Ch4 1
x3 a0a2a1
a3a2a4 a5a2a6a2a7a2a8a2a9 X
a10a12a11 E(X EX)2 a13 X a14a2a15a2a16a17a10a12a18a2a19 Var(X)a20
a15a2a16a2a21a2a22a2a23a2a24a2a25a2a26a2a27a2a28a2a29a31a30
Var(X) = EX2 (EX)2:
a32a2a33a2a34a36a35a38a37
a26a2a39a2a40a2a41a2a42a2a14a43a15a2a16a44a20
(1) a45a2a46a2a41a2a42
a47 X
a48a2a49a2a50a2a51
a13 p
a14a2a45a2a46a2a41a2a42a17a10a12a52
EX2 = 12 p + 02 q = p:
a53a2a54
VarX = EX2 (EX)2 = p p2 = pq:
(2) a55a2a56a2a41a2a42
a47 X
a48a2a49a2a50a2a51
a13 (n; p)
a14a2a55a2a56a2a41a2a42a17a10a12a52
EX2 =
nX
k=0
k2Ckn pkqn k
=
nX
k=1
k(k 1) n!k!(n k)!pkqn k +
nX
k=1
k n!k!(n k)!pkqn k
= n(n 1)p2
n 2X
k=0
Ckn 2pkqn 2 k + np
n 1X
k=0
Ckn 1pkqn 1 k
= n(n 1)p2 + np:
a53a2a54
a10
VarX = EX2 (EX)2 = n(n 1)p2 + np (np)2 = npq:
(3) Poisson a41a2a42
1
Ch4 2
a47 X P( )
a10a12a52
EX2 =
1X
k=0
k2
k
k! e
k
=
1X
k=1
k(k 1)
k
k! e
k +
1X
k=1
k
k
k! e
k
= 2
1X
k=2
k 2
(k 2)!e
k +
1X
k=1
k 1
(k 1)!e
k
= 2 +,
a53a2a54
a10
VarX = EX2 (EX)2 =,
(4) a57a2a58a2a41a2a42
a47 X
a48a2a49 (a; b) a59a2a14a2a57a2a58a2a41a2a42a17a10a12a52
EX2 = 1b a
Z b
a
x2dx = 13(b2 + ba + a2):
a60a38a61a2a62
VarX = EX2 (EX)2 = 112(b a)2:
(5) a63a2a51a2a41a2a42
a47 X
a48a2a49a2a50a2a51
a13
a14a2a63a2a51a2a41a2a42a17a10a12a52
EX2 =
Z 1
0
x2 e xdx
= 1 2
Z 1
0
t2e tdt
= 1 2 (3) = 2 2,
a53a2a54
Var = EX2 (EX)2 = 2 2 1 2 = 1 2,
2
Ch4 3
(6) a64a2a65a2a41a2a42
a47 X N( ; 2)
a10a12a52 EX = a10a12a66
VarX = E(X )2
=
Z 1
1
(x )2 1p2 e (x )2 2 dx
= 2 1p2
Z 1
1
t2e t
2
2
= 2:
(7) a67a2a68a2a41a2a42
a47 X ( ; )
a10a12a52
EX2 =
Z 1
0
x2
( )x
1e xdx
= 1 ( ) 2
Z 1
0
t +1e tdt
= ( + 2) ( ) = ( + 1) 2,
a53a2a54
a10
VarX = EX2 (EX)2 = 2,
a32a2a33a2a69a2a70a2a70
a15a2a16a2a14a2a71a2a72a17a20
a73a2a74 1 a; b
a13a2a75a2a76
a39a2a51a17a10a12a52
Var(a + bX) = b2VarX:
a73a2a74 2 Var(X + Y ) = VarX + VarY + 2E(X EX)(Y EY )
a20
a73a2a74 3
a77 X a78 Y a79a2a80a2a81a2a82a17a10a12a52
Var(X + Y ) = VarX + VarY:
a73a2a74 4 (EX)2 6 EX2; (E(X EX)(Y EY ))2 6 VarXVarY
a20
a83 3.1
a84 n a85a2a86a36a87a38a14a2a88a2a14 n a89a2a88a2a90a2a78 n a26a2a88a2a85a2a91a2a92
a6a2a7a94a93a96a95
a10a98a97
a93
a95a2a99a2a5
a14a2a51a101a100a102a14a2a15a2a16a44a20
a103
a18
Xi =
(
1; a104 i a89a2a88a2a90a2a78a2a104 i a26a2a88a2a85 a99a2a5a44a105
0; a106a2a107a17a10
i = 1; ; n
3
Ch4 4
a52 Xi
a108
a48a2a49a2a50a2a51
a13
1
n a14a2a45a2a46a2a41a2a42a17a10a12a66 EX = EX
2 = 1
na20
a18 N =
nX
i=1
Xi a13 a93a38a95a2a99a2a5 a14a2a51a101a100a109a10a110a22
EN =
nX
i=1
EXi = 1:
a111
EN2 = E(
nX
i=1
Xi)2 =
nX
i=1
EX2i +
X
i6=j
EXiXj:
a112a114a113 X
iXj a115
a54
a45a114a46a114a41a2a42a44a10a116a50a43a51
a13
1
n(n 1)a10a117a66 EXiXj =
1
n(n 1)a20
a53a114a54
a10
VarN =
nX
i=1
EX2i +
X
i6=j
EXiXj (EN)2 = 1:
a32a2a33a2a34a36a35
a25a2a26a2a118a2a119a2a14a2a86a2a120a43a29a44a10a110a107
a54a2a121 a33
a14a2a122a43a51a43a123a2a124a43a14a2a125a43a126a43a127a2a128a44a20
Chebyshev a129a114a130a114a131
a47 X
a14a114a132a114a133a114a15a114a16
a108a2a134a2a135
a10a136a52
a5
a75a2a76
a14 " > 0a10
a108
a22
P(jX EXj > ") 6 1"2 VarX:
a137 a138a2a139a2a140 a5a2a141a43a142a43a143
a14a145a144a43a146
a34a147a35a96a148a94a149
a20
a47 X
a22a2a150a2a151a2a152a2a51 p(x)a10a153a18 D =
fjx EXj > "ga10a12a52
P(jX EXj > ") =
Z
D
p(x)dx
6
Z
D
(x EX)2
"2 p(x)dx
6 1"2
Z 1
1
(x EX)2p(x)dx
= 1"2,
a60
a59
a33
a14 Chebyshev a86a2a120a2a29a17a10a12a154a2a155a2a156
a35a96a32a43a33
a14a43a25a2a26a43a118a43a119a2a157a43a126a31a30
VarX = 0 () P(X = c) = 1a20
a137 a158
a41a2a71a44a30a116a159 P(X = c) = 1 a160a17a10 EX = ca10 EX2 = c2a10a12a66
VarX = EX2 (EX)2 = 0:
4
Ch4 5
a161
a119a2a71a44a30a116a159 VarX = 0 a160a17a10
a60 Chebyshev
a86a2a120a2a29a17a10
a5
a75a2a76
a14 n
a108
a22
P(jX EXj > 1n) 6 0;
a162 a111
P(jX EXj > 1n) = 0;
a53a2a54 a60
fX 6= EXg =
1\
n=1
jX EXj > 1
n;
a163 a60a38a164a2a165a2a166
a151a2a14
a141a2a142
a71a43a154a43a167
P(X 6= EX) = 0;
P(X = EX) = 1:
a135
a138a2a139
a10 EX a54 a25a2a26a2a39a2a51a17a10 a53a2a54 a59a43a29a43a154a2a155a43a168a2a169 a99
P(X = c) = 1:
x4 a170 a0a2a1a2a171a2a172a2a173a2a174a2a175
a3a2a4 a47 X; Y a54 a164a2a165a2a176a36a177 ( ;F; P)
a59a2a14a2a45a2a26
a6a2a7a2a8a2a9
a10a110a11
E(X EX)(Y EY )
a13 X; Y
a14a2a178a2a15a2a16a17a10a12a18a2a19 Cov(X; Y ) a179 XY a20
a180a2a181
a22 VarX = Cov(X; X)a20
a3a114a4 a47 X = (X
1; ; Xn)
a54 ( ;F; P)
a59a114a14 n a182
a6a114a7a184a183a185a9
a10a116a52a2a11 n a186
a15a36a187
= ( ij)n n = ( XiXj)n n
a13 X
a14a2a178a2a15a2a16a36a187a188a20a189a123a145a190a43a25a43a26a43a191a94a187a96a14a145a132a43a133a43a192
a54
a107a43a193a145a26a43a194a43a195a43a196a43a132a43a133a145a197a43a198
a99
a14a43a199
a191a36a187a200a10a12a52a2a59a2a29a2a154a2a168a43a169
a13
= E(X EX)(X EX)0:
5
Ch4 6
a180a2a181 a54a2a201 a5
a11a2a191a36a187a200a10a12a21a43a154a43a155
a148a94a149
a10a110a107
a54a43a202a2a203
a123a2a14a44a20
a3a2a4 a47 X; Y a54
a45a2a26
a6a2a7a2a8a2a9
a10a12a204a43a15a2a16
a108
a86
a13 0
a10a12a52a2a11
= E(X EX)(Y EY )pVarXVarY = XYp
XX Y Y
a13 X; Y
a14a2a79a2a27a2a28a2a51a17a20
a27
a53
a79a2a27a2a28a2a51a17a10a12a22
a32a43a33
a14a43a123a43a125a31a30
a3a2a205 a47 X; Y
a14a2a79a2a27a2a28a2a51
a13
a10a12a52
(1) j j 6 1a20
(2) a206a2a207 X; Y a79a2a80a2a81a2a82a17a10a12a52
= 0:
(3) j j = 1 a14
a158
a119a2a208a2a209
a54
a30
a134a2a135
a39a43a51 a; b a210
a62
P(Y = aX + b) = 1:
a148
a30 (2) a180a2a181 a20 a32a2a33a2a148a36a149 (1)(3)a20
a112a2a113 a32a2a33
a14a2a27
a53a2a201
a51 t a14a2a211a2a56a2a29
E[(Y EY ) t(X EX)]2 = Y Y 2t XY + t2 XX > 0;
a53a2a54
a106a2a212a2a213a2a29a2a214
a53 0
a10a12a215a2a22
2XY 6 XX Y Y ;
a49
a111
a22 j j 6 1a20
a60
a59a2a21a2a154a2a167 j j = 1 a159a2a204a2a216a2a159
a134a2a135
a39a2a51 a a210
a62 t = a
a160a2a22
E[(Y EY ) a(X EX)]2 = 0;
a66a2a22
Var(Y aX) = 0;
P(Y aX = b) = 1:
(3) a62 a148 a20
6
Ch4 7
a59
a33
a14a2a123a2a125a2a217
a149a38a218
a10a189a159 X; Y a81a2a82a2a14a2a160a2a219a2a22 = 0a10a221a220a2a222a2a223
a69a2a224
a86a2a25a43a123a31a20
a32a2a33
a192
a54
a25a2a26a2a225a2a226a17a20
a83 4.1 a47 (X; Y )
a14a2a150a2a151a2a152a2a51
a13
p(x; y) =
8
<
:
1; x
2 + y2 6 1;
0; a106a2a227a17a20
a52a2a228a2a229a2a97
a62 EX = EY = 0
a20a12a230
a135
a97 XX; XY ; Y Y a20
XX =
ZZ
x2+y261
(x EX)2p(x; y)dxdy
=
ZZ
x2+y261
x
dxdy
= 14:
a87a38a125 Y Y = 14a20
XY =
ZZ
x2+y261
(x EX)(y EY )p(x; y)dxdy
=
ZZ
x2+y26
xy 1 dxdy
= 0
a66a2a22 =0a10a12a220
a54
a10 X; Y a86
a54
a79a2a80a2a81a2a82a2a14a17a20
a83 4.2 a47 (X; Y ) N(
1; 2; 21; 22; r)a10a12a97 X; Y a14a2a79a2a27a2a28a2a51a17a20
7
Ch4 8
a103 a231
a22 EX = 1; EY = 2; VarX = 21; VarY = 22a20
XY = E(X EX)(Y EY )
=
ZZ
R2
(x 1)(y 2) 12
1 2
p1 r2 exp 12(1 r2)
(x 1)2
21
2r(x 1)(y 2)
1 2 +
(y 2)2
22
dxdy
=
ZZ
R2
1 2uv
2 p1 r2 exp
1
2(1 r2)(u
2 2ruv + v2) dudv
= 1 2p2
Z 1
1
ve v22 dv
Z 1
1
up
2 p(1 r2)e
(u rv)22(1 r2) du
= 1 2p2
Z 1
1
rv2e v22 dv
= r 1 2:
a53a2a54
a79a2a27a2a28a2a51
a13
= XYp
XX Y Y
= r 1 2
1 2
= r:
a60a38a61
a21a2a154a2a155a2a156
a35 X; Y
a79a2a80a2a81a2a82 () = 0a20
a5 a53
a45a2a26
a6a2a7a2a8a2a9 X; Y
a10a110a232a2a233a2a234a2a133a2a40 aX + b a69a2a235 a166 Ya10
a115
a192
a54
a217a17a10a110a236
a237 a; b
a210
a62 aX + b
a238a2a239a2a240 Ya20
a112a2a113 a237 aX + b
a115
a54
a25a2a26
a6a2a7a2a8a2a9
a10
a53a43a54
a10a12a40
a32a2a33
a14a2a29a2a226
a69a2a241a2a242
a107a2a233a43a14a43a243a2a244a31a30
Q(a; b) = E[Y (aX + b)]2:
a230
a135
a119a2a97 a; ba10a12a210
a62 Q
a196
a237
a238a2a214a2a245a17a20
a40
a95
a15a2a246a17a10a12a22
Q(a; b) = E[y (aX + b)]2
= E[((Y EY ) a(X EX)) + (EY aEX b)]2
= E[(Y EY ) a(X EX)]2 + (EY aEX b)2
= Y Y 2a XY + a2 XX + (EY aEX b)2
= XX a XY XX 2 + Y Y
2
XY
XX + (EY aEX b)
2:
8
Ch4 9
a154a2a167a2a159
a = a = XY
XX; b = b = EY a EX
a160a17a10 Q a196
a237
a238a2a214a2a245a17a10a12a22
min
a;b
(a; b) = Q(a ; b ) = Y Y
2
XY
XX = Y Y (1
2
XY ):
a154a114a155
a70a184a35
a10 j XY ja247a114a239a114a240 1a10 a X + b a247a114a239a114a240 Ya10
a115
a192
a54
a217a17a10 j j a14a114a122a114a214a114a248a114a249
a218
X a78 Y a14a2a250a2a71a2a79a2a27a2a251a2a151a17a20
a138
a192
a54
a79a43a27a43a28a2a51a43a14
a164a43a165
a76
a190a44a20
9
x3 a0a2a1
a3a2a4 a5a2a6a2a7a2a8a2a9 X
a10a12a11 E(X EX)2 a13 X a14a2a15a2a16a17a10a12a18a2a19 Var(X)a20
a15a2a16a2a21a2a22a2a23a2a24a2a25a2a26a2a27a2a28a2a29a31a30
Var(X) = EX2 (EX)2:
a32a2a33a2a34a36a35a38a37
a26a2a39a2a40a2a41a2a42a2a14a43a15a2a16a44a20
(1) a45a2a46a2a41a2a42
a47 X
a48a2a49a2a50a2a51
a13 p
a14a2a45a2a46a2a41a2a42a17a10a12a52
EX2 = 12 p + 02 q = p:
a53a2a54
VarX = EX2 (EX)2 = p p2 = pq:
(2) a55a2a56a2a41a2a42
a47 X
a48a2a49a2a50a2a51
a13 (n; p)
a14a2a55a2a56a2a41a2a42a17a10a12a52
EX2 =
nX
k=0
k2Ckn pkqn k
=
nX
k=1
k(k 1) n!k!(n k)!pkqn k +
nX
k=1
k n!k!(n k)!pkqn k
= n(n 1)p2
n 2X
k=0
Ckn 2pkqn 2 k + np
n 1X
k=0
Ckn 1pkqn 1 k
= n(n 1)p2 + np:
a53a2a54
a10
VarX = EX2 (EX)2 = n(n 1)p2 + np (np)2 = npq:
(3) Poisson a41a2a42
1
Ch4 2
a47 X P( )
a10a12a52
EX2 =
1X
k=0
k2
k
k! e
k
=
1X
k=1
k(k 1)
k
k! e
k +
1X
k=1
k
k
k! e
k
= 2
1X
k=2
k 2
(k 2)!e
k +
1X
k=1
k 1
(k 1)!e
k
= 2 +,
a53a2a54
a10
VarX = EX2 (EX)2 =,
(4) a57a2a58a2a41a2a42
a47 X
a48a2a49 (a; b) a59a2a14a2a57a2a58a2a41a2a42a17a10a12a52
EX2 = 1b a
Z b
a
x2dx = 13(b2 + ba + a2):
a60a38a61a2a62
VarX = EX2 (EX)2 = 112(b a)2:
(5) a63a2a51a2a41a2a42
a47 X
a48a2a49a2a50a2a51
a13
a14a2a63a2a51a2a41a2a42a17a10a12a52
EX2 =
Z 1
0
x2 e xdx
= 1 2
Z 1
0
t2e tdt
= 1 2 (3) = 2 2,
a53a2a54
Var = EX2 (EX)2 = 2 2 1 2 = 1 2,
2
Ch4 3
(6) a64a2a65a2a41a2a42
a47 X N( ; 2)
a10a12a52 EX = a10a12a66
VarX = E(X )2
=
Z 1
1
(x )2 1p2 e (x )2 2 dx
= 2 1p2
Z 1
1
t2e t
2
2
= 2:
(7) a67a2a68a2a41a2a42
a47 X ( ; )
a10a12a52
EX2 =
Z 1
0
x2
( )x
1e xdx
= 1 ( ) 2
Z 1
0
t +1e tdt
= ( + 2) ( ) = ( + 1) 2,
a53a2a54
a10
VarX = EX2 (EX)2 = 2,
a32a2a33a2a69a2a70a2a70
a15a2a16a2a14a2a71a2a72a17a20
a73a2a74 1 a; b
a13a2a75a2a76
a39a2a51a17a10a12a52
Var(a + bX) = b2VarX:
a73a2a74 2 Var(X + Y ) = VarX + VarY + 2E(X EX)(Y EY )
a20
a73a2a74 3
a77 X a78 Y a79a2a80a2a81a2a82a17a10a12a52
Var(X + Y ) = VarX + VarY:
a73a2a74 4 (EX)2 6 EX2; (E(X EX)(Y EY ))2 6 VarXVarY
a20
a83 3.1
a84 n a85a2a86a36a87a38a14a2a88a2a14 n a89a2a88a2a90a2a78 n a26a2a88a2a85a2a91a2a92
a6a2a7a94a93a96a95
a10a98a97
a93
a95a2a99a2a5
a14a2a51a101a100a102a14a2a15a2a16a44a20
a103
a18
Xi =
(
1; a104 i a89a2a88a2a90a2a78a2a104 i a26a2a88a2a85 a99a2a5a44a105
0; a106a2a107a17a10
i = 1; ; n
3
Ch4 4
a52 Xi
a108
a48a2a49a2a50a2a51
a13
1
n a14a2a45a2a46a2a41a2a42a17a10a12a66 EX = EX
2 = 1
na20
a18 N =
nX
i=1
Xi a13 a93a38a95a2a99a2a5 a14a2a51a101a100a109a10a110a22
EN =
nX
i=1
EXi = 1:
a111
EN2 = E(
nX
i=1
Xi)2 =
nX
i=1
EX2i +
X
i6=j
EXiXj:
a112a114a113 X
iXj a115
a54
a45a114a46a114a41a2a42a44a10a116a50a43a51
a13
1
n(n 1)a10a117a66 EXiXj =
1
n(n 1)a20
a53a114a54
a10
VarN =
nX
i=1
EX2i +
X
i6=j
EXiXj (EN)2 = 1:
a32a2a33a2a34a36a35
a25a2a26a2a118a2a119a2a14a2a86a2a120a43a29a44a10a110a107
a54a2a121 a33
a14a2a122a43a51a43a123a2a124a43a14a2a125a43a126a43a127a2a128a44a20
Chebyshev a129a114a130a114a131
a47 X
a14a114a132a114a133a114a15a114a16
a108a2a134a2a135
a10a136a52
a5
a75a2a76
a14 " > 0a10
a108
a22
P(jX EXj > ") 6 1"2 VarX:
a137 a138a2a139a2a140 a5a2a141a43a142a43a143
a14a145a144a43a146
a34a147a35a96a148a94a149
a20
a47 X
a22a2a150a2a151a2a152a2a51 p(x)a10a153a18 D =
fjx EXj > "ga10a12a52
P(jX EXj > ") =
Z
D
p(x)dx
6
Z
D
(x EX)2
"2 p(x)dx
6 1"2
Z 1
1
(x EX)2p(x)dx
= 1"2,
a60
a59
a33
a14 Chebyshev a86a2a120a2a29a17a10a12a154a2a155a2a156
a35a96a32a43a33
a14a43a25a2a26a43a118a43a119a2a157a43a126a31a30
VarX = 0 () P(X = c) = 1a20
a137 a158
a41a2a71a44a30a116a159 P(X = c) = 1 a160a17a10 EX = ca10 EX2 = c2a10a12a66
VarX = EX2 (EX)2 = 0:
4
Ch4 5
a161
a119a2a71a44a30a116a159 VarX = 0 a160a17a10
a60 Chebyshev
a86a2a120a2a29a17a10
a5
a75a2a76
a14 n
a108
a22
P(jX EXj > 1n) 6 0;
a162 a111
P(jX EXj > 1n) = 0;
a53a2a54 a60
fX 6= EXg =
1\
n=1
jX EXj > 1
n;
a163 a60a38a164a2a165a2a166
a151a2a14
a141a2a142
a71a43a154a43a167
P(X 6= EX) = 0;
P(X = EX) = 1:
a135
a138a2a139
a10 EX a54 a25a2a26a2a39a2a51a17a10 a53a2a54 a59a43a29a43a154a2a155a43a168a2a169 a99
P(X = c) = 1:
x4 a170 a0a2a1a2a171a2a172a2a173a2a174a2a175
a3a2a4 a47 X; Y a54 a164a2a165a2a176a36a177 ( ;F; P)
a59a2a14a2a45a2a26
a6a2a7a2a8a2a9
a10a110a11
E(X EX)(Y EY )
a13 X; Y
a14a2a178a2a15a2a16a17a10a12a18a2a19 Cov(X; Y ) a179 XY a20
a180a2a181
a22 VarX = Cov(X; X)a20
a3a114a4 a47 X = (X
1; ; Xn)
a54 ( ;F; P)
a59a114a14 n a182
a6a114a7a184a183a185a9
a10a116a52a2a11 n a186
a15a36a187
= ( ij)n n = ( XiXj)n n
a13 X
a14a2a178a2a15a2a16a36a187a188a20a189a123a145a190a43a25a43a26a43a191a94a187a96a14a145a132a43a133a43a192
a54
a107a43a193a145a26a43a194a43a195a43a196a43a132a43a133a145a197a43a198
a99
a14a43a199
a191a36a187a200a10a12a52a2a59a2a29a2a154a2a168a43a169
a13
= E(X EX)(X EX)0:
5
Ch4 6
a180a2a181 a54a2a201 a5
a11a2a191a36a187a200a10a12a21a43a154a43a155
a148a94a149
a10a110a107
a54a43a202a2a203
a123a2a14a44a20
a3a2a4 a47 X; Y a54
a45a2a26
a6a2a7a2a8a2a9
a10a12a204a43a15a2a16
a108
a86
a13 0
a10a12a52a2a11
= E(X EX)(Y EY )pVarXVarY = XYp
XX Y Y
a13 X; Y
a14a2a79a2a27a2a28a2a51a17a20
a27
a53
a79a2a27a2a28a2a51a17a10a12a22
a32a43a33
a14a43a123a43a125a31a30
a3a2a205 a47 X; Y
a14a2a79a2a27a2a28a2a51
a13
a10a12a52
(1) j j 6 1a20
(2) a206a2a207 X; Y a79a2a80a2a81a2a82a17a10a12a52
= 0:
(3) j j = 1 a14
a158
a119a2a208a2a209
a54
a30
a134a2a135
a39a43a51 a; b a210
a62
P(Y = aX + b) = 1:
a148
a30 (2) a180a2a181 a20 a32a2a33a2a148a36a149 (1)(3)a20
a112a2a113 a32a2a33
a14a2a27
a53a2a201
a51 t a14a2a211a2a56a2a29
E[(Y EY ) t(X EX)]2 = Y Y 2t XY + t2 XX > 0;
a53a2a54
a106a2a212a2a213a2a29a2a214
a53 0
a10a12a215a2a22
2XY 6 XX Y Y ;
a49
a111
a22 j j 6 1a20
a60
a59a2a21a2a154a2a167 j j = 1 a159a2a204a2a216a2a159
a134a2a135
a39a2a51 a a210
a62 t = a
a160a2a22
E[(Y EY ) a(X EX)]2 = 0;
a66a2a22
Var(Y aX) = 0;
P(Y aX = b) = 1:
(3) a62 a148 a20
6
Ch4 7
a59
a33
a14a2a123a2a125a2a217
a149a38a218
a10a189a159 X; Y a81a2a82a2a14a2a160a2a219a2a22 = 0a10a221a220a2a222a2a223
a69a2a224
a86a2a25a43a123a31a20
a32a2a33
a192
a54
a25a2a26a2a225a2a226a17a20
a83 4.1 a47 (X; Y )
a14a2a150a2a151a2a152a2a51
a13
p(x; y) =
8
<
:
1; x
2 + y2 6 1;
0; a106a2a227a17a20
a52a2a228a2a229a2a97
a62 EX = EY = 0
a20a12a230
a135
a97 XX; XY ; Y Y a20
XX =
ZZ
x2+y261
(x EX)2p(x; y)dxdy
=
ZZ
x2+y261
x
dxdy
= 14:
a87a38a125 Y Y = 14a20
XY =
ZZ
x2+y261
(x EX)(y EY )p(x; y)dxdy
=
ZZ
x2+y26
xy 1 dxdy
= 0
a66a2a22 =0a10a12a220
a54
a10 X; Y a86
a54
a79a2a80a2a81a2a82a2a14a17a20
a83 4.2 a47 (X; Y ) N(
1; 2; 21; 22; r)a10a12a97 X; Y a14a2a79a2a27a2a28a2a51a17a20
7
Ch4 8
a103 a231
a22 EX = 1; EY = 2; VarX = 21; VarY = 22a20
XY = E(X EX)(Y EY )
=
ZZ
R2
(x 1)(y 2) 12
1 2
p1 r2 exp 12(1 r2)
(x 1)2
21
2r(x 1)(y 2)
1 2 +
(y 2)2
22
dxdy
=
ZZ
R2
1 2uv
2 p1 r2 exp
1
2(1 r2)(u
2 2ruv + v2) dudv
= 1 2p2
Z 1
1
ve v22 dv
Z 1
1
up
2 p(1 r2)e
(u rv)22(1 r2) du
= 1 2p2
Z 1
1
rv2e v22 dv
= r 1 2:
a53a2a54
a79a2a27a2a28a2a51
a13
= XYp
XX Y Y
= r 1 2
1 2
= r:
a60a38a61
a21a2a154a2a155a2a156
a35 X; Y
a79a2a80a2a81a2a82 () = 0a20
a5 a53
a45a2a26
a6a2a7a2a8a2a9 X; Y
a10a110a232a2a233a2a234a2a133a2a40 aX + b a69a2a235 a166 Ya10
a115
a192
a54
a217a17a10a110a236
a237 a; b
a210
a62 aX + b
a238a2a239a2a240 Ya20
a112a2a113 a237 aX + b
a115
a54
a25a2a26
a6a2a7a2a8a2a9
a10
a53a43a54
a10a12a40
a32a2a33
a14a2a29a2a226
a69a2a241a2a242
a107a2a233a43a14a43a243a2a244a31a30
Q(a; b) = E[Y (aX + b)]2:
a230
a135
a119a2a97 a; ba10a12a210
a62 Q
a196
a237
a238a2a214a2a245a17a20
a40
a95
a15a2a246a17a10a12a22
Q(a; b) = E[y (aX + b)]2
= E[((Y EY ) a(X EX)) + (EY aEX b)]2
= E[(Y EY ) a(X EX)]2 + (EY aEX b)2
= Y Y 2a XY + a2 XX + (EY aEX b)2
= XX a XY XX 2 + Y Y
2
XY
XX + (EY aEX b)
2:
8
Ch4 9
a154a2a167a2a159
a = a = XY
XX; b = b = EY a EX
a160a17a10 Q a196
a237
a238a2a214a2a245a17a10a12a22
min
a;b
(a; b) = Q(a ; b ) = Y Y
2
XY
XX = Y Y (1
2
XY ):
a154a114a155
a70a184a35
a10 j XY ja247a114a239a114a240 1a10 a X + b a247a114a239a114a240 Ya10
a115
a192
a54
a217a17a10 j j a14a114a122a114a214a114a248a114a249
a218
X a78 Y a14a2a250a2a71a2a79a2a27a2a251a2a151a17a20
a138
a192
a54
a79a43a27a43a28a2a51a43a14
a164a43a165
a76
a190a44a20
9
