Ch2 1
x6 a0a2a1a2a3a2a4a2a5a2a6
a7a9a8a9a10a9a11a9a12a2a13a2a14a16a15a2a17a16a18a2a19 F
a20a22a21a9a23a9a24a9a25a2a26
a14a2a27a16a18a2a19 F 1
a28a22a29a9a30a9a31a9a32a2a33a2a34
a35a2a36a2a37a2a38a2a39a2a40 a14a2a18a2a19
a20a42a41a16a43a44a20
a7a16a8 a39a16a40 a14a2a15a16a17a16a18a2a19
a20a42a45a47a46a49a48a2a50
a14
a34
a35a52a51
a53a2a54a56a55 F(x)
a57
a39 a15a2a17a2a18a2a19
a20
a34
a35 F 1(y)
F 1(y) = supfxjF(x) < yg; 0 < y < 1:
a58a2a59
a20 F 1(y) a57
a12a2a13a2a14a61a60a63a62 a39
a34
a10a2a11a16a12a16a13a65a64
a28
a50a2a66a16a45a47a46a68a67
a8 F 1 a39a2a69a2a70a2a71a2a72
a73,
a74a2a75
F 1(y) 6 x () y 6 F(x):
a76a78a77a2a79a2a80a82a81 a39a2a69a2a83a2a84 a14 a72a16a73
F 1(y) > x () y > F(x):
=)a51a86a85 F 1(y) > x a87 9x 0 a88a2a89 x < x < F 1(y)a28a91a90 a85 a15a2a17a2a18a2a19a2a14
a92a2a93a2a94
a21a2a87
F(x) 6 F(x ) < y:
(=a51a95a85 y > F(x) a96
a15a9a17a9a18a9a19a9a14a9a97a2a98a2a99a2a94
a87 9" > 0 a88a9a89 y > F(x+")a20
a8
a57a2a100
F 1(y) > x + " > x:
a59a2a101a2a102
a21a2a23a2a89
a38
a50a2a66
a14a2a103a16a104a2a105 a70a16a71 a14
a34a2a106
a51
a53a2a107a56a55 F(x)
a57
a15a2a17a2a18a2a19
a20 X U[0; 1]a20a95a108 Y = F 1(X) a14a2a15a2a17a2a18a2a19 a57
F( )a28
a76
P(Y 6 y) = P(F 1(X) 6 y)
= P(X 6 F(y))
= F(y):
F 1( ) a109a2a100a2a23a2a50
a103a2a110a2a94a2a111 a51
(1) F(F 1(y))) > y;
(2) F 1(F(y)) 6 y;
1
Ch2 2
(3) a112 F(x) a30 x = F 1(y) a113
a98a2a99
a20a95a108 F(F 1(y)) = y;
(4) F 1(y) a114 a98a2a99 a28
2
Ch3 3
a115a2a116a2a117 a118a2a119a2a120a2a121a2a122a2a123a2a124a2a125a2a3a2a4
x1 a126a2a127a2a128
a118a2a119a2a120a2a121a2a122a2a123a2a124a2a125a16a129a2a130a16a5a16a6
a53a131a54 a132a131a133a131a134a131a135a65a136 ( ;F; P)
a137 n a69a131a138a131a139a131a140a131a141 X1; X2; ; Xn a14a131a142a131a143
X = (X1; X2; ; Xn) a41 n a144
a138a2a139a82a145a68a141
a28
a105a2a146 a138a2a139
a29a2a147a65a148a49a149a131a149a16a150
a96
a38 a146 a69a16a138a16a139a131a140a16a141
a20a152a151a16a48a131a153a16a154
a14a16a155a16a156 (X; Y ) a102
a57a2a157
a69a2a138a2a139a2a140a2a141a2a158a2a159 a14 a39a2a69a16a138a2a139a47a145a68a141
a28
a138a16a139a82a145a49a141
a21a16a23a2a57a16a100a2a160a16a144
a14
a20a95a161a16a21a2a23a16a57
a162a2a163
a144
a14
a28
a39a2a69 n a14 a138a2a139a82a145a68a141 X = (X
1; X2; ; Xn) a21a2a23a2a164a2a165a2a48a2a50
a14a2a166a2a167 a51
! Rn;
! ! (X1(!); X2(!); ; Xn(!)):
a33a82a148
a20a95a168
a39a2a69 a15 a141a2a169
a57a2a21a16a170
a14
a20a42a171
f!jXi(!) 6 aig2F:
a172
a43a173a20
n\
i=1
f!jXi(!) 6 aig
= f!jX1(! 6 ai; X2(!) 6 a2; ; Xn(!) 6 ang2F:
a8
a57a173a20 P(X1 6 a1; X2 6 a2; ; Xn 6 an)
a102
a100a2a174
a35a2a175
a28
a176a68a177 a14
a20
a34
a35 n
a144a2a178a2a179
a15a2a17a2a18a2a19
a41
F(a1; a2; ; an) = P(X1 6 a1; X2 6 a2; ; Xn 6 an):
a176 a39
a144
a14a2a15a2a17a2a18a2a19 a39 a177
a20a180a178a16a179
a15a16a17a2a18a16a19
a100a2a23a16a50
a94a2a111 a51
(1) lim
ai! 1
F(a1; a2; ; an) = 0; i = 1; 2; ; n;
lima
1!+1a
2!+1
an!+1
F(a1; a2; ; an) = 1;
(2) F a7 a168
a69 a15 a141a2a169
a57
a92a2a93a2a181a16a182a2a14a52a183
(3) F a57
a97a2a98a2a99a2a14
a20a95a184a2a114a16a185a2a160a16a186
a30a44a28
3
Ch3 4
a50a2a66
a34
a35 a98a2a99a2a187a2a14 a138a2a139a82a145a49a141
a28
a53a2a54a188a132 n
a144
a138a2a139a82a145a68a141 (X
1; X2; ; Xn) a57
a98a2a99a2a187a2a14
a20a189a48a16a190a131a186
a30
a181a131a191
a21
a192 a18a2a19 p(x
1; x2; ; xn)a20a95a88a2a89
a7a2a193
a174
a14 x
1; x2; ; cna20
a169
a100
F(x1; x2; ; xn) =
Z
Z
D
p(t1; t2; ; tn)dt1 dtn:
a33a194a148
a20 D = f(t1; ; tn)jt1 6 x1; ; tn 6 xnga20a196a195
a132 p(x
1; ; xn) a41
X1; ; Xn a14 a178a2a179a2a197a2a198
a18a2a19
a28
a199a2a200
a164a82a46a201a20a202a178a2a179a2a197a2a198
a18a2a19
a195
a62a2a203 a39
a20a95a204
a140
a21
a19 a69 a156a16a14a2a18a16a19a2a205
a109a2a57a16a197a2a198
a18
a19
a28
a50a2a66a2a45a82a46a68a157
a69a2a206a2a70a2a71 a14a2a15a16a17
a28
(1) a207a2a208
a15a2a17
a132 (X
1; X2; ; Xn) a41 n a144a82a209a68a210 G a137
a14
a207a2a208
a15a2a17
a20a95a48a16a190
a33
a178a16a179a2a197a16a198a2a41
p(x1; ; xn) =
(
c; (x1; ; xn) 2 G;
0; a33a2a211,
a21a2a23a2a164a82a46a201a20a95a207a2a212
a15a16a17a16a213a2a214
a137
a102
a57a16a215a2a216
a133 a187
a28
(2) a146 a144a2a217a2a218
a15a2a17
a219
a34
a35a2a220
a144
a14
a28
a48a2a190 (X; Y ) a100a2a178a2a179a2a197a2a198
p(x; y) = 12
1 2
p1 2 exp 12(1 2)
(x 1)
2
21
2 (x 1)(y 2)
1 2 +
(y 2)2
22
,
a33a82a148
a20 1 < 1 < 1a20 1 < 2 < 1a20 1 > 0a20 2 > 0a20 1 < < 1a20a91a108
a132
(X; Y ) a221a2a222
a220
a144a2a217a2a218
a15a2a17 N(
1; 2; 21; 22; )a28
a59a2a101a2a223
a34
a35a2a224
a144
a14
a28
a132 n
a144
a138a16a139a47a145a49a141 (X
1; X2; ; Xn) a221a16a222 n a144a16a217a16a218
a15a16a17
a20a152a48a16a190a16a26a16a225a16a100a131a197
a198
p(x1; x2; ; xn) = 1(2 )n=2j j1=2 exp 12(x )0 1(x ) ;
4
Ch3 5
( 1; 2)
z = p(x; y)
x
y
z
O
a33a226a148
a20 x = (x1; x2; ; )0a20a228a227 = ( 1; 2; ; n)0 a57a9a229
a19 a145a230a141
a20 = ( ij)n n
a57 n a231a2a217
a34a2a232a82a233a201a28a95a234a2a235a2a236 X N( ; )a28
a199a2a200
a164a82a46a201a20a95a237
= ( 1; 2)0 ; =
21 1 2
1 2 22
!;
a102
a57
a220
a144a2a217a2a218
a15a2a17
a28
x2 a238a2a239a2a128
a118a2a119a2a120a2a121a2a122a2a240a2a241a16a3a16a4a2a5a16a6
a53a9a54 a132 n
a144
a138a9a139a226a145a230a141 X = (X
1; X2; Xn a57a9a242a9a243
a187a9a14
a20a22a48a2a190a2a26a16a244a2a237a2a245
a146
a21
a19 a69 a62 a176 a14a2a205
a28
a246a2a247 a14
a20a95a23
a220
a144a2a41a2a151a44a20
a33
a133a16a134 a15a16a17a16a248
a34
a35
a48a2a50
a51
pij = P((X1; X2) = (x1i; x2j)); i = 1; 2; ; j = 1; 2;,
a33a82a148 pij
a249a2a250
(1) pij > 0; i = 1; 2; ; j = 1; 2; ;
(2)
X
i;j
pij = 1:
a242a2a243
a187 a138a2a139a82a145a68a141
a148
a20a95a251a16a252
a14
a57
a146a16a253a2a15a16a17
a20
a33a16a34
a35
a48a16a50
a51
a132 (X
1; X2; ; Xr) a221a2a222a2a229
a19
a41 (n; p1; p2; ; pr) a14a2a146a2a253a2a15a2a17 a20a95a48a2a190
P((X1; ; Xr) = (k1; ; kr)) = n!k
1! kr!
pk11 pkrr ;
a33a82a148 k1; ; kn
a41
a181a2a191a2a142a2a19
a184
rX
i=1
ki = n;
rX
i=1
pi = 1a28
5
Ch3 6
a254 2.1
a255
a148
a100 2 a69a1a0a1a2 a20 3 a69a4a3a5a2 a20 5 a69a1a6a1a2a1a7 10 a69a1a2 a20 a222
a148
a100a1a8
a9 a14
a237 7 a10a52a20 a112
a235 X1; X2; X3
a15a1a11
a41a16a237
a38 a14 a0a12a2
a20
a3a13a2a12a14a12a6a12a2 a14a131a19a16a15
a20 a108
(X1; X2; X3) a57
a39
a242a2a243
a187 a138a16a139a47a145a49a141
a20 a184a131a221a16a222a16a229
a19
a41 (7; 0:2; 0:3; 0:5) a14a2a146a2a253a2a15
a17
a28
a53a131a54 a7a131a8 a39a131a69 a146
a144
a138a131a139a194a145 a141 X = (X
1; X2; ; Xn)a20a49a26
a14 a39 a110a131a15 a141
(Xi1; ; Xik); 1 6 i1 < i2 < < ik; 1 6 k < n a158a2a159a2a39a2a69a18a17 a14 a138a2a139a82a145a68a141 a20
a103
a69a18a17 a14 a138a2a139a82a145a68a141 a14a2a15a2a17a16a18a16a19 a102 a132
a41a1a19a1a20
a15a16a17a2a18a16a19
a28
a7a2a8
a242a2a243
a187 a138a2a139a2a140a2a141
a20a42a23
a220
a144a2a41a16a151a44a20
a55 (X; Y ) a14
a178a2a179
a15a2a17
a41
P((X; Y ) = (xi; yj)) = pij; i = 1; 2; ; j = 1; 2; ;
a108 X a14 a19a18a20
a15a2a17
a41
P(X = xi) =
X
j
P((X; Y ) = (xi; yj)) =
X
j
pij; i = 1; 2;,
a176a68a177
a20 Y a14 a19a18a20
a15a2a17
a41
P(Y = yj) =
X
i
pij; j = 1; 2;,
a7a16a8a16a98a16a99a16a187 a138a16a139a16a140a16a141
a20a86a26
a14
a19a12a20
a15a131a17a131a18a16a19
a161a131a100a131a197a131a198
a18a131a19
a28
a23
a220
a144a131a41a131a151a226a20
a55 (X; Y ) a14
a178a2a179a2a197a2a198
a18a2a19
a41 p(x; y)a20a95a108a18a19a18a20a2a197a2a198
a18a2a19
a41
pX(x) =
Z 1
1
p(x; y)dy; pY (y) =
Z 1
1
p(x; y)dx:
a76 a21a18a22a18a23a18a24 a15a2a17a2a18a2a19a2a14
a34
a35
a20
FX(t) = P(X < t)
= P(X < t; 1 < y < 1)
=
ZZ
( 1;t) ( 1;1)
p(x; y)dxdy
=
Z t
1
Z 1
1
p(x; y)dy dx:
a23a25a24
a197 a198
a18 a19 a14
a34
a35
a21a9a87 pX(x) =R1 1 p(x; y)dya28 a176 a106 pY (y) =R1 1 p(x; y)dxa28
a50a2a66a2a164
a39a2a69
a151a18a26
a28
6
Ch3 7
a254 2.1 a55 (X; y) N(
1; 2; 21; 22; )a20a28a27 X; Y
a14
a19a18a20a2a197a2a198
a28
a29 a21a18a22a18a30a18a31
a197a2a198
a18a2a19
pX(x) = R1 1 p(x; y)dy
= 12
1 2
p1 2 exp 12(1 2) (x 1)
2
2
Z 1
1
exp 12(1 2) 2 (x 1)(y 2)
1 2
+ (y 2)
2
22
dy
a7 a192 a15a18a32
a148a34a33
a15a18a35a18a36a18a37a18a38a16a142
a106
a20a42a89
pX(x) = 12
1 2
p1 2 exp 12(1 2) (x 1)
2
2 +
2(x 1)2
2(1 2) 21
Z 1
1
exp 12(1 2) y 2
2
(x 1)
1
2 dy
= 1p2
1
exp (x 1)
2
2 21
1p
2 2p1 2
Z 1
1
exp 12(1 2) 2
2
y
2
2(x 1)
1
2 dy
= 1p2
1
exp (x 1)
2
2 21
,
a171 X N( 1; 21)a28 a176 a106 Y N( 2; 22)a28
a87a18a39
a175
a19a18a20
a15a2a17
a20
a39a16a69 a105 a21a1a22 a14a4a40 a73
a57a52a20a42a41
a62
a41a1a43
a34
a178a16a179
a15a16a17a45a44a47a46a18a48
a57
a49
a34
a14
a28
a50a2a66a2a45a82a46
a39a16a69
a242a16a243a18a50a1a51
a14
a151a1a26
a28
a254 2.2 a39a2a69
a197a18a52
a14
a255a18a26
a148
a8a2a100 2 a69a53a3a34a2 a20 3 a69a18a6a18a2 a28a95a235
=
(
1; a54
a39
a10a18a55
a38a53a3a34a2 ;
0; a54
a39
a10a18a55
a38a18a6a18a2 ;
=
(
1; a54
a220
a10a18a55
a38a53a3a34a2 ;
0; a54
a220
a10a18a55
a38a18a6a18a2 ;
a56a18a57
a21a2a87a44a20
a30
a100a1a8
a9 a14 a162
a8
a9
a157a1a58a1a50a1a59a16a50a52a20 a14 a169 a221a2a222a2a229
a19
a41 0.4 a14 a220 a253
a15a2a17
a20a28a60a2a100a18a8
a9 a14 a162
a8
a9a5a61 ( ; ) a14
a178a2a179
a15a2a17
a57
a62 a176 a14
a28
7
x6 a0a2a1a2a3a2a4a2a5a2a6
a7a9a8a9a10a9a11a9a12a2a13a2a14a16a15a2a17a16a18a2a19 F
a20a22a21a9a23a9a24a9a25a2a26
a14a2a27a16a18a2a19 F 1
a28a22a29a9a30a9a31a9a32a2a33a2a34
a35a2a36a2a37a2a38a2a39a2a40 a14a2a18a2a19
a20a42a41a16a43a44a20
a7a16a8 a39a16a40 a14a2a15a16a17a16a18a2a19
a20a42a45a47a46a49a48a2a50
a14
a34
a35a52a51
a53a2a54a56a55 F(x)
a57
a39 a15a2a17a2a18a2a19
a20
a34
a35 F 1(y)
F 1(y) = supfxjF(x) < yg; 0 < y < 1:
a58a2a59
a20 F 1(y) a57
a12a2a13a2a14a61a60a63a62 a39
a34
a10a2a11a16a12a16a13a65a64
a28
a50a2a66a16a45a47a46a68a67
a8 F 1 a39a2a69a2a70a2a71a2a72
a73,
a74a2a75
F 1(y) 6 x () y 6 F(x):
a76a78a77a2a79a2a80a82a81 a39a2a69a2a83a2a84 a14 a72a16a73
F 1(y) > x () y > F(x):
=)a51a86a85 F 1(y) > x a87 9x 0 a88a2a89 x < x < F 1(y)a28a91a90 a85 a15a2a17a2a18a2a19a2a14
a92a2a93a2a94
a21a2a87
F(x) 6 F(x ) < y:
(=a51a95a85 y > F(x) a96
a15a9a17a9a18a9a19a9a14a9a97a2a98a2a99a2a94
a87 9" > 0 a88a9a89 y > F(x+")a20
a8
a57a2a100
F 1(y) > x + " > x:
a59a2a101a2a102
a21a2a23a2a89
a38
a50a2a66
a14a2a103a16a104a2a105 a70a16a71 a14
a34a2a106
a51
a53a2a107a56a55 F(x)
a57
a15a2a17a2a18a2a19
a20 X U[0; 1]a20a95a108 Y = F 1(X) a14a2a15a2a17a2a18a2a19 a57
F( )a28
a76
P(Y 6 y) = P(F 1(X) 6 y)
= P(X 6 F(y))
= F(y):
F 1( ) a109a2a100a2a23a2a50
a103a2a110a2a94a2a111 a51
(1) F(F 1(y))) > y;
(2) F 1(F(y)) 6 y;
1
Ch2 2
(3) a112 F(x) a30 x = F 1(y) a113
a98a2a99
a20a95a108 F(F 1(y)) = y;
(4) F 1(y) a114 a98a2a99 a28
2
Ch3 3
a115a2a116a2a117 a118a2a119a2a120a2a121a2a122a2a123a2a124a2a125a2a3a2a4
x1 a126a2a127a2a128
a118a2a119a2a120a2a121a2a122a2a123a2a124a2a125a16a129a2a130a16a5a16a6
a53a131a54 a132a131a133a131a134a131a135a65a136 ( ;F; P)
a137 n a69a131a138a131a139a131a140a131a141 X1; X2; ; Xn a14a131a142a131a143
X = (X1; X2; ; Xn) a41 n a144
a138a2a139a82a145a68a141
a28
a105a2a146 a138a2a139
a29a2a147a65a148a49a149a131a149a16a150
a96
a38 a146 a69a16a138a16a139a131a140a16a141
a20a152a151a16a48a131a153a16a154
a14a16a155a16a156 (X; Y ) a102
a57a2a157
a69a2a138a2a139a2a140a2a141a2a158a2a159 a14 a39a2a69a16a138a2a139a47a145a68a141
a28
a138a16a139a82a145a49a141
a21a16a23a2a57a16a100a2a160a16a144
a14
a20a95a161a16a21a2a23a16a57
a162a2a163
a144
a14
a28
a39a2a69 n a14 a138a2a139a82a145a68a141 X = (X
1; X2; ; Xn) a21a2a23a2a164a2a165a2a48a2a50
a14a2a166a2a167 a51
! Rn;
! ! (X1(!); X2(!); ; Xn(!)):
a33a82a148
a20a95a168
a39a2a69 a15 a141a2a169
a57a2a21a16a170
a14
a20a42a171
f!jXi(!) 6 aig2F:
a172
a43a173a20
n\
i=1
f!jXi(!) 6 aig
= f!jX1(! 6 ai; X2(!) 6 a2; ; Xn(!) 6 ang2F:
a8
a57a173a20 P(X1 6 a1; X2 6 a2; ; Xn 6 an)
a102
a100a2a174
a35a2a175
a28
a176a68a177 a14
a20
a34
a35 n
a144a2a178a2a179
a15a2a17a2a18a2a19
a41
F(a1; a2; ; an) = P(X1 6 a1; X2 6 a2; ; Xn 6 an):
a176 a39
a144
a14a2a15a2a17a2a18a2a19 a39 a177
a20a180a178a16a179
a15a16a17a2a18a16a19
a100a2a23a16a50
a94a2a111 a51
(1) lim
ai! 1
F(a1; a2; ; an) = 0; i = 1; 2; ; n;
lima
1!+1a
2!+1
an!+1
F(a1; a2; ; an) = 1;
(2) F a7 a168
a69 a15 a141a2a169
a57
a92a2a93a2a181a16a182a2a14a52a183
(3) F a57
a97a2a98a2a99a2a14
a20a95a184a2a114a16a185a2a160a16a186
a30a44a28
3
Ch3 4
a50a2a66
a34
a35 a98a2a99a2a187a2a14 a138a2a139a82a145a49a141
a28
a53a2a54a188a132 n
a144
a138a2a139a82a145a68a141 (X
1; X2; ; Xn) a57
a98a2a99a2a187a2a14
a20a189a48a16a190a131a186
a30
a181a131a191
a21
a192 a18a2a19 p(x
1; x2; ; xn)a20a95a88a2a89
a7a2a193
a174
a14 x
1; x2; ; cna20
a169
a100
F(x1; x2; ; xn) =
Z
Z
D
p(t1; t2; ; tn)dt1 dtn:
a33a194a148
a20 D = f(t1; ; tn)jt1 6 x1; ; tn 6 xnga20a196a195
a132 p(x
1; ; xn) a41
X1; ; Xn a14 a178a2a179a2a197a2a198
a18a2a19
a28
a199a2a200
a164a82a46a201a20a202a178a2a179a2a197a2a198
a18a2a19
a195
a62a2a203 a39
a20a95a204
a140
a21
a19 a69 a156a16a14a2a18a16a19a2a205
a109a2a57a16a197a2a198
a18
a19
a28
a50a2a66a2a45a82a46a68a157
a69a2a206a2a70a2a71 a14a2a15a16a17
a28
(1) a207a2a208
a15a2a17
a132 (X
1; X2; ; Xn) a41 n a144a82a209a68a210 G a137
a14
a207a2a208
a15a2a17
a20a95a48a16a190
a33
a178a16a179a2a197a16a198a2a41
p(x1; ; xn) =
(
c; (x1; ; xn) 2 G;
0; a33a2a211,
a21a2a23a2a164a82a46a201a20a95a207a2a212
a15a16a17a16a213a2a214
a137
a102
a57a16a215a2a216
a133 a187
a28
(2) a146 a144a2a217a2a218
a15a2a17
a219
a34
a35a2a220
a144
a14
a28
a48a2a190 (X; Y ) a100a2a178a2a179a2a197a2a198
p(x; y) = 12
1 2
p1 2 exp 12(1 2)
(x 1)
2
21
2 (x 1)(y 2)
1 2 +
(y 2)2
22
,
a33a82a148
a20 1 < 1 < 1a20 1 < 2 < 1a20 1 > 0a20 2 > 0a20 1 < < 1a20a91a108
a132
(X; Y ) a221a2a222
a220
a144a2a217a2a218
a15a2a17 N(
1; 2; 21; 22; )a28
a59a2a101a2a223
a34
a35a2a224
a144
a14
a28
a132 n
a144
a138a16a139a47a145a49a141 (X
1; X2; ; Xn) a221a16a222 n a144a16a217a16a218
a15a16a17
a20a152a48a16a190a16a26a16a225a16a100a131a197
a198
p(x1; x2; ; xn) = 1(2 )n=2j j1=2 exp 12(x )0 1(x ) ;
4
Ch3 5
( 1; 2)
z = p(x; y)
x
y
z
O
a33a226a148
a20 x = (x1; x2; ; )0a20a228a227 = ( 1; 2; ; n)0 a57a9a229
a19 a145a230a141
a20 = ( ij)n n
a57 n a231a2a217
a34a2a232a82a233a201a28a95a234a2a235a2a236 X N( ; )a28
a199a2a200
a164a82a46a201a20a95a237
= ( 1; 2)0 ; =
21 1 2
1 2 22
!;
a102
a57
a220
a144a2a217a2a218
a15a2a17
a28
x2 a238a2a239a2a128
a118a2a119a2a120a2a121a2a122a2a240a2a241a16a3a16a4a2a5a16a6
a53a9a54 a132 n
a144
a138a9a139a226a145a230a141 X = (X
1; X2; Xn a57a9a242a9a243
a187a9a14
a20a22a48a2a190a2a26a16a244a2a237a2a245
a146
a21
a19 a69 a62 a176 a14a2a205
a28
a246a2a247 a14
a20a95a23
a220
a144a2a41a2a151a44a20
a33
a133a16a134 a15a16a17a16a248
a34
a35
a48a2a50
a51
pij = P((X1; X2) = (x1i; x2j)); i = 1; 2; ; j = 1; 2;,
a33a82a148 pij
a249a2a250
(1) pij > 0; i = 1; 2; ; j = 1; 2; ;
(2)
X
i;j
pij = 1:
a242a2a243
a187 a138a2a139a82a145a68a141
a148
a20a95a251a16a252
a14
a57
a146a16a253a2a15a16a17
a20
a33a16a34
a35
a48a16a50
a51
a132 (X
1; X2; ; Xr) a221a2a222a2a229
a19
a41 (n; p1; p2; ; pr) a14a2a146a2a253a2a15a2a17 a20a95a48a2a190
P((X1; ; Xr) = (k1; ; kr)) = n!k
1! kr!
pk11 pkrr ;
a33a82a148 k1; ; kn
a41
a181a2a191a2a142a2a19
a184
rX
i=1
ki = n;
rX
i=1
pi = 1a28
5
Ch3 6
a254 2.1
a255
a148
a100 2 a69a1a0a1a2 a20 3 a69a4a3a5a2 a20 5 a69a1a6a1a2a1a7 10 a69a1a2 a20 a222
a148
a100a1a8
a9 a14
a237 7 a10a52a20 a112
a235 X1; X2; X3
a15a1a11
a41a16a237
a38 a14 a0a12a2
a20
a3a13a2a12a14a12a6a12a2 a14a131a19a16a15
a20 a108
(X1; X2; X3) a57
a39
a242a2a243
a187 a138a16a139a47a145a49a141
a20 a184a131a221a16a222a16a229
a19
a41 (7; 0:2; 0:3; 0:5) a14a2a146a2a253a2a15
a17
a28
a53a131a54 a7a131a8 a39a131a69 a146
a144
a138a131a139a194a145 a141 X = (X
1; X2; ; Xn)a20a49a26
a14 a39 a110a131a15 a141
(Xi1; ; Xik); 1 6 i1 < i2 < < ik; 1 6 k < n a158a2a159a2a39a2a69a18a17 a14 a138a2a139a82a145a68a141 a20
a103
a69a18a17 a14 a138a2a139a82a145a68a141 a14a2a15a2a17a16a18a16a19 a102 a132
a41a1a19a1a20
a15a16a17a2a18a16a19
a28
a7a2a8
a242a2a243
a187 a138a2a139a2a140a2a141
a20a42a23
a220
a144a2a41a16a151a44a20
a55 (X; Y ) a14
a178a2a179
a15a2a17
a41
P((X; Y ) = (xi; yj)) = pij; i = 1; 2; ; j = 1; 2; ;
a108 X a14 a19a18a20
a15a2a17
a41
P(X = xi) =
X
j
P((X; Y ) = (xi; yj)) =
X
j
pij; i = 1; 2;,
a176a68a177
a20 Y a14 a19a18a20
a15a2a17
a41
P(Y = yj) =
X
i
pij; j = 1; 2;,
a7a16a8a16a98a16a99a16a187 a138a16a139a16a140a16a141
a20a86a26
a14
a19a12a20
a15a131a17a131a18a16a19
a161a131a100a131a197a131a198
a18a131a19
a28
a23
a220
a144a131a41a131a151a226a20
a55 (X; Y ) a14
a178a2a179a2a197a2a198
a18a2a19
a41 p(x; y)a20a95a108a18a19a18a20a2a197a2a198
a18a2a19
a41
pX(x) =
Z 1
1
p(x; y)dy; pY (y) =
Z 1
1
p(x; y)dx:
a76 a21a18a22a18a23a18a24 a15a2a17a2a18a2a19a2a14
a34
a35
a20
FX(t) = P(X < t)
= P(X < t; 1 < y < 1)
=
ZZ
( 1;t) ( 1;1)
p(x; y)dxdy
=
Z t
1
Z 1
1
p(x; y)dy dx:
a23a25a24
a197 a198
a18 a19 a14
a34
a35
a21a9a87 pX(x) =R1 1 p(x; y)dya28 a176 a106 pY (y) =R1 1 p(x; y)dxa28
a50a2a66a2a164
a39a2a69
a151a18a26
a28
6
Ch3 7
a254 2.1 a55 (X; y) N(
1; 2; 21; 22; )a20a28a27 X; Y
a14
a19a18a20a2a197a2a198
a28
a29 a21a18a22a18a30a18a31
a197a2a198
a18a2a19
pX(x) = R1 1 p(x; y)dy
= 12
1 2
p1 2 exp 12(1 2) (x 1)
2
2
Z 1
1
exp 12(1 2) 2 (x 1)(y 2)
1 2
+ (y 2)
2
22
dy
a7 a192 a15a18a32
a148a34a33
a15a18a35a18a36a18a37a18a38a16a142
a106
a20a42a89
pX(x) = 12
1 2
p1 2 exp 12(1 2) (x 1)
2
2 +
2(x 1)2
2(1 2) 21
Z 1
1
exp 12(1 2) y 2
2
(x 1)
1
2 dy
= 1p2
1
exp (x 1)
2
2 21
1p
2 2p1 2
Z 1
1
exp 12(1 2) 2
2
y
2
2(x 1)
1
2 dy
= 1p2
1
exp (x 1)
2
2 21
,
a171 X N( 1; 21)a28 a176 a106 Y N( 2; 22)a28
a87a18a39
a175
a19a18a20
a15a2a17
a20
a39a16a69 a105 a21a1a22 a14a4a40 a73
a57a52a20a42a41
a62
a41a1a43
a34
a178a16a179
a15a16a17a45a44a47a46a18a48
a57
a49
a34
a14
a28
a50a2a66a2a45a82a46
a39a16a69
a242a16a243a18a50a1a51
a14
a151a1a26
a28
a254 2.2 a39a2a69
a197a18a52
a14
a255a18a26
a148
a8a2a100 2 a69a53a3a34a2 a20 3 a69a18a6a18a2 a28a95a235
=
(
1; a54
a39
a10a18a55
a38a53a3a34a2 ;
0; a54
a39
a10a18a55
a38a18a6a18a2 ;
=
(
1; a54
a220
a10a18a55
a38a53a3a34a2 ;
0; a54
a220
a10a18a55
a38a18a6a18a2 ;
a56a18a57
a21a2a87a44a20
a30
a100a1a8
a9 a14 a162
a8
a9
a157a1a58a1a50a1a59a16a50a52a20 a14 a169 a221a2a222a2a229
a19
a41 0.4 a14 a220 a253
a15a2a17
a20a28a60a2a100a18a8
a9 a14 a162
a8
a9a5a61 ( ; ) a14
a178a2a179
a15a2a17
a57
a62 a176 a14
a28
7
