y =  76 108 116 . . . 115 105  · 

49 82 85 . . . 67 84

 7.8
10.5
;;
det(X T X) = 411214952

XTR? (X
T
X)−1
76
108
116
...
115
105
49
82
85
...
67
84




=




 


154.6

 
 = 16023.3  .

11322.8


Z=H?R Q?_REQ;

1.877
−2.192 · 10−2
5.157 · 10−3
3.896 · 10−4 −2.447 · 10−4  .
(X T X)−1 =  −2.192 · 10−2
−3
5.157 · 10
−2.447 · 10−4
2.785 · 10−4


−2.595
b =  0.08319  .
0.02955
£ž ;®ž¥ ·@QQ?2
KVS^VY QYQE FVZ}VV@H ?`S?`@@RUE HVS_RHSVY
FVYGV?HVU GQYEQ;
DEV HVS_RHSVYB >]^F]E A=^F=?S\E >X>==E ž Z½VV? EVZVSAVFVA EVS =}RYTE\
VVY}REAVV QYGQ?YQF E[[?@ERU FVZ}VV AXEA}==? 8;89 ž¬ HQEEQQ?B >]^F]E Z`½
F=ERa}RYH\E H[^_RE ž ?Q`EHQQ? EVZVSAVFVA QYGQ?YQYH 8;87¬­­ HQEEQQ?
EVZVSAVFRUS HX@ HX@ [>[[YEV;
£ž ;«ž¥ G= £ž ;«7¥ HQZ¶·QSQQ? ?`S?`@@RUE @H=EA=?HTRY=SA@=E aQVÅÅRR`EH[[A
G= ZVA?VZ}RUE aQVÅÅRR`EHRUS QY¶·;
y = −2.595 + 0.08319 · x1 + 0.02955 · x2
X 1 = 99.6, X 2 = 69.0, y = 7.73, Sx1 = 75.68, Sx2 = 89.51, Sy = 9.68,
89.51
75.68
= 0.65, b02 = 0.02955
= 0.273
b01 = 0.08319
9.68
9.68
99.6
69.0
ϑ1 = 0.08319
= 1.071, ϑ2 = 0.02955
= 0.264.
7.73
7.73
DASVV? aQVÅÅRR`EH[[ARUE XHS\S H=UYG=?Y=^=Y2 A=^F=?S\E >X>==EB Z`F=ERa½
}RYH\E H[^_RE F=?S=Y>=E EVS EVS S , S EVS}VV? EVZVSAVFVA EVS =}RYTE\
VVY}REAVV QYGQ?YQF E[[?@ERU AXEA=} FVZ}VV F=?S=Y>=E 0.65 · S , 0.273 · S
FVZ}VVSVV?B F=?RE AVV?F FQ·? [>[[YVYH £AXEA=} XHS==@==¥ ž% EVZVSAVFVA
x1
x2
y
y
°È ù›’Š Œš‰Âššº– —œŽ—œºº”™Š ԔŠÂ”›Žšš
»ª
E[[?@ QYGQ?YQYH ž;8«ž% G= 8;7®¡%½R=? HX@ HX@ EVZVSAVEV SV@VE [S ~Z; DEAVV@
[>^VY âA=^F=?S\E >X>==E â SV@VE [>[[YVYH âZ`F=ERa}RYH\E H[^_RE â SV@VE
[>[[YVYHRUS GQA^QY E[[?@ QYGQ?YQF FVZ}VVEA RY[[ E]Y]] [>[[Y} G=UE=;
==
=
b , b , b  ? Z`H?[[ARUS QYQFAQQ (X X) Z H?R\E X?^XXS QYQFS[USVV?
=
==
;
A ? F @R@H`ZVV@ QY} GQYQF\S HVZAVSYV`
0
1
T
2
−1

20b0 + 1992b1 + 1380b2 = 154.6

1992b0 + 204130b1 + 142480b2 = 16023.3

1380b0 + 142480b1 + 103232b2 = 11322.8
G¥ ¤=?==F HX@Y=F H=GYR\S >QFRQ} HQQQQ FRU`;
(y − y )
«® x¡¬ y­;8 (y«;¡­−7¬Y ) y­;ž« e =8;8
89
ž89 97 ž8; ®;®8¡¬ 9;9ž
;7 7ž7
žž® 9­ ¬;« ;998¬ ¬;­«
8;8ž««
; ;; ;; ; ;;; ;;;
;;;
;;;
žž­ ®« «;9 8;88¡¬ 9;¬­
ž; 7®¡
ž8­ 9¡ ž8;­ «;®«7¬ 9;®7
;­7®
P
¬
;«8
ž
½ ½ ½
7 ½
77; ¬¡«
DFYVVA ?`S?`@@RUE HVS_RHSVYRUE RYV?FRUYVF T=A^=?\S _=YS=€;
K=GYR==@2 Q = 93.7021, Q = 22.3947, Q = Q−Q = 93.7021−22.3947 = 71.3074
£ž ;««¥ ·@QQ?2
ž
7
;;;
ž¬
78
i
xi1
i2
i
2
F =
2
i
1
71.3074(20 − 2 − 1)
= 27.1,
22.3947.2
2
i
xi
xi
i
2
2
F0.05;2;17 = 3.59, F > F0.05;2;17
XTR? S=?S=E =^@=E ?`S?`@@RUE HVS_RHSVY @XA=Y} GXU ?Q`@@RUS [EVZ_RYHVU½
SVV? RYV?FRUY} G=UE=;
ÖAQQ ?`S?`@@RUE b , b aQVÅÅRR`EHXXA\E RHSVYHVU T=E=?\S _=YS=€; £ž ;®®¥
·@QQ?2
22.3947
S˜ˆ” =
= 1.399, S˜ˆ” = 1.182,
20 − 2 − 1
ž ;®¬¥ HQZ¶·QS =_RSY=^=Y2
C = 3.896 · 10 , C = 2.785 · 10 HXY £
1
2
2
−4
11
tb1 =
22
−4
0.02955
0.08319
√
√
= 3.57 , tb2 =
= 1.5
1.182 3.896 · 10−4
1.182 2.785 · 10−4
; LUZA ­%½\E RHSVF H[^_ERU FX^WA b ½aQVÅÅRR`EH RHSVYHVU
GR_ S=?T G=UE=; DEV EW ]S]SA@]E H[[^?RUE FX^WA E[[?@ QYGQ?YQYH\E FVZ½
}VVEA Z`F=ERa}RYH\E H[^_RES ZVA?VF[U E]Y]]HVU SV} HQQQF [EAV@S[U
SV@VE [S;
LHSVYHVU GR_ aQVÅÅRR`EHRUS ?`S?`@@RUE HVS_RHSVYVV@ F=@=} ?`S?`@@RUE
=?=Z`H?XXA\S ]]E HQQE\ FX^W@=STA\E FX^WA A=FRE GQAAQS; O=E=U HQ½
FRQYAQYA X FX^W@=STRUS F=@=} A=FRE HQQQQ FRU^VY ?`S?`@@RUE HVS_RHSVY
t0.05;17 = 2.11
2
2
לŽ—œºº”™Š ԔŠÂ”›Žšš
»È
FVYGV?HVU GQYEQ; w]Y]] G=S=H=U FX^W@=STRUS F=@=Fb
==
=
=
=
A [EAV@YVY @ UH U T E=?\E @XA=YS==E AVV? HXYSXX?Y=^=Y >QFREQ; w]Y]]
G=S=H=U T HQAQ?FQU [EAV@YVYHVUSVV? H[[ERUS ?`S?`@@RUE HVS_RHSVYA [YAVV}
GQYEQ;
^¥ x = 100, x = 60 [`A £]]?]]? FVYGVYB X = (1, 100, 60)¥ F=Z==?=E
FX^W@=STRUE XHS= y ½RUS GQA¶·;
; ž ;«¡¥ ·@QQ?2
y = −2.595 + 0.008319 · 100 + 0.02955 · 60 = 7, 497 £H¥ £
y = −3.142 + 0.1091x1
1
T
0
2
x0
x0

X T0 (X T X)−1 X 0 = (1, 100, 60)·

 
1
−2.192 · 10−2
5.157 · 10−3
3.896 · 10−4 −2.447 · 10−4  ·  100  = 0.07404
60
−2.447 · 10−4
2.785 · 10−4
1.877
·  −2.192 · 10−2
5.157 · 10−3
√
= 1.82 0.07404 = 0.322
£H¥; £ž ;« ¥ HQZ¶·QSQQ? RHSVF >=^@?\S G=USXXYb
7.497 − 2.11 · 0.322 ≤ M (Y ) ≤ 7.497 + 2.11 · 0.322 GX~X
6.82 ≤ M (Y ) ≤ 8.18 (H)
£ž ;«®¥B £ž ;«­¥ HQZ¶·QSQQ? S=E==?TRY@=E XHS\E RHSVF >=^@?\S G=USXXYG=Y2
√
S = 1.182 1 + 0.7404 = 1.225 (H)
4.92 ≤ y ≤ 10.08 (H).
˜ˆ”
G=Y2
S
x0
x0
y0
∗
0
ö ÷ ø ù^
h × üü ‡‡
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GQYEQ;
þ5 þ H % q u & / #”Ôšš$ ¸
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( u & % ' u ' 1 ' 1 ,
N
X
xi p i .
i=1
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Ú ž 7
Ÿ? A[E2
8
ž
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x
8;7 8;¡ 8; 8;ž
p
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åƒóíu w = 50 G
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ìôuñ£zn = z¥GPñèî £ó¥G
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2
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X
i=1
(xi − M (ξ))2 pi .
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Ú ž 7
Ÿ? A[E2
8 ž 7
x
8;
8;¡
8;
8;ž
p
7
i
i
D(ξ) = 0.81
Ÿ? A[E2 D(ξ) = 0.81
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#”Ôšš$ ¸°
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;
k=2
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N
X
xki pi .
i=1
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Ú ž 7
Ÿ? A[E2
8 ž 7
x
8;7 8;¡ 8; 8;ž
p
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ìôuñõó£zÍôìíu½ ðƒðñóu= zBíòð2«2 ¥
{óî;
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νk = 2.5
i
þ5 5 & I- - q u v 7 u v q ' q % ( u & % ' u ' 1 ' 1 ,
µk =
N
X
(xi − M (ξ))k pi .
i=1
Æ
Æ¦
ؕº˜•›Ä ӗ’Ž—•‰‰µµ
#”Ôšš$ ¸¸
N = 4, k = 3
¡
Ú ž 7
Ÿ? A[E2
8 ž 7
x
8;
8;¡
8;
8;ž
7
p
i
i
Ÿ? A[E2 µ = 0.144
ìƒQìèð âɈÎZ ÑùÑZÉâG
åƒóíu =­8G
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êñQôó
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ñQôó
2=žG
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íòð2=íòð+ÙY lôm
ñóîG
ìôuñõó£zåñóuìèõ ðƒðñóu= zB íòð2«2 ¥
{óî;
k
µk = 0.144
¸» ‘’——œ›²Õ”™Š ˜’šii”Õ”œŠ–
þ5 K
ê
' %1 [s u v
Æ°
&'-/ / usu% N
P
yi /N
xi ·
i=1
i=1
s i=1
R= s
2
P
N
N
N
N
2
P 2
P
P
xi −
yi2 −
xi /N ·
yi /N
N
P
xi yi −
P
N
#”Ôšš$ ¸» N = 5
¡ ­
Ú ž 7
¡;ž ¡;¬ Ÿ? A[E2 R = 0.9987280191
8;¬­ 7;ž
x
¡;8­ ­;9 9;ž ¬;7
y 7
ìƒQìèð â‘ù××aÎZ ‘ùrÉÉÎÎZÉâG
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ì 2=í ìu £ £ B ¥½í 죣 ¥¥ÔÚ¥G
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ìôuñõó£zP= zBì2«2ž8¥
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i=1
i
i
i=1
i=1
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ؕº˜•›Ä ӗ’Ž—•‰‰µµ
þ5 L & I- - q u v 7 u v q ' q % ( 1 t v ' u ' 1 ' 1 ,
µk =
#”Ôšš$ ¸Æ
µ2 =
Zb
Zb
(x − M (ξ))k f (x)dx
a
k = 2, a = 0, b = 1, f (x) =
(x − M (ξ))2
(b − a)2
dx
=
,
b−a
12
ìƒQìèð âɈÎZ ÑùÑZÉâG
a
åƒóíu ñ=8;888žG
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Í2= Ô£ ñ½èõ¥
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24.3
25.3
26.3
27.3
28.3
15.5
16.3
17.2
18.1
19.0
19.9
20.8
21.7
22.7
23.6
12.4
13.2
14.0
14.8
15.7
16.5
17.3
18.1
18.9
19.8
10.9
11.6
12.3
13.1
13.8
14.6
15.4
16.2
16.9
17.7
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12.4
13.1
13.8
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15.3
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92.0
104.2
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128.3
140.2
50.9
63.7
76.2
88.4
100.4
112.3
124.1
135.8
47.0
59.3
71.4
83.3
95.0
106.6
118.1
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43.8
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67.5
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