Ch6 1
a0a2a1a2a3 *
a4a2a5a2a6a2a7a2a8
x1 a9a2a10a2a11a2a12
a13a2a14a2a15a2a16a2a17a19a18a19a20a22a21a19a23a22a24a26a25a28a27a19a29a22a30a19a31 t = 0 a30a2a32a2a33a2a34a2a35a19a36 a
a37 a a38a2a39
a15a41a40a42a25
a43a45a44a45a46a45a47a45a48 a35a2a30a50a49a51a25a52a32a2a53a2a54 a43a2a55a2a56 p
a57a59a58a60a57a59a61a45a62
a20a2a21 a48 a35a63a25 a43a2a55a2a56 q = 1 p
a57a65a64a50a57a65a61a2a62
a20a2a21 a48 a35a67a66a69a68a2a70a19a71a2a72a19a73a2a74a19a75a2a76a19a17a2a18a19a77a2a78a19a79
a62
a66a69a20a19a80a2a81a19a82a2a83a19a84a2a85a19a13
a14a2a86a2a87a2a88a2a70a2a71a2a72a63a89
(1) a90a2a91a2a92
a18a2a77a2a78a2a79
a62
a93a2a94a2a23a2a24a2a29a2a30a2a31 0
a95a2a96
a24a50a97a65a98a67a25a69a99
Xn = a23a2a24a2a29a2a30a2a31 t = n a30a2a18a2a35a2a36 ;
a13a2a14 X
n
a18a2a53a2a100 P(X
n = k)
a25a69a101a2a102a2a103 jkj > n a30 P(X
n = k) = 0
a66a69a104a2a30a67a25a69a105 n
a106a50a107 a103 (n + k)=2 a106
a57a65a58a50a57a65a61a2a62
a25 (n k)=2 a106
a57a65a64a50a57a65a61a2a62
a66a69a34
a38a19a108
a103
P(Xn = k) =
(
C n+k2n pn+k2 q n k2 ; jkj 6 na109 k; n a110a2a111a2a112a2a113a50a114 ;
0; a115a2a116,
(2) a117a2a118a2a119a2a120
a18a2a77a2a78a2a79
a62
a93a2a94a2a23a2a24a2a29 0 a30a2a31a2a35a2a34 x = aa25a122a121a2a29 x = 0
a123 x = a + b = c a33a2a124a2a103a2a20
a118a45a119a45a120
a25
a115
a107 a; b > 0a66a126a125
a108a45a38a45a127
a25a129a128a2a23a19a24a2a130a2a131 0
a132 c a107 a18a45a20a45a24a45a30a63a25 a108a19a133a2a134a2a61
a62
a25a69a135
a118a2a119a2a136
a66a138a137a19a23a19a24a2a135 x = 0
a123 x = c a118a2a119
a18 a55a2a56 a66
a139 a43a45a140 a97a141a25a142a68
a108a45a38a2a143a2a144
a71a2a72a63a66a145a99 r = q=pa25a142a146a45a23a45a24a45a135 0 a24
a118a45a119
a18 a55a45a56a45a147a2a148
pk = ppk+1 + qpk 1;
a34
a38
a139 a43a2a149a2a150
pk =
8
>><
>>:
ra ra+b
1 ra+b ; r 6= 1;
b
a + b; r = 1:
a114a65a151
a18a67a25a69a23a2a24a2a135 c a24
a118a2a119
a18 a55a2a56 a74
p k =
8
><
>:
1 ra
1 ra+b ; r 6= 1;
a
a + b; r = 1:
1
Ch6 2
(3) a117a2a152a2a153a2a120
a18a2a77a2a78a2a79
a62
a93a2a94a2a23a2a24a2a29 0 a30a2a31a2a35a2a34 x = aa25a122a121a2a29 x = 0
a123 x = a + b = c a33a2a124a2a103a2a20
a152a45a153a45a120
a25
a115
a107 a; b > 0a25a126a125
a108a45a38a45a127
a25a129a128a19a23a2a24a19a131a2a154 0 a24a45a30a67a25a126a86a45a20 a106
a108a19a155
a102
a57a65a156a2a61
a62
a131 1a66
a114a65a151
a25a69a128a2a23a2a24a2a131a19a154 c a24a2a30a67a25a69a86a2a20 a106
a155
a102
a57a158a157a2a61a19a62
a131 c 1a66
a159 a29a2a13a2a14a50a160a65a17a2a18a2a77a2a78a2a79
a62
a25 a55a2a56a19a161a50a162a158a163 a103a19a164a2a165a19a166a2a167a63a66 a159 a29 a43 a15a2a16
a58a2a168
a16
a17
a39
a24
a117a2a118a2a119a2a120
a18a2a77a2a78a19a79
a62
a74a19a169a63a66
a27a45a170a45a171a45a24 i
a172 i + 1 a18a45a173a45a17a45a18 a162a65a174 a74 qp i; i = 0; 1; 2; a25a129a175a45a17 a162a59a163a45a176a2a150
0 a24 a162a65a163 V0 = 1a25 c a24 a162a65a163 Vc = 0a66a177a68a2a30a63a25a177a178a2a34 0 a172 c a107 a49a65a18a2a179a2a180a2a20a2a24 k a24
a18 a162a65a163 V
k
a147a2a148
Vk 1 Vk
q
p
k 1 = Vk VK+1 q
p
k ;
a39a2a181a2a182
a150
Vk = pVk+1 + qVk 1;
a161 a17a45a87a45a18
a117a45a118a45a119a2a120
a18a2a77a2a78a2a79
a62a2a113a50a183a65a184
a139a2a185 V
k
a161 p
k
a147a45a148
a113a60a114
a18a45a186a45a187a63a25a145a188
Vk = pka66
a178a2a34a2a20a2a80a2a18a50a160a65a125 a139 a43a2a189a2a190a2a191a2a55a2a56a19a189 a94 a162a65a174 a25a69a102 a44a2a192 a94 a162a158a163 a25a69a193 a55a19a56a2a161a194a162
a163 a178a2a195a2a196a2a197a67a66
x2 a9a2a10a2a11a2a12a2a198a2a199a2a200a2a201
a178a2a34a2a15a2a16a2a17
a39
a24a2a18a2a77a19a78a2a79
a62
a25a177a13a19a14
a95 0 a24a50a97a65a98a67a25a138a202a2a203 2n a106 a44a2a204a50a205 0 a24
a18 a55a2a56
a37
a202a2a203
a110
a15 a106a19a206 a139a19a207 a204a194a205 0 a24a41a40a42a25
P(X2n = 0) = Cn2npnqn:
a178 n a137
a172
a103
1X
n=1
P(X2n = 0) =
1X
n=1
Cn2npnqn = 11 p4pq:
a128 p 6= q a30a67a25 4pq < (p + q)2 = 1a25a138a188a2a77a2a78a2a79
a62
a202a2a203 0 a24a2a18a2a208a2a15a2a74a2a103
a91
a18
a55a2a56 a74 1a66a209a121a2a128 p = q = 1=2 a30a67a25a209a17a2a210a2a211a2a34
a90a2a212
a165a67a25a69a125
a108a19a38a2a127
a25a69a77a19a78a2a79
a62
a83a19a202
a203 0 a24
a90a2a212a2a213
a208a67a66
2
Ch6 3
a214a2a215 a73a2a20a2a21a2a24
a38a2a216
a204 a18a26a25a177a217a19a218
a95
a68a19a20a19a24a50a97a158a98a19a18a19a77a19a78a19a79
a62
a83
a90a19a212
a208 a204a50a205
a68a2a20a2a24a67a66
a95
a20a2a24 v a97a65a98a2a18a2a77a2a78a2a79
a62a2a38a2a216
a204 a18a19a219a2a220a19a221a19a222a2a74
1X
n=1
P(Xn = v) = 1:
a29a2a68a2a223 a206 a189 a97a65a224a50a225a141a66
a171a2a86a2a197 a140a2a226a2a227 a18a2a228a2a229a67a66 a226a2a227 a103a50a230a158a21a2a186
a57
a25a69a128a50a230a158a21a2a186
a57
a18 a55a2a56 a206
a113
a211a2a18a19a30
a231a63a25a177a232a2a233a2a224a50a225a65a234a2a103a19a24a19a235
a38a19a236a19a216
a204 a18a26a66a237a121a19a128a194a230a65a21a19a186
a57
a55a19a56 a235
a113
a211a19a18a19a30a19a231a26a25a177a13a19a14
a95
a179a2a238a2a20a2a24 v a97a65a98a2a18a2a77a2a78a2a79
a62
a25a69a103
P(X2n = v) =
nX
i=0
(2n)!
k!k!(n k)!(n k)!
1
4
2n
= 14 2n (2n)!(n!)2
nX
k=0
CknCn kn
= 14 2n(Cn2n)2:
a34
a38
a25 a226a2a227 a18a2a239a2a24a2a17a19a18a2a77a19a78a19a79
a62a19a38a2a216
a204 a18a63a66
a240
a114a65a151
a18a2a186a2a241 a139 a43 a224a50a225a141a25a177a242 a227a2a43 a17a2a18a19a239a19a24a2a17a19a18a19a77a2a78a19a79
a62a19a38a19a236a2a216
a204 a18a67a66
a178 Zd a17a2a18a2a77a2a78a2a79
a62
a25 a46 a20 a106 a235 a139 a43 a240 a20a2a21
a57a158a243a2a244a19a245
a66a138a99a19a246 n a106 a18
a57a65a243
a74
n
a25a177a146
n a38a2a38a2a247a2a248a50a114
a53a2a100a19a18a2a77a19a78a19a249
a243
a25
a115a19a250a19a251
a74 f e
i; i = 1; ; dg
a25
a115
a107 e
i
a74a2a252a2a246 i a21a2a186
a57
a17a2a18 a48 a35
a57a158a243
a66a253a104a22a30a26a25a253a246 n a106 a23a2a24a2a18a2a35a19a36 X
n a108a2a38 n
a21
a247a2a248a50a114
a53a2a100a2a18a2a77a2a78a2a249
a243
a18
a172
a89
Xn = 1 + 2 + + n:
a254a2a54a2a18a67a25a69a128 jxi
n a255
a124a2a21
a57a65a243
a18 a55a2a56 a235
a113
a211a19a18a2a30a19a231a63a25a138a125
a108a19a38a2a127
a25
P( n = ei) = P( n = ei) = 12d;
a104a2a30a2a73a2a77a2a78a2a79
a62
a74 Zd a17a2a18a1a0 a48 a77a2a78a2a79
a62
a25a2a0a19a99a2a74 SRWa66
a17a45a20a4a3a4a5a45a131
a136
a160a65a17a2a77a2a78a2a79
a62
a18 a55a2a56a2a161a50a162a1a6a8a7 a18 a162a65a163 a103a2a166a1a9a63a25a11a10a65a74a2a32a2a82a2a235
a147a2a148a1a12
a172
a186a2a187a67a66a69a86a19a87a19a193a1a13a15a14a1a16 a240 a162a65a163 a197a1a17a15a18a19a77a2a78a19a79
a62
a18a20a19a8a21a63a66
3
Ch6 4
a99 N = N(!) = fa77a2a78a2a79
a62
a204a50a205 0 a24a2a18a2a208a2a15 ga25a69a146a2a103
N =
1X
n=1
1(Xn=0)(!):
a22a65a17a2a20a1a3 a189 a18a2a94
a181
a25a69a103
P(N = 1) = 1 () EN =
1X
n=1
P(Xn = 0) = 1:
a94a1a23
i = inf
n
fn > 0,Xn = ig; +i = inf
n
fn > 0,Xn = ig;
a146 0 a24
a216
a204
a182
+
0 < 1
a66
a178a2a34 Zd a17a2a18a2a77a2a78a2a79
a62
a25 a13a2a14
a95 0 a24a50a97a65a98a2a18 SRW a37
a139 a43a1a24 a203a1a25
a61
a193a19a179
a238a2a20a2a24a2a249a1a26 0 a24a194a40 a25 a189a2a46 a221a19a173a2a175a19a17 a162a158a174 1a25a209a175 a162a65a163a2a176a2a150 0 a24 a162a65a163 a74 1a25
a90a2a212a1a27
a33 a162a65a163 a74 0a66a122a146 a139 a43 a224a50a225 a25 0 a24
a216
a204 a18a2a219a2a220a19a221a19a222
a38a19a95 0
a24a2a131
a90a2a212a2a96
a24a19a18a19a211a15a28
a162a65a174 a74
a90a2a212
a165a67a66
4
a0a2a1a2a3 *
a4a2a5a2a6a2a7a2a8
x1 a9a2a10a2a11a2a12
a13a2a14a2a15a2a16a2a17a19a18a19a20a22a21a19a23a22a24a26a25a28a27a19a29a22a30a19a31 t = 0 a30a2a32a2a33a2a34a2a35a19a36 a
a37 a a38a2a39
a15a41a40a42a25
a43a45a44a45a46a45a47a45a48 a35a2a30a50a49a51a25a52a32a2a53a2a54 a43a2a55a2a56 p
a57a59a58a60a57a59a61a45a62
a20a2a21 a48 a35a63a25 a43a2a55a2a56 q = 1 p
a57a65a64a50a57a65a61a2a62
a20a2a21 a48 a35a67a66a69a68a2a70a19a71a2a72a19a73a2a74a19a75a2a76a19a17a2a18a19a77a2a78a19a79
a62
a66a69a20a19a80a2a81a19a82a2a83a19a84a2a85a19a13
a14a2a86a2a87a2a88a2a70a2a71a2a72a63a89
(1) a90a2a91a2a92
a18a2a77a2a78a2a79
a62
a93a2a94a2a23a2a24a2a29a2a30a2a31 0
a95a2a96
a24a50a97a65a98a67a25a69a99
Xn = a23a2a24a2a29a2a30a2a31 t = n a30a2a18a2a35a2a36 ;
a13a2a14 X
n
a18a2a53a2a100 P(X
n = k)
a25a69a101a2a102a2a103 jkj > n a30 P(X
n = k) = 0
a66a69a104a2a30a67a25a69a105 n
a106a50a107 a103 (n + k)=2 a106
a57a65a58a50a57a65a61a2a62
a25 (n k)=2 a106
a57a65a64a50a57a65a61a2a62
a66a69a34
a38a19a108
a103
P(Xn = k) =
(
C n+k2n pn+k2 q n k2 ; jkj 6 na109 k; n a110a2a111a2a112a2a113a50a114 ;
0; a115a2a116,
(2) a117a2a118a2a119a2a120
a18a2a77a2a78a2a79
a62
a93a2a94a2a23a2a24a2a29 0 a30a2a31a2a35a2a34 x = aa25a122a121a2a29 x = 0
a123 x = a + b = c a33a2a124a2a103a2a20
a118a45a119a45a120
a25
a115
a107 a; b > 0a66a126a125
a108a45a38a45a127
a25a129a128a2a23a19a24a2a130a2a131 0
a132 c a107 a18a45a20a45a24a45a30a63a25 a108a19a133a2a134a2a61
a62
a25a69a135
a118a2a119a2a136
a66a138a137a19a23a19a24a2a135 x = 0
a123 x = c a118a2a119
a18 a55a2a56 a66
a139 a43a45a140 a97a141a25a142a68
a108a45a38a2a143a2a144
a71a2a72a63a66a145a99 r = q=pa25a142a146a45a23a45a24a45a135 0 a24
a118a45a119
a18 a55a45a56a45a147a2a148
pk = ppk+1 + qpk 1;
a34
a38
a139 a43a2a149a2a150
pk =
8
>><
>>:
ra ra+b
1 ra+b ; r 6= 1;
b
a + b; r = 1:
a114a65a151
a18a67a25a69a23a2a24a2a135 c a24
a118a2a119
a18 a55a2a56 a74
p k =
8
><
>:
1 ra
1 ra+b ; r 6= 1;
a
a + b; r = 1:
1
Ch6 2
(3) a117a2a152a2a153a2a120
a18a2a77a2a78a2a79
a62
a93a2a94a2a23a2a24a2a29 0 a30a2a31a2a35a2a34 x = aa25a122a121a2a29 x = 0
a123 x = a + b = c a33a2a124a2a103a2a20
a152a45a153a45a120
a25
a115
a107 a; b > 0a25a126a125
a108a45a38a45a127
a25a129a128a19a23a2a24a19a131a2a154 0 a24a45a30a67a25a126a86a45a20 a106
a108a19a155
a102
a57a65a156a2a61
a62
a131 1a66
a114a65a151
a25a69a128a2a23a2a24a2a131a19a154 c a24a2a30a67a25a69a86a2a20 a106
a155
a102
a57a158a157a2a61a19a62
a131 c 1a66
a159 a29a2a13a2a14a50a160a65a17a2a18a2a77a2a78a2a79
a62
a25 a55a2a56a19a161a50a162a158a163 a103a19a164a2a165a19a166a2a167a63a66 a159 a29 a43 a15a2a16
a58a2a168
a16
a17
a39
a24
a117a2a118a2a119a2a120
a18a2a77a2a78a19a79
a62
a74a19a169a63a66
a27a45a170a45a171a45a24 i
a172 i + 1 a18a45a173a45a17a45a18 a162a65a174 a74 qp i; i = 0; 1; 2; a25a129a175a45a17 a162a59a163a45a176a2a150
0 a24 a162a65a163 V0 = 1a25 c a24 a162a65a163 Vc = 0a66a177a68a2a30a63a25a177a178a2a34 0 a172 c a107 a49a65a18a2a179a2a180a2a20a2a24 k a24
a18 a162a65a163 V
k
a147a2a148
Vk 1 Vk
q
p
k 1 = Vk VK+1 q
p
k ;
a39a2a181a2a182
a150
Vk = pVk+1 + qVk 1;
a161 a17a45a87a45a18
a117a45a118a45a119a2a120
a18a2a77a2a78a2a79
a62a2a113a50a183a65a184
a139a2a185 V
k
a161 p
k
a147a45a148
a113a60a114
a18a45a186a45a187a63a25a145a188
Vk = pka66
a178a2a34a2a20a2a80a2a18a50a160a65a125 a139 a43a2a189a2a190a2a191a2a55a2a56a19a189 a94 a162a65a174 a25a69a102 a44a2a192 a94 a162a158a163 a25a69a193 a55a19a56a2a161a194a162
a163 a178a2a195a2a196a2a197a67a66
x2 a9a2a10a2a11a2a12a2a198a2a199a2a200a2a201
a178a2a34a2a15a2a16a2a17
a39
a24a2a18a2a77a19a78a2a79
a62
a25a177a13a19a14
a95 0 a24a50a97a65a98a67a25a138a202a2a203 2n a106 a44a2a204a50a205 0 a24
a18 a55a2a56
a37
a202a2a203
a110
a15 a106a19a206 a139a19a207 a204a194a205 0 a24a41a40a42a25
P(X2n = 0) = Cn2npnqn:
a178 n a137
a172
a103
1X
n=1
P(X2n = 0) =
1X
n=1
Cn2npnqn = 11 p4pq:
a128 p 6= q a30a67a25 4pq < (p + q)2 = 1a25a138a188a2a77a2a78a2a79
a62
a202a2a203 0 a24a2a18a2a208a2a15a2a74a2a103
a91
a18
a55a2a56 a74 1a66a209a121a2a128 p = q = 1=2 a30a67a25a209a17a2a210a2a211a2a34
a90a2a212
a165a67a25a69a125
a108a19a38a2a127
a25a69a77a19a78a2a79
a62
a83a19a202
a203 0 a24
a90a2a212a2a213
a208a67a66
2
Ch6 3
a214a2a215 a73a2a20a2a21a2a24
a38a2a216
a204 a18a26a25a177a217a19a218
a95
a68a19a20a19a24a50a97a158a98a19a18a19a77a19a78a19a79
a62
a83
a90a19a212
a208 a204a50a205
a68a2a20a2a24a67a66
a95
a20a2a24 v a97a65a98a2a18a2a77a2a78a2a79
a62a2a38a2a216
a204 a18a19a219a2a220a19a221a19a222a2a74
1X
n=1
P(Xn = v) = 1:
a29a2a68a2a223 a206 a189 a97a65a224a50a225a141a66
a171a2a86a2a197 a140a2a226a2a227 a18a2a228a2a229a67a66 a226a2a227 a103a50a230a158a21a2a186
a57
a25a69a128a50a230a158a21a2a186
a57
a18 a55a2a56 a206
a113
a211a2a18a19a30
a231a63a25a177a232a2a233a2a224a50a225a65a234a2a103a19a24a19a235
a38a19a236a19a216
a204 a18a26a66a237a121a19a128a194a230a65a21a19a186
a57
a55a19a56 a235
a113
a211a19a18a19a30a19a231a26a25a177a13a19a14
a95
a179a2a238a2a20a2a24 v a97a65a98a2a18a2a77a2a78a2a79
a62
a25a69a103
P(X2n = v) =
nX
i=0
(2n)!
k!k!(n k)!(n k)!
1
4
2n
= 14 2n (2n)!(n!)2
nX
k=0
CknCn kn
= 14 2n(Cn2n)2:
a34
a38
a25 a226a2a227 a18a2a239a2a24a2a17a19a18a2a77a19a78a19a79
a62a19a38a2a216
a204 a18a63a66
a240
a114a65a151
a18a2a186a2a241 a139 a43 a224a50a225a141a25a177a242 a227a2a43 a17a2a18a19a239a19a24a2a17a19a18a19a77a2a78a19a79
a62a19a38a19a236a2a216
a204 a18a67a66
a178 Zd a17a2a18a2a77a2a78a2a79
a62
a25 a46 a20 a106 a235 a139 a43 a240 a20a2a21
a57a158a243a2a244a19a245
a66a138a99a19a246 n a106 a18
a57a65a243
a74
n
a25a177a146
n a38a2a38a2a247a2a248a50a114
a53a2a100a19a18a2a77a19a78a19a249
a243
a25
a115a19a250a19a251
a74 f e
i; i = 1; ; dg
a25
a115
a107 e
i
a74a2a252a2a246 i a21a2a186
a57
a17a2a18 a48 a35
a57a158a243
a66a253a104a22a30a26a25a253a246 n a106 a23a2a24a2a18a2a35a19a36 X
n a108a2a38 n
a21
a247a2a248a50a114
a53a2a100a2a18a2a77a2a78a2a249
a243
a18
a172
a89
Xn = 1 + 2 + + n:
a254a2a54a2a18a67a25a69a128 jxi
n a255
a124a2a21
a57a65a243
a18 a55a2a56 a235
a113
a211a19a18a2a30a19a231a63a25a138a125
a108a19a38a2a127
a25
P( n = ei) = P( n = ei) = 12d;
a104a2a30a2a73a2a77a2a78a2a79
a62
a74 Zd a17a2a18a1a0 a48 a77a2a78a2a79
a62
a25a2a0a19a99a2a74 SRWa66
a17a45a20a4a3a4a5a45a131
a136
a160a65a17a2a77a2a78a2a79
a62
a18 a55a2a56a2a161a50a162a1a6a8a7 a18 a162a65a163 a103a2a166a1a9a63a25a11a10a65a74a2a32a2a82a2a235
a147a2a148a1a12
a172
a186a2a187a67a66a69a86a19a87a19a193a1a13a15a14a1a16 a240 a162a65a163 a197a1a17a15a18a19a77a2a78a19a79
a62
a18a20a19a8a21a63a66
3
Ch6 4
a99 N = N(!) = fa77a2a78a2a79
a62
a204a50a205 0 a24a2a18a2a208a2a15 ga25a69a146a2a103
N =
1X
n=1
1(Xn=0)(!):
a22a65a17a2a20a1a3 a189 a18a2a94
a181
a25a69a103
P(N = 1) = 1 () EN =
1X
n=1
P(Xn = 0) = 1:
a94a1a23
i = inf
n
fn > 0,Xn = ig; +i = inf
n
fn > 0,Xn = ig;
a146 0 a24
a216
a204
a182
+
0 < 1
a66
a178a2a34 Zd a17a2a18a2a77a2a78a2a79
a62
a25 a13a2a14
a95 0 a24a50a97a65a98a2a18 SRW a37
a139 a43a1a24 a203a1a25
a61
a193a19a179
a238a2a20a2a24a2a249a1a26 0 a24a194a40 a25 a189a2a46 a221a19a173a2a175a19a17 a162a158a174 1a25a209a175 a162a65a163a2a176a2a150 0 a24 a162a65a163 a74 1a25
a90a2a212a1a27
a33 a162a65a163 a74 0a66a122a146 a139 a43 a224a50a225 a25 0 a24
a216
a204 a18a2a219a2a220a19a221a19a222
a38a19a95 0
a24a2a131
a90a2a212a2a96
a24a19a18a19a211a15a28
a162a65a174 a74
a90a2a212
a165a67a66
4
