Ch6 1
x3 a0a2a1a2a3a2a4a2a5a2a6a2a7a2a8a2a9a2a10
a11a13a12 a14 X
n(n = 0; 1; 2; ) a15a13a16a13a17a13a18a13a19a2a20a2a21a23a22a25a24a2a16a26a17a2a27 Ia28a30a29a13a27a13a31a13a32a13a33
a34a36a35
a28a38a37a2a39a2a40a2a41a2a42a2a15 i0; i1; ; in 1; i; j 2 Ia28a38a43a2a44
P(Xn+1jXn = i; X0 = i0; X1 = i1; ; Xn 1 = in 1) = P(Xn+1 = jjXn = i);
a45
a29a13a46a13a47a13a48a26a49 fXn; n > 0g a27a13a50a13a51a13a52a13a20a2a53a26a54a23a22a25a55a26a29a2a51a57a56a58a54
a35a60a59
a37a2a39a26a61a2a50a26a62a63a28a64a40
a41a2a42a2a15 n a43a2a44
P(Xn+1 = jjXn = i) = P(X1 = jjX0 = i);
a45
a29a2a65a2a51a57a56a66a54a2a67a2a68a2a69a2a15
a59a71a70a26a72a26a73a26a74a2a75a26a76a2a77a26a78
a68a2a69a26a51a79a56a66a54
a59
a40a2a51a57a56a66a54 fXn; n > 0ga28a38a80a2a81a82a22a25a50a2a62
a35a84a83a2a85a26a86a2a87
a27
pij = P(X1 = jjX0 = i):
a88a2a89
a15a90a28 (pij)n n a29a2a27
a83a2a85a2a86a2a87a57a91a92a59
a93
a68a90a28
a77a2a78 fX
ng a15a2a44a2a94a2a95a2a96a2a97a2a98a2a99
P(X0 = i0; X1 = i1; ; Xn = in)
= P(X0 = i0)P(X1 = i1jX0 = i0)P(X2 = i2jX1 = i1; X0 = i0)
P(Xn = injXn 1 = in 1; ; X0 = i0)
= P(X0 = i0)pi0i1pi1i2 p(in 1in):
a100
a67a90a28
a70a2a101a2a102a2a103 P(X
0 = i0) a104a2a105a90a28a38a96a2a97a2a98a2a99a2a106a26a107a26a50a2a108a26a80
a103a63a59
a29 P(X0 =
i0) a27a2a51a57a56a66a54a2a15a2a109a2a98a2a99
a59
a40
a83a2a85a2a86a2a87
a28a38a110a2a111a2a44
(1) 8i; j 2 Ia28 pij > 0;
(2) 8i 2 Ia28
X
j2I
pij = 1:
a40
a100
a51a57a56a66a112 fXn; n >ga28a38a29
pn(ij) = P(Xn = jjX0 = i)
1
Ch6 2
a27 n a62
a83a2a85a2a86a2a87
a28
a88a2a89
a15a63a28a71a29 (pn(ij))n n a27 n a62
a83a2a85a57a91a92a59
a40 n a62
a83a2a85a2a86a2a87
a28a38a44a2a113a2a114a26a15 K-C a115a2a116a63a117
pn(ij) =
X
k2I
pm(ik)pn m(kj); 81 6 m 6 n 1:
a118
a40 Xm a15a2a31a2a32a2a61a2a119a2a98a2a120
pn(ij) = P(Xn = jjX0 = i)
=
X
k2I
P(Xn = j; Xm = kjX0 = i)
=
X
k2I
P(Xm = kjX0 = i)P(Xn = jjXm = k; X0 = i)
=
X
k2I
P(Xm = kjX0 = i)P(Xn = jjXm = k)
=
X
k2I
pm(ik)pn m(kj):
a121a2a122a2a123a57a124
a28 n a62
a83a2a85a2a86a2a87a2a125
a67a2a68a2a69a2a15
a59a127a126a58a128
a114a2a15 K-C a115a2a116a2a20
a101
(pn(ij)) = (pij)n:
a113a2a114a2a129a57a130a66a131a2a132a2a51a57a56a66a54a2a15a2a133a26a134
a59
a135 3.1 I = f ; 2; 1; 0; 1; 2; g
a67a2a136a2a137a2a138a2a21a2a139a2a97a90a28
a83a26a85a2a86a26a87
pij =
8
>><
>>:
p; j = i + 1;
q; j = i 1;
0; a140a2a141 ;
a72
a106a2a67a2a142a2a143
a128
a15a2a144a2a94a2a145a26a46a26a47a2a146a26a147a63a28a71a140
a83a2a85a79a91
a27
P =
0
BB
BB
BB
B@
q 0 p 0 0
0 q 0 p 0
0 0 q 0 p
1
CC
CC
CC
CA
:
a135 3.2 I = f0; 1; ; cg
a28 c > 2a59 a40 1 6 i 6 c 1a28 pij
a148
a128
a133a150a149
a88a150a151
a28a153a152
p00 = pcc = 1:
2
Ch6 3
a72
a67a2a154a2a44a2a155a2a132a2a156a2a157a2a158 0 a159 c a15a2a46a2a47a2a146a2a147a90a28a38a140
a83a26a85a57a91
a27
P =
0
BB
BB
BB
BB
BB
@
1 0 0 0
q 0 p 0
0 q 0 p
0 q 0 p
0 0 0 1
1
CC
CC
CC
CC
CC
A
:
a135 3.3
a160
a128
a133a57a149 p00 = pcc = 1 a161a2a162 p01 = pc(c 1) = 1a28a64a163a2a164a2a15a2a67a2a44a2a155a2a132
a165a2a166
a158a2a15a2a46a36a47a26a146a36a147
a59a168a167a26a169a26a170a36a171
a164a173a172
a34 [0; c]
a155a2a174a2a15a2a68a2a175a176a28a168a113a26a50a36a62a26a50a36a80a26a177
a165a26a166
a178a66a179 a59
a140
a83a2a85a57a91
a27
P =
0
BB
BB
BB
BB
BB
@
0 1 0 0
q 0 p 0
0 q 0 p
0 q 0 p
0 0 1 0
1
CC
CC
CC
CC
CC
A
:
a135 3.4 Ehrenfest
a180a2a181
a59a182a14 fX
n; n > 0g a67a2a50a2a51a57a56a66a54a176a28 I = f0; 1; ga28
a152a2a40 i 2 Ia28a38a44
pi(i+1) = c ic ; pi(i 1) = ic:
a183
a20a2a184a2a185a2a50a2a132a2a186a2a187a2a15a2a180a2a181
a179a2a188a2a189
a117a64a50a26a132a2a190a26a191
a121a26a192
a149a58a193a2a44 c a132a2a187a194a22a196a195a66a197a2a15
a198a13a199a13a200
a197a13a15
a35
a28a84a46a2a47a2a15a2a201a57a149a66a202a2a50a2a132a2a187a63a28a153a203a2a161a2a162a2a204a2a50a2a205a2a206a2a197a2a15a2a187a2a207
a178 a59a153a45 a121a2a192
a149
a15a57a195a66a187a2a21a2a106a2a67
a128
a114a2a15 Xn
a59
a40
a100 n
a62
a83a13a85a13a86a13a87a150a91 P(n)
a28a208a50a13a209a13a210a13a211a13a160 P a83a13a212 a27 Jordan a213a13a214a13a181
a179a13a215 a59
x4 a216a2a217a2a218a2a219
a40
a100
a51a150a56a220a54 fXn; n > 0ga28
a14
a140
a83a13a85a13a86a26a87a57a91
a27 P = (pij)n na28
a70
a129a13a80a13a109a13a221
a98a13a99a13a222 X0 a15a13a98a13a99a13a105a90a28 Xn a15a13a98a13a99
a125
a106a13a108a13a80
a103a63a59
a154a2a137a13a223a2a224a63a28
a126
a136
a86a2a225a13a226
a28a227a24 Xn
3
Ch6 4
a15a2a98a2a99a2a49a57a228a66a229a2a27 na28
a45
0n+1 = 0nP;
a100
a67a2a106a2a44
0n = 00P n:
a11a2a12
a29a2a98a2a99a2a49 = (pi)i2I a27a2a51a57a56a66a54 fXn; n > 0g a15a2a230a2a231a2a98a2a99a2a49a63a28a232a37a2a39
a184
a183
a27a2a109a2a221a2a98a2a99a2a68a90a28 X0; X1; ; Xn; a151 a98a2a99 a59
a121a2a122a2a233
a130a92a28 a27a2a230a2a231a2a98a2a99a2a49a2a15a2a234a2a235a2a236a26a237a26a67
a183a26a238a26a239
a113a26a114a2a15a26a115a26a116a176a117
0 = 0P:
a40
a100
a50a2a209a2a15a2a51a57a56a66a54a90a28a64a142a2a240a2a241a26a185
a83a26a85a57a91
a106a2a20a26a184a2a120a79a130
a183
a15a2a230a26a231a2a98a26a99
a59
a113a2a114a26a242
a243
a50a2a205a2a244a2a245a2a15a2a51a57a56a66a54a63a28a71a20a26a246a2a51a79a56a58a54
a59
a11a13a12
a50a13a132a13a51a150a56a220a54 fXn; n > 0g a29a13a27a13a20a13a246a13a15a63a28a30a37a13a39a26a247
a70
a98a26a99 = f igi2Ia28
a248
a163
ipij = jpji; 8i; j 2 I:
a93
a68a90a28a38a29a2a98a2a99 a27a2a65a2a51a57a56a66a54a2a15a2a20a2a246a2a98a2a99
a59
a121a2a122 a215
a130a92a28a38a20a2a246a2a98a2a99a26a67a2a249a26a250a26a98a2a99
a59
a40
a100
a20a2a246a2a51a57a56a66a54a90a28a38a20a26a184a2a37a26a113
a215
a130a58a20a2a246a26a98a26a99a63a28
a125
a106a2a67a26a249a26a250a2a98a26a99a176a117
a251a2a167
a129a2a31a2a32a2a213a2a252a63a28
a248
a163 pn(n+1) > 0 a22
a72
a50a2a209a2a67a2a20a2a184a2a253a2a164a26a15
a35a60a59
a202 0 = 1a28
a111a2a105a2a37a2a113a2a254a2a255a63a117
n+1 = npn(n+1)p
(n+1)n;
a24 =
1X
n=0
na28
a45
a20a2a246a2a98a2a99a2a27
n = n1X
n=0
n
= n,
a135 4.1 a77a2a78
a44a2a155a2a132
a165a2a166
a158a2a15a26a40a26a29a2a46a26a47a26a146a26a147 a22
a1a3a2 a128a3a4
a133 3.3a35 a28
a93
a68a2a44
p = q = 1=2a59
a202 0 = 1a28
a45
1 = 0 p01p
10
= 2 0 = 2; n = n 1 p(n 1)np
n(n 1)
= 12 n 1:
4
Ch6 5
a152
a167 1 6 i < n 1
a68a90a28a38a44
i+1 = ipi(i+1)p
(i+1)i
= i = 1 = 2:
a100
a67a2a44 n = 12 n 1 = 1a28 =
nX
i=1
i = 2na59
a5
0 = n =
1
2na28 i =
1
na28 i = 1; ; i 1
a59
a20a2a184a7a6
a123
a28 = ( 0; 1; ; n) a67a2a51a57a56a66a54a2a15a2a20a2a246a2a98a2a99
a59
5
x3 a0a2a1a2a3a2a4a2a5a2a6a2a7a2a8a2a9a2a10
a11a13a12 a14 X
n(n = 0; 1; 2; ) a15a13a16a13a17a13a18a13a19a2a20a2a21a23a22a25a24a2a16a26a17a2a27 Ia28a30a29a13a27a13a31a13a32a13a33
a34a36a35
a28a38a37a2a39a2a40a2a41a2a42a2a15 i0; i1; ; in 1; i; j 2 Ia28a38a43a2a44
P(Xn+1jXn = i; X0 = i0; X1 = i1; ; Xn 1 = in 1) = P(Xn+1 = jjXn = i);
a45
a29a13a46a13a47a13a48a26a49 fXn; n > 0g a27a13a50a13a51a13a52a13a20a2a53a26a54a23a22a25a55a26a29a2a51a57a56a58a54
a35a60a59
a37a2a39a26a61a2a50a26a62a63a28a64a40
a41a2a42a2a15 n a43a2a44
P(Xn+1 = jjXn = i) = P(X1 = jjX0 = i);
a45
a29a2a65a2a51a57a56a66a54a2a67a2a68a2a69a2a15
a59a71a70a26a72a26a73a26a74a2a75a26a76a2a77a26a78
a68a2a69a26a51a79a56a66a54
a59
a40a2a51a57a56a66a54 fXn; n > 0ga28a38a80a2a81a82a22a25a50a2a62
a35a84a83a2a85a26a86a2a87
a27
pij = P(X1 = jjX0 = i):
a88a2a89
a15a90a28 (pij)n n a29a2a27
a83a2a85a2a86a2a87a57a91a92a59
a93
a68a90a28
a77a2a78 fX
ng a15a2a44a2a94a2a95a2a96a2a97a2a98a2a99
P(X0 = i0; X1 = i1; ; Xn = in)
= P(X0 = i0)P(X1 = i1jX0 = i0)P(X2 = i2jX1 = i1; X0 = i0)
P(Xn = injXn 1 = in 1; ; X0 = i0)
= P(X0 = i0)pi0i1pi1i2 p(in 1in):
a100
a67a90a28
a70a2a101a2a102a2a103 P(X
0 = i0) a104a2a105a90a28a38a96a2a97a2a98a2a99a2a106a26a107a26a50a2a108a26a80
a103a63a59
a29 P(X0 =
i0) a27a2a51a57a56a66a54a2a15a2a109a2a98a2a99
a59
a40
a83a2a85a2a86a2a87
a28a38a110a2a111a2a44
(1) 8i; j 2 Ia28 pij > 0;
(2) 8i 2 Ia28
X
j2I
pij = 1:
a40
a100
a51a57a56a66a112 fXn; n >ga28a38a29
pn(ij) = P(Xn = jjX0 = i)
1
Ch6 2
a27 n a62
a83a2a85a2a86a2a87
a28
a88a2a89
a15a63a28a71a29 (pn(ij))n n a27 n a62
a83a2a85a57a91a92a59
a40 n a62
a83a2a85a2a86a2a87
a28a38a44a2a113a2a114a26a15 K-C a115a2a116a63a117
pn(ij) =
X
k2I
pm(ik)pn m(kj); 81 6 m 6 n 1:
a118
a40 Xm a15a2a31a2a32a2a61a2a119a2a98a2a120
pn(ij) = P(Xn = jjX0 = i)
=
X
k2I
P(Xn = j; Xm = kjX0 = i)
=
X
k2I
P(Xm = kjX0 = i)P(Xn = jjXm = k; X0 = i)
=
X
k2I
P(Xm = kjX0 = i)P(Xn = jjXm = k)
=
X
k2I
pm(ik)pn m(kj):
a121a2a122a2a123a57a124
a28 n a62
a83a2a85a2a86a2a87a2a125
a67a2a68a2a69a2a15
a59a127a126a58a128
a114a2a15 K-C a115a2a116a2a20
a101
(pn(ij)) = (pij)n:
a113a2a114a2a129a57a130a66a131a2a132a2a51a57a56a66a54a2a15a2a133a26a134
a59
a135 3.1 I = f ; 2; 1; 0; 1; 2; g
a67a2a136a2a137a2a138a2a21a2a139a2a97a90a28
a83a26a85a2a86a26a87
pij =
8
>><
>>:
p; j = i + 1;
q; j = i 1;
0; a140a2a141 ;
a72
a106a2a67a2a142a2a143
a128
a15a2a144a2a94a2a145a26a46a26a47a2a146a26a147a63a28a71a140
a83a2a85a79a91
a27
P =
0
BB
BB
BB
B@
q 0 p 0 0
0 q 0 p 0
0 0 q 0 p
1
CC
CC
CC
CA
:
a135 3.2 I = f0; 1; ; cg
a28 c > 2a59 a40 1 6 i 6 c 1a28 pij
a148
a128
a133a150a149
a88a150a151
a28a153a152
p00 = pcc = 1:
2
Ch6 3
a72
a67a2a154a2a44a2a155a2a132a2a156a2a157a2a158 0 a159 c a15a2a46a2a47a2a146a2a147a90a28a38a140
a83a26a85a57a91
a27
P =
0
BB
BB
BB
BB
BB
@
1 0 0 0
q 0 p 0
0 q 0 p
0 q 0 p
0 0 0 1
1
CC
CC
CC
CC
CC
A
:
a135 3.3
a160
a128
a133a57a149 p00 = pcc = 1 a161a2a162 p01 = pc(c 1) = 1a28a64a163a2a164a2a15a2a67a2a44a2a155a2a132
a165a2a166
a158a2a15a2a46a36a47a26a146a36a147
a59a168a167a26a169a26a170a36a171
a164a173a172
a34 [0; c]
a155a2a174a2a15a2a68a2a175a176a28a168a113a26a50a36a62a26a50a36a80a26a177
a165a26a166
a178a66a179 a59
a140
a83a2a85a57a91
a27
P =
0
BB
BB
BB
BB
BB
@
0 1 0 0
q 0 p 0
0 q 0 p
0 q 0 p
0 0 1 0
1
CC
CC
CC
CC
CC
A
:
a135 3.4 Ehrenfest
a180a2a181
a59a182a14 fX
n; n > 0g a67a2a50a2a51a57a56a66a54a176a28 I = f0; 1; ga28
a152a2a40 i 2 Ia28a38a44
pi(i+1) = c ic ; pi(i 1) = ic:
a183
a20a2a184a2a185a2a50a2a132a2a186a2a187a2a15a2a180a2a181
a179a2a188a2a189
a117a64a50a26a132a2a190a26a191
a121a26a192
a149a58a193a2a44 c a132a2a187a194a22a196a195a66a197a2a15
a198a13a199a13a200
a197a13a15
a35
a28a84a46a2a47a2a15a2a201a57a149a66a202a2a50a2a132a2a187a63a28a153a203a2a161a2a162a2a204a2a50a2a205a2a206a2a197a2a15a2a187a2a207
a178 a59a153a45 a121a2a192
a149
a15a57a195a66a187a2a21a2a106a2a67
a128
a114a2a15 Xn
a59
a40
a100 n
a62
a83a13a85a13a86a13a87a150a91 P(n)
a28a208a50a13a209a13a210a13a211a13a160 P a83a13a212 a27 Jordan a213a13a214a13a181
a179a13a215 a59
x4 a216a2a217a2a218a2a219
a40
a100
a51a150a56a220a54 fXn; n > 0ga28
a14
a140
a83a13a85a13a86a26a87a57a91
a27 P = (pij)n na28
a70
a129a13a80a13a109a13a221
a98a13a99a13a222 X0 a15a13a98a13a99a13a105a90a28 Xn a15a13a98a13a99
a125
a106a13a108a13a80
a103a63a59
a154a2a137a13a223a2a224a63a28
a126
a136
a86a2a225a13a226
a28a227a24 Xn
3
Ch6 4
a15a2a98a2a99a2a49a57a228a66a229a2a27 na28
a45
0n+1 = 0nP;
a100
a67a2a106a2a44
0n = 00P n:
a11a2a12
a29a2a98a2a99a2a49 = (pi)i2I a27a2a51a57a56a66a54 fXn; n > 0g a15a2a230a2a231a2a98a2a99a2a49a63a28a232a37a2a39
a184
a183
a27a2a109a2a221a2a98a2a99a2a68a90a28 X0; X1; ; Xn; a151 a98a2a99 a59
a121a2a122a2a233
a130a92a28 a27a2a230a2a231a2a98a2a99a2a49a2a15a2a234a2a235a2a236a26a237a26a67
a183a26a238a26a239
a113a26a114a2a15a26a115a26a116a176a117
0 = 0P:
a40
a100
a50a2a209a2a15a2a51a57a56a66a54a90a28a64a142a2a240a2a241a26a185
a83a26a85a57a91
a106a2a20a26a184a2a120a79a130
a183
a15a2a230a26a231a2a98a26a99
a59
a113a2a114a26a242
a243
a50a2a205a2a244a2a245a2a15a2a51a57a56a66a54a63a28a71a20a26a246a2a51a79a56a58a54
a59
a11a13a12
a50a13a132a13a51a150a56a220a54 fXn; n > 0g a29a13a27a13a20a13a246a13a15a63a28a30a37a13a39a26a247
a70
a98a26a99 = f igi2Ia28
a248
a163
ipij = jpji; 8i; j 2 I:
a93
a68a90a28a38a29a2a98a2a99 a27a2a65a2a51a57a56a66a54a2a15a2a20a2a246a2a98a2a99
a59
a121a2a122 a215
a130a92a28a38a20a2a246a2a98a2a99a26a67a2a249a26a250a26a98a2a99
a59
a40
a100
a20a2a246a2a51a57a56a66a54a90a28a38a20a26a184a2a37a26a113
a215
a130a58a20a2a246a26a98a26a99a63a28
a125
a106a2a67a26a249a26a250a2a98a26a99a176a117
a251a2a167
a129a2a31a2a32a2a213a2a252a63a28
a248
a163 pn(n+1) > 0 a22
a72
a50a2a209a2a67a2a20a2a184a2a253a2a164a26a15
a35a60a59
a202 0 = 1a28
a111a2a105a2a37a2a113a2a254a2a255a63a117
n+1 = npn(n+1)p
(n+1)n;
a24 =
1X
n=0
na28
a45
a20a2a246a2a98a2a99a2a27
n = n1X
n=0
n
= n,
a135 4.1 a77a2a78
a44a2a155a2a132
a165a2a166
a158a2a15a26a40a26a29a2a46a26a47a26a146a26a147 a22
a1a3a2 a128a3a4
a133 3.3a35 a28
a93
a68a2a44
p = q = 1=2a59
a202 0 = 1a28
a45
1 = 0 p01p
10
= 2 0 = 2; n = n 1 p(n 1)np
n(n 1)
= 12 n 1:
4
Ch6 5
a152
a167 1 6 i < n 1
a68a90a28a38a44
i+1 = ipi(i+1)p
(i+1)i
= i = 1 = 2:
a100
a67a2a44 n = 12 n 1 = 1a28 =
nX
i=1
i = 2na59
a5
0 = n =
1
2na28 i =
1
na28 i = 1; ; i 1
a59
a20a2a184a7a6
a123
a28 = ( 0; 1; ; n) a67a2a51a57a56a66a54a2a15a2a20a2a246a2a98a2a99
a59
5
