1 Ô0´ Ê â v)[jpkc.mku.edu.cn/jpkc/gllnet/jy/1.pdf · 2009. 3. 26. · 2 þ _ Ê È g{(h,t)...
TRANSCRIPT
1
.
..
. 10
E . E
1 2 3 .
E . .
1E H T
),( HH ),( TH ),( HT ),( TT .
2E
0 1 2 n
3E ),( 00 yx
)()(:),( 220
20 ryyxxyx ≤−+− .
4E 10 3
.3 4 5 6 7 8 9 10
..
. 1E
),( HH
2
),( TH ),( HT ),( TT ),( TH ),( HT
),( TT ..
. ie ),3,2,1 =i
. E .
6,,1 61 == ee
. .
.
Ω φ .
E E E Ω . Ω E
. Ω = ,, 21 ee .
..
..
1.1 A B A BBA ⊂ .
A Ω⊂A A⊂φ .
1.2 A B BA ⊂ AB ⊂ A B BA = .
1.3 A B A
. BAC += BAC = .
A ,, 642 eeeA = C
3 , 63 eeC = CA+ 2 3 4 6
,,, 6432 eeeeCA =+ . A AA =φ Ω=ΩA .
BA ⊂ BBA = .
1.4 A B C A B
ABC = BAC = .
A ,, 642 eeeA = B
3
,, 531 eeeB = φ=BA 6eCA = 3eCB = .
A φφ =A AA =φ . BA ⊂ ABA = .
1.5 A B A B A B
φ=AB .
A B . 1.6 A B
B A BA = .
B A A B A B. 1.6
A B Ω=+ BA φ=AB BA = .
A B .
1.7 A B C A B
BAC −= .
BA− A B
BA− = BA Venn BA− ABA−
BA− = BA = ABA−
1 ABBA +=+ BAAB= BA−
2 CBACBA ++=++ )()( 1.1
3 ACABCBA +=+ )(
4 De.Morgen BABA =+ BAAB +=
BABA =+ .
BA + A B BA + A B
BA BABA ⊂+ . BA A B
BA BA + BA + BABA +⊂ . BABA =+ .
..
A B
4
.
[ 1] CBA ,,
1 BA, C CAB CAB −
2 CBA ,, BCACBACAB ++
3 CBA ,, ABCBCACBACAB +++
4 CBA ,, CBA ++
BCACBACAB ++ + ABCBCACBACAB +++
..
. .
.
—— .
n n A An A
.nnA A )(Afn .
1 )(Af n ≥ 0
2 )(Ωnf =1
3 BA, )()()( BfAfBAf nnn +=+
A A. A A .
A . ,5 10
0 1 100 500 1000.
1
5
n Hn )(Hfn
2048 1061 0.5181
4040 2048 0.5069
12000 6019 0.5016
24000 12012 0.5005
1 n )(Hfn
2 n )(Hfn n
)(Hfn . n )(Hfn 0.5 0.5.
.
. 1.8 A p
p A )(AP .
1 )(AP ≥ 0
2 )(ΩP =1
3 BA, )()()( BPAPBAP +=+
.
. 1 2
.
Ω = ,, 21 ee ne )()()( 21 nePePeP === .
1.9 n A k A
nk
AP =)(
(Laplace) 1812 ..
..
.
6
[ 1] 6 4 2(1) 2 .
BA, .
BA + .
..
6 666× 66 × . A 4
4 44 ×A 44 × . B 22 × .
94
6644
)( =××=AP
91
6622
)( =××=BP
A B
)()()( BPAPBAP +=+ =95
[ 2] 7 . A 7 36
6 7 (1,6),(2,5),(3,4), (4,3),(5,2),(6,1)
61
366
)( ==AP
. .
..
. .
n i im
∏=
n
iim
1
.
n i im
=
n
iim
1
.
. n r
.
rnnnn =⋅
7
)1()2)(1( +−−−⋅ rnnnn rnP .
rnP =
)!(!rn
n−
n m mkkk ,,, 21
nkkk m =+++ 21 , n!!!
!
21 mkkkn
.
n r .rnC .
n r 1A r
2A 1A n r 2A r
)!(!
!rn
nrCP r
nr
n −=⋅=
!)!(
!rrn
nC r
n ⋅−=
[ 3] A
B C D .
DCBA ,,, .
33 =27
271
31
)(3
==AP
272
32
)(3
==BP
92
3)(
3
11
12
13 ==
CCCCP
278
3)(
3
12
12
12 ==
CCCDP
8
[ 4] 5 4 4 2 . : A = 4 2 10 1 2 3 4 9 10 .
4 410C . 4 2 2
4 25
212
24
15 )( CCCC +
2113)(
)( 410
25
212
24
15 =
+=
CCCCC
AP
[ 5] 15 15 1 2 .
A B .
!5!5!5!15
1 !3 12
!4!4!4!12
!4!4!4!12!3
9125
!5!5!5!15!4!4!4!12!3
)( ==AP
2 3 12
!5!5!2!12
!5!5!2!123×
916
!5!5!5!15!5!5!2!123
)( =
×
=BP
.
..
12 .
1.10 Ω Ω ΩS A
Ω⊂A A A AS A
Ω
=SS
AP A)(
.
9
.
[ 6] 0 T )( Ttt ≤
. 0 T
.
yx, TyTx ≤≤≤≤ 0,0 xoy
),( yx .
tyx ≤− Ω A
22
22
)1(1)(
)(Tt
TtTT
AP −−=−−=
1.2 [ 7] a .
yxayx −−,, Ω
>>
<+<
00
0
y
x
ayx
1.3 1.3
2
21
aS AOB =∆ .
+−>+
<<
<<
)(2
0
20
yxayx
ay
ax
DCE
2)2
(21 a
S DCE =∆
10
a
2
2
21
)2
(21
a
a
p = =41
.
.
. σ - . 1.11 Ω F Ω F
1 F∈Ω ;
(2) FA∈ FA∈
3 FAi ∈ ni ,,2,1 = FAi
n
i∈
=1 ;
F F . 1.12 F 1 2)
)3( ′ FAi ∈ ,2,1=i FAii
∈∞
=1
F σ - .
1.12 F∈φ FAi ∈ ,,2,1 =i FAi
n
i∈
=1 FAi
i∈
∞
=1
Ω σ - . Ω F Ω σ - .
σ - . Ω Ω
σ - F F,Ω σ -
. .
1.13 F,Ω FA∈ )(AP
1 FA∈ 1)(0 ≤≤ AP
2 Ω 1)( =ΩP
3 FAi ∈ ,2,1=i φ=ji AA ji ≠
11
∞
=
∞
==
11)()(
iii
iAPAP
P F,Ω )(AP A .
6.
Ω F Ω σ - P F .
),,( PFΩ .
§3
1 0)( ≥AP .
(2) 1)( =ΩP .
(3) A B )()()( BPAPBAP +=+ .
.
(4) A )(1)( APAP −= .
: A A 3 )()()( APAPAAP +=+
Ω=+ AA 1)( =ΩP
1)()( =+ APAP
)(1)( APAP −= .
5 BA ⊂ )()()( APBPABP −=−
BA ⊂ )( ABAB −+=
3 )()()( ABPAPBP −+=
)()()( APBPABP −=− .
5
BA ⊂ )()( BPAP ≤
A B )()()( ABPBPABP −=− .
12
6 A B
)()()()( ABPBPAPBAP −+=+
BA + A ABB −
)( ABBABA −+=+
3
)()()]([)( ABBPAPABBAPBAP −+=−+=+
BAB ⊂ )()()( ABPBPABP −=−
)()()()( ABPBPAPBAP −+=+ .
6
A B )()()( BPAPBAP +≤+
A B C
)()()()()()()()( ABCPBCPACPABPCPBPAPCBAP +−−−++=++ .
7 ,,, 21 AA
)(lim)lim( nnnnAPAP
∞→∞→=
.
1 ,, 21 AA
⊂⊂⊂ 321 AAA
+−+−+==∞
=∞→)()(lim 23121
1AAAAAAA n
nnn
+−+−+=∞→
)()()()lim( 23121 AAPAAPAPAP nn
5 ),3,2,1()()()( 11 =−=− ++ nAPAPAAP nnnn
+−+−+=∞→
)]()([)]()([)()lim( 23121 APAPAPAPAPAP nn
)(lim nnAP
∞→= .
2 ,, 21 AA
⊃⊃⊃ 321 AAA
,,, 312111 AAAAAA −−− 1
13
)(lim))(lim( 11 nnnnAAPAAP −=−
∞→∞→
)()()(lim 11
11
1 nn
nn
nnAAAAAA
∞
=
∞
=∞→=−=−
)()(1
11
1 nn
nn
AAAA∞
=
∞
===
nnnn
AAAA∞→
∞
=−=−= lim1
11
=−∞→
))(lim( 1 nnAAP )lim( 1 nn
AAP∞→
−
1AAn ⊂ 1lim AAnn⊂
∞→
5 )lim()()lim( 11 nnnnAPAPAAP
∞→∞→−=−
)()()( 11 nn APAPAAP −=−
)lim()( 1 nnAPAP
∞→− )()([lim 1 nn
APAP −=∞→
)(lim)( 1 nnAPAP
∞→−=
)(lim)lim( nnnnAPAP
∞→∞→= .
. [ 1] 4 1
A
A .
A
12516
5)( 3
11
14
14 == CCC
AP
12516
1)(1)( −=−= APAP
)(AP )(AP
)(1)( APAP −= .
[ 2] BA,21
41
BA − .
1 A B 2 AB ⊂
14
3 81
)( =ABP .
1 A B BA ⊂ ABA =
21
)()()( ===− APBAPBAP
2 AB ⊂
41
41
21
)()()( =−=−=− BPAPBAP
383
81
21
)()()()( =−=−=−=− ABPAPABAPBAP
[ 3] 1 9 910 .
1 1A 5 2A
21 AAA =
)(1)()( 2121 AAPAAPAP −==
)(1 21 AAP +−=
)()()(1 2121 AAPAPAP +−−=
333 )94
()95
()98
(1 +−−=
786.01−= 214.0=
.
§4 Bayes
.
[ 1] 500 40 10 1 2
.
: 1 A101
50050
)( ==AP
2 1B 2 B A
)|( BAP .
15
. B
A .
25010
)|( =BAP
)()(
25010
)|(50025050010
BPABP
BAP ===
1.14 BA, 0)( >BP
)()(
)|(BP
ABPBAP = 1.4.1
B A .
1 A 0)|( ≥BAP
2 1)|( =Ω BP
3 ,, 21 AA
∞
=
∞
==
11)|()|(
iii
iBAPBAP
. [ 2] 20 0.8, 30 0.5
20 10 A 20 B 30
A B . .
85
8.05.0
)()(
)()(
)|( ====APBP
APABP
ABP
83
85
1)|(1)|( =−=−= ABPABP
)()(
)|(BP
ABPBAP =
16
)|()()()|()( ABPAPBPBAPABP == (1.4.2)
.
(1.4.2) . CBA ,, 0)( >ABP (
0)( >AP )
)()|()|()( APABPABCPABCP = 1.4.3
nAAA ,,, 21 n 0)( 121 >−nAAAP
)()|()|()|()( 112221112121 APAAPAAAAPAAAAPAAAP nnnnn −−−= 1.4.4
[ 3] 10 7 3 .4 .
A = 2 B = 4
401
71
)|()()( 310
23
17 =×==C
CCABPAPABP
Bayes .
[ 4]
0.4,0.9, 0.5,
[ ] .90% .
.
B 1A 2A 1A
2A 21 AA = 12 AA = 1A + 2A = Ω
BBB =Ω= 1(A + 2A = 1BA + 2BA
1A 2A = Φ 1BA 2BA = Φ
)(BP = 1(BAP + 2BA =P 1BA +P 2(BA
17
)|()()( 111 ABPAPBAP = )|()()( 222 ABPAPBAP =
5.0)( 1 =AP )( 2AP= )|( 1ABP
0.4 )|( 2ABP 0.9
65.09.05.04.05.0)( =×+×=BP
1.15 nAAA ,,, 21
1 Ω=+++ nAAA 21
2 ),,2,1,,( njijiAA ji =≠Φ=
nAAA ,,, 21 .
Ω ..
1 .1 nAAA ,,, 21 B
)|()()(1
i
n
ii ABPAPBP
== 1.4.5
1.4.5.
[ 5] 5 3 4 32 2 2 .
B 0 1 2
1A 2A 3A 1A 2A 3A
BBB =Ω= 1(A + 2A )3A+ = 1BA + 2BA + 3BA
)(BP = 1(BAP + 2BA )3BA+ = )( 1BAP + )( 2BAP )( 3BAP+
)|()( 11 ABPAP= + )|()( 22 ABPAP + )|()( 33 ABPAP
+⋅= 29
24
28
23
CC
CC
+⋅ 29
25
28
15
13
CC
CCC
29
26
28
25
CC
CC
⋅
⋅=16853
18
Bayes B iA
.
1.2 nAAA ,,, 21 B 0)( >BP ,
),,2,1()|()(
)|()()|(
1
niABPAP
ABPAPBAP
i
n
ii
iii ==
=
(1.4.6)
[ 6] 25% 35% 40% 5% 4% 2% 1
2
321 ,, AAA B 321 ,, AAA
. 1 .
)|()()( 11 ABPAPBP = + )|()( 22 ABPAP
02.040.004.035.005.025.0 ×+×+×= 0345.0=
2 B 321 ,, AAA .
Bayes
36.0)|( 1 ≈BAP , 41.0)|( 2 ≈BAP 23.0)|( 3 ≈BAP
B .
. B iA )|( BAP i
),,2,1()|()(
)|()()|(
1
niABPAP
ABPAPBAP
i
n
ii
iii ==
=
Bayes . Bayes Bayes.
[ 7]
§5
BA, 0)( >BP , )|( BAP . )|()( BAPAP ≠ ,
19
B A )|()( BAPAP =
BA, .
)()|()( BPBAPABP = )|()( BAPAP =
)()()( BPAPABP =
1.16 BA, )()()( BPAPABP = BA, .
0)( =AP 0)( =BP .
0)( >BP BA, )|()( BAPAP = .
1.3 BA, A B A B A B .
A B A B .
BAABA = )()()( BAPABPAP +=
BA, )()()( BPAPABP =
)()()( ABPAPBAP −=
)()()( BPAPAP −=
))(1)(( BPAP −=
)()( BPAP=
A B .
)()( BAPBAP +=
)(1 BAP +−=
)()()(1 ABPBPAP +−−=
)()()()(1 BPAPBPAP +−−=
)](1)][(1[ BPAP −−=
)()( BPAP=
20
A B
1.17 nAAA ,,, 21 n s
( ns ≤≤2 ), niii s ≤<<<≤ 211
)()()()(2121 ss iiiiii APAPAPAAAP =
n 1232 −−=+++ nCCC nnnnn .
CBA ,,
)()()( BPAPABP =
)()()( CPAPACP =
)()()( CPBPBCP =
)()()()( CPBPAPABCP =
CBA ,, CBA ,, CBA ,, .
CBA ,,
.
[ 1] ,,, 4321 ωωωω=Ω41
)( =iP ω , 21 ωω=A , 31 ωω=B
, 41 ωω=C CBA ,, .
21
)()()( === CPBPAP 1ω=== BCACAB 1ω=ABC
41
)()()( === BCPACPABP41
)( =ABCP
)()(21
21
41
)( BPAPABP =×==
)()(21
21
41
)( CPAPACP =×==
)()(21
21
41
)( CPBPBCP =×==
CBA ,, .
)()(21
21
21
41
)( CPBPABCP =××≠=
CBA ,, .
21
.
. [ 2]
.. .
0.5
iA i ,2,1=i A
iB i ,2,1=i B
iC i ,2,1=i C .
,,,,,, 765432165432154321432132121 BBCABCACCABCAAABCABBCACCAAA
,,,,,, 765432165432154321432132121 AACAACBCCBACBBBACBAACBCCBBB
++++= )(()()()( 654321654321321321 CCBACBPCCABCAPCCBPCCAPCP
+×+×+×=963 2
12
21
221
2
3
3
2112
12
−×=
72=
145
)72
1(21
))(1(21
)()( =−=−== CPBPAP
72
,145
,145
.
[ 3]
. .
.
. .
22
n )10( << rr ,
. 1 n .
iA i , )( iAP = r A .
⇔
nn AAAAAAA 2121 =∩∩=
)()( 21 nAAAPAP = )()()( 21 nAPAPAP = = nr .
n B
nAAAB ∪∪∪= 21
)()( 21 nAAAPBP ∪∪∪= = )(1 21 nAAAP +++−
= )(1 21 nAAAP − = nr)1(1 −−
[ 4] A B C CBA ,,
0.3,0.2,0.2, .
CBA ,, CBA ,, D
)()()()()( ABCPBCPAPBCAPDP −+=+=
)()()()()()( CPBPAPCPBPAP −+=
=1-0.672 =0.328
[ 5] KL MN r
KL MN KLR MNR , iC = iA
iD = iB ni ,,2,1 =
)( 2121 nnKL DDDCCCPR +=
)()( 2121 nn DDDPCCCP += )( 2121 nn DDDCCCP −
nn rr 22 −=
)2( nn rr −=
23
)]())([( 2211 nnMN DCDCDCPR +++=
)()( 2211 DCPDCP ++= )( nn DCP +
nrr )2( 2−=
nn rr )2( −=
2≥n MNKL RR < .
§6 ]1[
.
1 2
3 AA,
4 pqAPpAP −=== 1)(,)( .
Bernoullin n n .
3
. . n .
.
. n
n . n A k
)(kPn .
1.4 nkqpCkP knkknn ,2,1)( == − pqp −=<< 1,10
p A .
24
n A k , kn −k A , kn − A
nkkk AAAAAA 2121 ++
knknkkk ppAAAAAPp −
+ −== )1()( 121
nkqpCkP knkknn ,2,1)( == −
[ 1] 5 p ,
1 2 5 2 2 5 3 4 .
: iA i
(1) 2 5 2 5
252 )( pAAP = .
2 2 5 , .
3254321 )1()( ppAAAAAP −=
3 1 32255 )1()2( ppCP −=
4
55 )1(1)0(1 pP −−=−
[ 2] n A p n A
. A a b
++= −22200 nn
nn qpCqpCa
++= −− 33311 nn
nn qpCqpCb
nqpba )(1 +==+
npqba )( −=−
])21(1[21
])(1[21 nn ppqb −−=−−=
[1] "!#$ $ %'& ()*+",-(./* 0213 45
627 8 9:;<2=?>A@B CDE:F>HGI2J KLMON 5 > PQSRUT VWDX9YD 2ZM 627 > 3 [ \ D] ^_ 3 `a >2bHcd e (f2g
p8
25
2?hi j P2k?> Hl >H@m2noD29:2p=?>Hqn D2mrs ("/ tu > 2 D9 e v w x2Q M 5Oy M >Oz 8 : >H^|2w ~ i2 >8 O O p?A D O 9 i k 8
§7
[ 1] 1651 ,
.32 3 6 3 4
2 6 1 464
1 . 2 .
3 4 664 .
4 4 3216 64
.
..
..
A = B = C =BAC += A B .
43
21
21
21
)()()()( =×+=+=+= BPAPBAPCP
41
)(1)( =−= CPCP
. [ 2] r
1−r . n np nnp
∞→lim .
nq n 1−r
1)1( =−+ nn qrp )1(1
1nn p
rq −
−= 1
n 1−n 1−r
1,1
1)1( 11 ≥=
−×−= −− nq
rqrp nnn 2
1 2
26
=
≥−−
= −
1
1),1(1
1
0
1
p
npr
p nn
=−
−−
= −111
11
nn prr
p =−−
−− − )1(
)1(1
11
22 nprr
01
2 )1(1
)1()1(
11
1p
rrr nn
−−++
−−
−= −
2],)1
1(1[
1 ≥−
−= nrr
n
rpnn
1lim =
∞→.
. .
[ 3] 25
. n
. iA = i niAP i ,,2,1,361
)( ==
P = )( 21 nAAAP +++
= )(1 21 nAAAP ++−
= )(1 21 nAAAP −
=n
−3635
1
P =n
3635
n
−3635
1n
>3635
21
3635 <
n
61.2436ln35ln
2ln ≈−
−>n
25 . [ 4] N
r P .
27
: 21=p . r 1+N
N N rN −
=
−
− 21
21
21
2
rNNN
rNC12
2 21
+−
−
rN
NrNC
12
2 21
2+−
−
=rN
NrNCP
rNN
rNC−
−
=2
2 21
0=rN
NNCP
2
2 21
= Nr =N
P
=21
.
[ 5] α β ba +
βα ≤≤ ba , a b .
1 2 3 .
B = ba + a b
1 baC ++βα B baCC βα
ba
ba
C
CCBP +
+
=βα
βα)(
2 baP ++βα B )!( baCC ba +βα
ba
ba
P
baCCBP +
+
+=
βα
βα )!()(
ba
ba
C
CC++
=βα
βα
3 ),(~βα
α+
+ baBX
baabaCBP )()()(
βαβ
βαα
++= +