Ch2 1
a0a2a1a2a3 a4a2a5a2a6a2a7a2a8a2a9a2a10a2a11a2a12a2a13
x1 a4a2a5a2a6a2a7
a14a2a15a17a16 ( ;F; P)
a18a2a19a2a20a2a21a2a22a24a23a26a25 X(!) a18a2a27a2a28a2a29 a30a2a31a2a32a2a33a2a34a2a35a36a25a38a37
a39a2a40a2a41a2a42
a32a2a35 xa25a44a43a2a45
f!jX(!) 6 xg 2 F;
a46a48a47 X(!)
a18 ( ;F; P) a30a48a31a48a19a48a49a48a50a48a51a48a52a48a53a36a54a56a55a48a57a58a18a48a59a60a25a61a50a58a51a48a52a58a53a48a18a58a62a48a20a58a21
a22a24a23a64a63a2a32a2a35a2a65 Ra30a2a31a2a66a2a67a2a68a2a69a70a54
a45a2a71a2a72a2a73a2a74a2a75a2a57a2a76a48a35a48a77a2a45a48a78a60a25a79a37a48a75a48a80a36a81a79a82a48a83a60a25a79a45a48a71
a46
a18a48a84a48a85a48a86a48a87a48a88a2a89a48a90
a91
a33a70a25a44a37a2a92a94a93a61a95a2a96a98a97a100a99a2a101a48a31a2a102a48a103a36a54
a104a2a105a2a106a24a107a64a108
a49a2a109a2a110a70a54
a111 1.1
a112a2a113a2a19a2a114a2a115a2a116a70a25
a46
X(!) =
(
1; ! = !1 = a117a119a118a64a120a2a121a2a30a36a122 ;
0; ! = !1 = a117a119a118a64a120a2a121
a104
a122
a18a2a19a2a49a2a50a2a51a2a52a2a53a70a54
a111 1.2
a112a2a113a2a123a2a114a2a115a2a116a70a25
a46
X(!) = a117a119a118a64a120a2a121a2a30a2a31a2a115a2a116a2a35a60a122 ;
Y (!) = a117a119a118a64a120a2a121
a104
a31a2a115a2a116a2a35a60a122
a43a2a18a2a50a2a51a2a52a2a53a70a54
a111 1.3
a83a2a124a2a112a2a113a2a19a2a114a2a115a2a116a70a25a126a125a48a63a2a127a48a19a48a128
a107a100a129
a118a64a120a48a121a2a30a36a25
a46
X(!) = a117a131a130a2a132a2a112a2a113a2a128a2a35a36a122
a18a2a19a2a49a2a50a2a51a2a52a2a53a70a54
a111 1.4
a133a2a134a2a135a2a136a2a137a2a138a2a139a48a140 5 a141a2a142a2a45a2a19a2a143a2a136a2a137a2a144a48a145a60a25a38a37
a39
a133a2a92a48a63a48a146a48a147a48a137
a138a2a31a2a148a24a23a64a18a2a50a2a51a2a31a70a25a126a149a48a150a36a25a126a151a48a152a48a137a2a31a48a148a98a23 X(!) a18a2a19a2a49a2a50a2a51a2a52a2a53a70a54
a40a2a153a2a154
a82a2a31a2a50a2a51a2a52a2a53a70a25a155a144a2a156a48a141a2a84a48a123a2a157a48a158a2a159
a154
a82a48a160a2a161a36a54a44a37
a39
a19a48a49a2a50a48a51a2a52a48a53
X(!) a66a163a162a163a164a163a31a163a33a163a18a163a66a163a35a163a49a70a25
a46a2a47 X(!)
a18a163a165a163a166a163a167a163a50a163a51a163a52a163a53a168a54a170a169a2a165a2a166a163a167a2a31a2a50a2a51
a52a2a53a2a171a24a172a64a173a48a174a60a25a56a175a176a97a100a177a48a83a58a178a48a31a58a55a48a18a48a32a58a179a98a180a100a181a176a97a100a177a58a156a48a182a48a31a58a18a48a183a48a184a58a167a48a50a58a51a48a52a48a53a60a54
a29a2a185a2a186a70a25a44a187a2a160a2a161a2a165a48a166a48a167a2a31a48a50a2a51a48a52a48a53a36a54
1
Ch2 2
x2,a188a2a189a2a190
a4a2a5a2a6a2a7
a165a2a166a2a167a2a50a2a51a2a52a2a53 X(!) a191a2a164a2a66a2a35a2a49a2a33a36a25a38a192a48a84 x1; x2; ; xna54a79a193a2a194a2a31a36a25a79a195
a164a2a196a2a49a2a33a2a31a2a20a2a21a2a141a2a103a2a192a48a84
pi = P(X(!) = xi) = P(f!,X(!) = xkg); i = 1; 2;,
a47 p
1; p2; a84 X a31a163a20a163a21a163a141a163a197a163a198a70a25a170a195a163a199a2a200a2a31a2a201a2a202a163a85 X a164a163a33a163a31a163a20a163a21a163a141a163a197a163a203a163a204a36a54
a104a2a105a2a205a2a206a2a108a2a207
a156a2a182a2a31a2a20a2a21a48a141a2a197a48a208a48a175a2a102a48a209a36a54
(1) a123a2a210a2a141a2a197a212a211 Bernoulli a141a2a197a176a213
a37
a39
a50a2a51a2a52a2a53 X a31a2a141a2a197a2a37
a104a36a214
P(X = 1) = p; P(X = 0) = q;
a175a24a97 0 < p < 1; q = 1 pa25
a46a2a47 X
a215a2a62a2a216a2a35a2a84 p a31a2a123a2a210a2a141a2a197a70a54
a123a163a210a163a141a163a197a163a18a163a177a163a217a2a218a2a31a2a19a2a49a2a141a163a197a2a157a36a54
a41a2a219
a19a2a49a2a191a2a45a2a123
a207
a66a2a162a2a220
a39
a31a2a50a2a51a2a52
a53a70a25a61a221a64a37a2a92a2a31a2a102a2a103a36a25a223a222a100a224a48a18a48a225
a104a48a226
a152a36a25a227a43a2a66a48a228a48a229a2a123a48a210a48a141a2a197a48a230a2a231a48a232a36a54
(2) a233a2a234a2a141a2a197
a37
a39
a50a2a51a2a52a2a53 X a31a2a141a2a197a2a37
a104a36a214
P(X = k) = Cknpkqn k; k = 0; 1; 2; ; n;
a175a24a97 0 < p < 1; q = 1 pa25
a46a2a47 X
a215a2a62a2a216a2a35a2a84 (n; p) a31a2a233a2a234a2a141a2a197a36a25a79a235a2a236a48a229
a192a2a237
X B(n; p)
a230a2a201a2a202a70a54
a238a2a239
a25a44a216a2a35a2a84 p a31a2a123a2a210a2a141a2a197a70a25a44a57a2a18a2a233a48a234a48a141a2a197a98a97a64a31a48a19a48a49a2a110a48a157 B(1; p)a54
a230a2a240a2a19
a104a2a104a2a105
a31a2a109a2a110a70a54
a111 2.1
a133a2a69a2a241a2a242a2a241a24a97 10 a243a2a31a2a20a2a21a2a84 0.4a25
a129a2a244a2a245
a31a2a69a2a85 5 a246a36a25
a46
a151a2a247
a45a2a123a2a128a2a248a24a97a64a31 10 a243a2a31a2a20a2a21a2a84
p = C25 0:42 0:63:
a220
a39a2a249a2a250
a18a2a233a2a234a2a141a2a197a2a31a2a251a48a252a36a54
a129
a29
a106a98a107
a19a2a49a98a221a100a253a2a19a48a254a2a31a98a180a100a181
a214
2
Ch2 3
a255a1a0
a218a2a128a3a2a5a4a98a97 a72a48a73 A a246a3a6a2a31a2a20a2a21a48a18 pa25 a149a2a150 n a128a2a83a2a124
a244a2a245
a2a5a4a176a97a100a72
a73 A a247
a250
a246a3a6 k a128a2a31a2a20a2a21a2a57a2a18
P(a72a2a73 Aa247a2a246a3a6 ka128 ) = Cknpkqn k; k = 0; 1; 2; ; n:
a7a1a8
a25a44a29 n a128a2a83a2a124
a244a2a245
a2a3a4a24a97 a25a44a192a48a50a48a51a48a52a2a53
X = Aa246a3a6a2a31a2a128a2a35,
a46 X B(n; p)
a54
(3) a9a3a10a212a211 Poissona213 a141a2a197
a37
a39
a50a2a51a2a52a2a53 X a31a2a20a2a21a2a141a2a197a2a37
a104a36a214
P(X = k) =
k
k! e
k; k = 0; 1; 2; ( > 0);
a46a2a47 X
a215a2a62a2a216a2a35a2a84 a31a3a9a3a10a2a141a2a197a70a25a44a192a3a11 X P( )a54
a40a2a153
a233a2a234a2a141a2a197a70a25a13a12a3a14a5a15a3a16
a104a48a105
a31a5a17a3a18a48a203a48a204
a214
np = ; n ! 1:
a8
a148a70a25
Cknpkqn k = n!k!(n k)! n k 1 n
=
k
k!
n(n 1) (n k + 1)
nk
1
n
n
1
n
k
!
k
k! e
(n ! 1) ;
a19
P(X = k) =
k
k! e
k; k = 1; 1; 2;,
a229a2a157a3a20a2a31a3a21a3a22a2a66a2a228a3a23a2a63
a104a48a105
a31a48a220
a39 a214
a24 np ! ( > 0; n ! 1)
a25
a46
a45
Cknpkqn k !
k
k! e
k (n ! 1) ;
3
Ch2 4
a19
a9a3a10a2a141a2a197a2a18a2a233a2a234a2a141a2a197a5a25 np ! (n ! 1) a203a2a204
a104
a31a3a17a3a18a2a141a2a197a70a54
a26
a229a2a185a2a49a2a220
a39
a25a44a66a2a229a5a9a3a10a48a141a48a197a2a230a5a11a2a233a48a234a48a141a2a197a48a31a5a27a3a20a5a28a3a29a36a54
(4) a108 a219 a141a2a197
a29a2a72a2a73 A a29a2a218a2a128a3a2a3a4a24a97a64a246a3a6a2a31a2a20a48a21a2a84 p a31
a244a2a245
a83a2a124a3a2a3a4a24a97 a25a126a192
X = Aa30a2a128a2a246a3a6a2a148a2a31a3a2a3a4a2a128a2a35 ;
a31a3a32
a4a3a33a70a25 X a154 a45a2a37
a104
a31a2a20a2a21a2a141a2a197
a214
P(X = k) = qk 1p; k = 1; 2; ;
a185a2a49a2a20a2a21a2a141a2a197
a47
a84
a108 a219
a141a48a197a70a54
a108 a219
a141a2a197a2a45a3a34a2a192a3a35a48a102a60a25a227a55a48a57a48a18a48a59a60a25a227a29a5a36 m a128a3a2a3a4a24a97 A a43a2a19a2a125a3a37a2a45a2a246a3a6
a31a39a38a163a73
a104
a25 a152a163a63 A a246a39a6a163a130a163a132a163a178a163a31a39a2a3a4a2a128a2a35 X a40 X a141a163a197a163a193a42a41 a54 a90a39a43a36a25 a66a2a228a3a33
a222 a25a44a29a2a164a2a33a2a84a3a44a2a200a48a35a48a31a2a165a48a166a2a167a48a141a48a197a24a97a26a25
a108 a219
a141a48a197a2a18a5a45a2a19a48a31a5a34a2a192a5a35a2a102a48a31a48a141a2a197a36a54
(5) a46
a108 a219
a141a2a197
a16
a19a39a47a42a41 a157a3a48a3a49a2a135 N a49a70a25 a175 a97 a45 M a49a163a128a39a49a70a25
a129
a62a24a97
a41
a164 n a49 a211 a84a163a86a163a87
a192a36a25a51a50a2a27 n 6 N Ma213 a25
a46
a185 n a49a24a97a64a130a3a52a2a31a2a128a5a49a48a35 X a18a2a19a2a49a2a165a2a166a2a167a2a50a48a51a48a52
a53a70a25a44a175a2a20a2a21a2a141a2a197a2a37
a104a60a214
P(X = m) = C
m
MC
n m
N M
CnN ; m = 0; 1; 2; ; l;
a175a24a97 l = min(M; n)a54
a185a2a49a2a20a2a21a2a141a2a197
a47
a84a3a46
a108 a219
a141a2a197a36a54
a104a2a105
a240a3a46
a108 a219
a141a2a197a2a31a5a17a5a18a48a203a2a204a60a54
a40a48a153a48a41a48a42 a106
a27a48a31 n; m( 6 m 6 n)a25a126a37
a39
a25 N ! 1a148a70a25 M=N ! p > 0a25a44a55a2a57a2a18a2a59a70a25a13a48a3a49a3a53a2a35a3a54a56a55
a153
a34a3a57a48a90a48a128a3a49a56a58
a31a24a221a64a109a3a54a59a55
a153 p
a25
a46
a45
CmMCn mN M
CnN ! C
m
n p
mqn m (N ! 1):
a60 a61a3a62a3a63a3a64
a49a3a65a3a66a2a35a2a141a2a103a5a67a3a65a5a66a48a134a2a252a5a68a3a69a36a25
a239a5a70
a200a48a89a2a164a5a17a5a18
a19
a23a36a54
(6) a71a2a233a2a234a2a141a2a197
a29
a244a2a245
a83a2a124a3a2a3a4a24a97 a25a44a72a48a73 A a246a3a6a2a31a2a20a2a21a2a84 pa25
a40 a106
a27a2a31 ra25a13a15a3a16
X = A a127 r a128a2a246a3a6a2a148a2a31a3a2a3a4a2a128a2a35a70a25
4
Ch2 5
a31a3a32
a4a3a33a70a25 X a154 a45a2a37
a104
a31a2a20a2a21a2a141a2a197
a214
P(X = k) = Cr 1k 1qk rpr; k = r; r + 1;,
a129
a29a3a12a3a14a3a72a2a27
Y = X r;
a15a3a16 Y a31a2a20a2a21a2a141a2a197a70a25
a40a2a153a74a73 a239
a35 ka25
P(Y = k)P(X = k + r) = Cr 1k+r 1qkpr:
Y a31a2a141a2a197
a47
a84a2a216a2a35a2a84 (r; p) a31a3a71a2a233a2a234a2a141a2a197a70a54
a66a2a228a2a240
a107
a25a44a30
a105
a31 X a31a2a141a2a197a2a18 r a49
a244a2a245
a31
a108 a219
a141a2a197a2a31a3a75a36a54
a15a3a16a3a71a2a233a2a234a2a252 (1 x)r a31 Taylor a68a3a69a2a252
a214
(1 x)r =
1X
k=0
( r)( r 1) ( r (k 1))
k! ( x)
k
=
1X
k=0
(r + k 1) (r + 1)r
k! x
k
=
1X
k=0
Cr 1k+r 1xk;
a185a2a55a2a57a2a18
a47
a84a3a71a2a233a2a234a2a141a48a197a48a31a3a76
a7
a54
x3 Poisson a77a3a78
a229 X(t) a201a2a202a2a62 0 a69a3a79a36a25a79a133a3a80a2a27a2a72a48a73a48a246a5a6a48a31a5a53a48a128a48a35a60a25a38a57a48a66a48a228
a47
a147a48a145a5a81a48a84
a28a2a35a2a145a3a81a70a54
a40a2a153
a28a48a35a48a145a3a81a36a25a126a45
(1) X(t) a18a2a169a3a71a2a200a2a35a83a82
(2) a41a2a42 0 6 t1 6 t2a25a44a45
X(t1) 6 X(t2):
a32a2a179a2a30a70a25a44a185a2a186a2a31 X(t) a194a2a147a3a84a3a85 X(t; !).
a14a2a15 a47
a28a2a35a2a145a3a81 X(t) a84
a154
a45a2a216a2a35 a31 Poisson a145a3a81a70a25a44a37
a39
5
Ch2 6
(1) X(0) = 0;
(2) a40a2a41a2a42 a31 0 6 t1 6 t2 6 tna25
X(t2) X(t1); X(t3) X(t2); ; X(tn) X(tn 1)
a18a2a193a3a86
a244a2a245
a31a212a211
a244a2a245a5a87
a53a48a102a176a213a88a82
(3) a40a2a41a2a42 a31 s; t > 0a25a44a45
X(s + t) X(s) P( t);
a19
a215a2a62a2a216a2a35a2a84 t a31 Poisson a141a2a197a70a54
Poisson a145a3a81a2a45a2a19a2a49a2a152a3a89a2a31a2a27a2a28a2a37
a104a60a214
a16
a28a2a35a2a145a3a81 fX(t); t > 0 a90a3a91
(1) X(0) = 0a82
(2) a154 a45
a244a2a245a3a87
a53a2a102a83a82
(3) a40a2a41a2a42 a31 s; t > 0a25 X(s + t) X(s) a31a2a141a2a197a2a76 s a34a2a78a212a211 a148a3a92a2a102a176a213a93a82
(4) a40a2a41a2a42 a31 s; t > 0a25a44a45
P(X(t + 4t) X(t) = 1) = 4t + o(4t);
P(X(t + 4t) X(t) > 2) = o(4t);
a46 fX(t); t > 0g
a84a2a19a2a216a2a35a2a84 a31 Poisson a145a3a81a70a54
a104a2a105
a33a24a222a64a123a2a49a2a27a2a28a2a31a2a152a3a89a2a102a70a54a44a62a48a127a2a19a48a49a2a27a48a28a3a21
a107
a127a48a233a2a49a48a18
a238a48a239
a31a36a25a44a191a48a132
a33a24a222a64a127a2a233a2a49a2a162a3a21
a107
a127a2a19a48a49a36a54
a192 pk(t) = P(X(t) = k)a25
a46 a40a2a41a2a42
a31 4t > 0a25a44a45
pk(t + 4t) = P(X(t + 4t) = k)
= P(X(t) = k; X(t + 4t) = k)
+P(X(t) = k 1; X(t + 4t) = k)
+P(X(t) 6 k 2; X(t + 4t) = k)
= P(X(t) = k; X(t + 4t) X(t) = 0)
+P(X(t) = k 1; X(t + 4t) X(t) = 1)
+P(X(t + 4t) = k; X(t + 4t) X(t) > 2):
6
Ch2 7
a94 (2)(3)(4)
a66
a0
pk(t) = P(X(t) = k)P(X(t + 4t) X(t) = 0)
+P(X(t) = k 1)P(X(t + 4t) X(t) = 1)
+P(X(t + 4t) = k; X(t + 4t) X(t) > 2
= pk(t)(1 4t) + pk 1(t) 4t + o(4t):
a153
a18a2a45
pk(t + 4t) pk(t) = pk(t) 4t + pk 1(t) 4t + o(4t);
p0k(t) = pk(t) + pk 1(t); k > 1:
a25 k = 0 a148a70a25a95a41a64a30a2a141a3a96a2a45
p00(t) = p0(t):
a66a2a228a3a97
a107
p0(t) = e t:
a98
a229a3a99a3a100a3a101a2a57a2a66a2a228a3a21
a107 p
k(t)
a214
pk(t) = ( t)
k
k! e
t; k = 0; 1; 2;,
a102a3a103
a4a3a33a70a25a44a130a3a104
a107
a31a48a141a2a197a5a90a5a91a2a127a48a19a2a49a48a27a48a28a36a54a106a105a48a123a2a49a48a27a48a28a2a18a48a152a3a89a48a31a36a54
7