Ch3 1
x3 a0a2a1a2a3
a4a2a5a2a6a2a7a2a8a2a9a2a10a2a11a2a12a2a13a2a14a2a15a2a16a2a17a19a18a2a20a19a21a23a22a25a24a2a4a19a26a2a4a19a27a2a28a19a29a2a30a19a31a23a22a25a32a2a6a19a33
a10a2a11a2a34a2a35a2a36a2a37a2a15a2a16a2a17a2a18a19a20a19a21a39a38
a40a2a41 a42 X
1; ; Xn a43a2a44a2a45a2a46a48a47
( ;F; P) a31a2a15 n a28a2a34a2a35a2a36a2a37a39a22a50a49a2a51a2a52
a43
a16a2a17a2a18a2a20a2a15a2a53a2a54a2a55a2a56a19a57 n a28a2a58a2a59 x
1; ; xn a60a2a61
P(X1 6 x1; ; Xn 6 xn) = P(X1) P(Xn):
a58a63a62a63a31a39a22a64a34a63a35a2a36a2a37a19a15a2a18a2a20a19a21a2a65
a43
a13a19a14a2a15a2a18a19a20a2a21a23a22a66a31a2a67a19a68a2a69a2a70a19a13a2a14 fX
1 6
x1g; ;fXn 6 xng a16a2a17a2a18a2a20a39a38
a71a2a70a2a34a2a35a2a36a2a37a2a15a2a16a2a17a2a18a2a20
a61a2a72
a9a19a15a2a21a19a73a75a74
a53a63a54 X
1; ; Xn
a16a63a17a63a18a63a20a39a22 f
1(x); ; fn(x) a60a63a43
a6a63a76 Borel
a77a63a78a63a79
a59a39a22
a80 f
1(X1); ; fn(Xn)
a65
a43
a16a2a17a2a18a2a20a2a15a39a38
a55a81a70a81a82a81a83a81a84a81a34a81a35a81a36a81a37 X
1; ; Xn
a22a85a42a81a86a81a87a81a88
a79
a59a81a89a63a90 p
1(x1); ; pn(xn)
a22
a80
X1; ; Xna16a2a17a2a18a2a20
()
nY
i=1
pi(xi)
a43
(X1; ; Xn)a15a2a91a2a92a2a87a2a88,
a31a63a6a63a93a63a94a96a95a97a12a63a98a2a76a2a99a2a100a2a89a2a101a63a15a2a102a2a103a2a89a2a101a23a22a104a24a2a4a2a105a2a106a2a51a2a15a2a107a2a28a2a89a2a37a2a108a2a109a2a18
a20a39a38
a110 3.1 a42 (X; Y ) N(
1; 2; 21; 22;
a22a111a80 X; Y a16a2a17a2a18a2a20a2a15a2a112a2a26a19a113a19a14
a43
= 0a38
a114 X; Y a15a2a102a2a103a2a87a2a88
a79
a59a2a89a2a90a2a115
PX(x) = 1p2
1
exp (x 1)
2
2 21; P
Y (y) =
1p
2 2 exp
(y 2)2
2 22
,
a116 = 0 a109a39a22
a61
p(x; y) = pX(x)pY (y):
a77a2a117 X; Y a16a2a17a2a18a2a20a39a22a25a112a2a89a2a21a19a118a2a119a23a38
1
Ch3 2
a120 X; Y a16a2a17a2a18a2a20a23a22a122a121a123a70 p(x; y); p
X(x); pY (y) a60a2a43
a82a2a83
a79
a59a23a22a111a124a2a55a19a56a19a57
a15 (x; y)
a60a2a61
pX(x) pY (y) = p(x; y)
a125 a90a2a15a2a126 (X; Y ) = (
1; 2)
a22
a61
1
2 1 2 =
1
2 1 2p1 2,
a121a123a127a2a128a48a95 = 0a38a25a129a2a26a2a21a2a118a2a119a39a38
a27a2a8a2a130a2a131a2a12a2a87a2a88
a79
a59a2a6a2a28a2a21a19a73a75a74a25a53a2a54 p
1(x; y); p2(x; y) a60a2a43
(X; Y ) a15a2a87
a88
a79
a59a39a22a25a132a2a51a2a52
a60
a4 (x
0; y0) a133
a82a2a83a39a22a25a80a2a129
a61
p1(x0; y0) = p2(x0; y0):
x4 a134a2a135a2a136a2a137a2a138a2a139a2a140a2a141a2a142a2a143
a42 n
a144
a34a63a35a96a145a97a37 X = (X
1; ; Xn)
a15a63a91a63a92a63a89a63a101
a79
a59a2a115 F
X(x1; ; xn)
a22
a146
Y1 = g1(X1; ; Xn);
Y2 = g2(X1; ; Xn);
Ym = gm(X1; ; Xn);
a86a48a147 g
i(x1; ; xn); i = 1; ; m a60a2a43
n a76 Borel a77a2a78a2a79
a59a39a22a25a24a2a4a2a26a19a148a149a95 m
a144
a34a2a35a48a145a123a37 Y = (Y
1; ; Ym)
a15a2a91a2a92a2a89a2a101
a79
a59 F
Y (y1; ; ym)
a38
a6a81a150a81a15a81a151a81a152a81a53
a72
a74a85a153 C = f(x
1; ; xn)jgi(x1; ; xn) 6 yi; i = 1; ;g
a22
a80
FY (y1; ; ym) = P(g1(X1; ; Xn) 6 y1; ; gm(X1; ; Xn) 6 ym)
= P((X1; ; Xn) 2 C):
a116 X
a61
a91a2a92a2a87a2a88 p(x
1; ; xn)
a109a39a22
a61
FY (y1; ; ym) =
Z
Z
C
p(x1; ; xn)dx1 dxn:
2
Ch3 3
a72
a9a2a94a48a95a123a154a2a28a2a155a2a156a39a38
a110 4.1 a42 (X; Y ) a15a2a91a2a92a2a87a2a88a2a115 p(x; y)a22a25a148 Z = X + Y a15a2a89a2a101a39a38
a157 a158a2a159a2a160a2a161 Z a15a2a89a2a101
a79
a59
FZ(z) = P(X + Y 6 z)
=
ZZ
x+y6z
p(x; y)dxdy
=
Z 1
1
dx
Z z x
1
p(x; y)dy
=
Z 1
1
dx
Z z
1
p(x; u x)du
=
Z z
1
du
Z 1
1
p(x; u x)dx
a27a2a162a48a163 Z
a164
a61
a87a2a88
a79
a59
pZ(z) =
Z 1
1
p(x; z x)dx:
a121a123a55a2a49a2a21
a77a2a117
a22 p
Z (z) =
Z 1
1
p(z y; y)dya38
a110 4.2 a42 X; Y a16a2a17a2a18a2a20a23a22a111a132
a60a19a165a19a166
a89a19a101 N( ; 2)a22a111a148 X + Y a15a2a89a2a101
a87a2a88
a79
a59 g(z)a38
a157 a121a123a31a2a155a2a158a2a159a2a160a2a161
a77
a118
g(z) = 1p2 p2 exp (z 2 )
2
4 2
,
a27a2a162a48a163 X + Y N(2 ; 2 2)a38
a110 4.3 a42 X; Y a16a2a17a2a18a2a20a2a132
a60a2a165a2a166
a89a2a101 N(0; 1)a22a167a148 Z = pX2 + Y 2 a15
a89a2a101a2a87a2a88a39a38
a157 a130a2a168a2a169a2a170a2a36a2a171
(
x = r cos ;
y = r sin ; r > 0; 0 6 6 2,
3
Ch3 4
FZ(z) = P(pX2 + Y 2 6 z)
=
ZZ
pX2+Y 26z
1
2 exp
1
2(x
2 + y2) dxdy:
=
Z 2
0
Z z
0
1
2 e
r22 rdr
=
Z z
0
re r22 dr
a172 a127a39a22
pZ(z) =
(
ze z22 ; z > 0;
0; a86a2a51,
a27a2a28a2a89a2a101a2a49a2a115 Rayleigh a89a2a101a39a38
a31a63a9a63a15
a60a63a43
2 ! 1 a15a63a173a63a174a39a22
a72
a9a2a26a2a10a2a11 n ! n a15a63a173a63a174a39a22a176a175 n = 2 a115a63a155a39a38
a42 (X; Y )
a61
a91a81a92a81a87a81a88 p(x; y)a22a85a132a75a177a179a178 A
a180a181a77
a175a81a182a81a183a81a184a81a9a48a185a187a186a63a188 P((X; Y ) 2
A) = 1a189
a55a2a36a2a190 ( )
a189 (
u = f(x; y);
v = g(x; y);
a116 (x; y) 2 A a109a39a22 (u; v) a15a2a191a2a178a2a115 Ga22a25a132 ( ) a186a2a188a23a74
(1) A ( ) ! G
a43
a6a2a6a2a55a2a192a2a15
a189
(2) f; g a4 A a147
a61
a82a2a83a2a193a2a194a2a59
a189
(3) @(u; v)@(x; y) a4 A a147 a133a2a133a2a195
a115 0
a189
a80 U(= f(X; Y )); V (= g(X; Y ))
a164
a61
a91a2a92a2a87a2a88
a79
a59a23a74
q(u; v) =
8<
:
p(x(u; v); y(u; v))
@(x; y)
@(u; v); (u; v) 2 G;
0; a86a2a51,
a86a96a147a196a22 x(u; v); y(u; v)
a43
a121 ( )
a197a63a198
a15a63a199
a79
a59a23a22 a146 @(u; v)
@(x; y)
a115a63a16a63a192a63a15a63a200
a77a48a201a123a202
a203 a67a39a38
a110 4.4 a42 X; Y a16a2a17a2a18a2a20a23a22a204a132
a60a205a165a19a166
N(0; 1)a22 (R; )
a43
a184a2a9a2a31a2a34a19a35a19a206
(X; Y ) a16a2a192a2a15a2a168a2a207a39a208a25a168a2a209a23a22a167a210
a61
(
X = R cos ;
Y = R sin ; R > 0; 0 6 6 2,
4
Ch3 5
a148 (R; ) a15a2a91a2a92a2a87a2a88a39a38
a157 (X; Y ) a131 (R; ) a15a2a36a2a190 ( ) a115
8
<
:
R = pX2 + Y 2;
= arctan YX ;
a16a2a192a2a15a2a200
a77a48a201a123a202
a203 a67
@(u; v)
@(x; y)
= 1
r,
@(x; y)
@(u; v)
= r:
a70
a43
a22 (R; ) a15a2a91a2a92a2a87a2a88
a79
a59a2a115
q(r; ) = 12 e 12 (x2+y2)
@(x; y)
@(u; v)
= 12 e 12 r2 r; r > 0; 0 < < 2,
a211
a77
a175a81a106a75a95a196a22 R
a212
a43
a16a81a17a81a18a81a20a63a15a39a22 R
a165a81a166
Rayleigha89a81a101a213a22 a146
a165a81a166
(0; 2 )
a31a2a15a2a214a2a215a2a89a2a101a39a38
a27a205a28a205a155a205a156a205a15a205a216a205a54 a211
a77
a175a213a130a213a105a205a217a213a218a213a18a205a20a213a15a213a107a213a28a205a99a213a100a213a34a205a35a213a59a96a22
a164a213a219a213a220
a152a213a53
a72
a74a50a221a2a217a2a218a2a107a2a28a2a16a19a17a19a18a2a20a19a15 (0; 1) a15a2a214a2a215a2a89a2a101a2a15a2a34a2a35a2a59 U
1; U2
a22a25a222a2a223a2a126
X = ( 2lnU1)12 cos 2 U2;
Y = ( 2lnU1)12 sin 2 U2;
a80 X; Y
a43
a16a2a17a2a18a2a20a2a15 N(0; 1) a34a2a35a2a59a39a38
a159
a72
a105a2a224
a144
a99a2a100a2a89a2a101a2a15a2a225a19a21
a79
a59a19a15a2a89a19a101a23a38
a40a2a226 a42 X N( ; )a22 B
a43
a56a2a6 n
a227a2a228a2a229
a171a2a230a48a231a196a22a25a80
Y = BX N(B ; B B0):
a119a48a163a123a158a2a159a2a160a2a161 Y a15a2a89a2a101
a79
a59
a77
a118a39a38
a232a2a94a48a95a123a6a2a28a2a71a2a70a2a224
a144
a99a2a100a19a89a2a101a19a15a19a102a2a103a19a89a2a101
a79
a59a2a15
a198a19a233
a38
a40a2a226 a42 X N( ; )a22 a120
=
1 0
0 2
!;
5
Ch3 6
a80
X1 N( 1; 1); X2 N( 2; 2):
a86a48a147a196a22
1 a43
m a227a2a220
a231a196a22
2 a43
n m a227a2a220
a231a196a22 a146 X
1; X2; 1; 2 a43
a16a2a192a2a15 a203
a145a123a37a39a38
a121a97a31a63a9a63a15a63a107a63a28
a198a2a233a2a234a2a77
a175
a229
a95a123a6a2a150a2a173a2a174a2a15a2a224
a144
a99a2a100a2a89a2a101a2a15a2a102a2a103a2a89a2a101a2a12a23a38
a40a2a226 a42 X N( ; )a22a25a55 a32
a202
a89a2a235
=
11 12
21 22
!;
a80
X1 N( 1; 1); X2 N( 2; 2):
a86a48a147a196a22
1 a43
m a227a2a220
a231a196a22
2 a43
n m a227a2a220
a231a196a22 a146 X
1; X2; 1; 2 a43
a16a2a192a2a15 a203
a145a123a37a39a38a25a236a2a130a2a230a48a231a123a15a19a36a19a190
a234a19a77
a175a19a119a149a163a237a38
a55a63a70 n a28a63a18a63a20a96a238a97a89a19a101a2a15
a165a19a166
N(0; 1) a15a63a34a63a35a63a36a63a37a39a22a25a51a2a52a19a15a2a184 a220a19a212
a165a19a166
a15
a89a2a101a2a4a2a239a2a160a2a8a2a9a2a192a2a130a2a240a19a241a23a38
a72
a9a19a105a2a160a19a161a2a27a19a28a19a89a2a101a23a38
a42 X
1; ; Xn
a16a63a17a63a18a63a20a63a132a96a238a123a89a2a101a23a22
a165a2a166
N(0; 1)a22a242a148 Y = X21 + + X2n
a15a2a89a2a101a2a87a2a88
a79
a59a39a38
a157 a153 D
y = f(x1,; xn)jx21 + + x2n 6 yg
a22a25a80
P(Y 6 y) = P(X21 + X2n 6 y)
=
Z
Z
Dy
1
(p2 )n exp
x21 + + x2n
2
dx
1 dxn
a32
a202 n a144a2a243
a169a2a170a2a36a2a190
8
>>>
>>>
<
>>>
>>>
:
x1 = r cos 1;
x2 = r sin 1 cos 2;
xn 1 = r sin 1 sin 2 sin n 2 cos n 1;
xn = r sin 1 sin 2 sin n 2 sin n 1;
a86a48a147a237a22 r > 0; 0 6
1; 2; ; n 2 < ; 0 6 n 1 < 2
a38a244a70
a43
a31a2a9a2a15a19a245a19a89a149a177a246a178
a234
a171a2a115
D y = f(r; 1; ; n 1)j0 < r 6 py; 0 6 1; 2; ; n 2 < ; 0 6 n 1 < 2 g:
6
Ch3 7
a70
a43
a31a2a9a2a15a2a245a2a89a2a67a2a36a2a115
P(Y 6 y) =
Z
Z
D y
1
(p2 )n e
r22 rn 1f( 1; ; n 1)drd 1 d n 1
= c
Z py
0
rn 1e r22 dr
= c Ry0 sn2 1e s2 ds:
a70
a43
a22 Y
a61
a87a2a88
a79
a59
pY (y) = cy n2 1e y2 ;
a86a247a147 c a115a213a248a213a6a213a171a213a249a213a59a96a38
a201a251a250
a175a39a31a213a87a39a88a39a252 (n
2;
1
2)
a15a213a87a213a88
a79
a59a96a22
a77a213a117 Y
(n2; 12)a38
a31a2a9a2a118a2a131a2a15a2a89a2a101a2a49a2a115a254a253a123a121a246a88a2a115 n a15a2a255
a220
a89a2a101a39a22a25a153a1a0 2(n)a38
a222a2a223a39a22a25a94a48a95a123a6a2a28
a61
a71a19a255
a220
a89a19a101a2a15
a198a2a233
a38
a40a2a226 a42 X
1; ; Xn
a18a2a20a48a238a123a89a2a101 N(0; 1)a22a25a80
(1) X = 1n(X1 + + Xn) N(0; 1);
(2)
nX
i=1
(Xi X)2 2(n 1);
(3) X a252
nX
i=1
(Xi X)2 a16a2a17a2a18a2a20a39a38
x5 a2a1a3a1a4a1a5a2a137
a40a81a41 a42 X
1; ; Xn
a18a81a20a75a238a179a89a81a101 F( )a22a85a55a7a6a81a28 ! 2 a22a9a8 X
1(!); ; Xn(!)
a166a1a10
a131a1a11a1a12a1a13a23a74
X(1)(!) 6 X(2)(!) 6 X(n)(!);
a27a7a14a7a15a81a52a81a118a81a131 na28a7a16a81a15a81a34a81a35a81a36a63a37 X
(1)(!); ; X(n)(!)
a22 a49a81a51a81a52a81a115 X
1; ; Xn
a15a1a17a1a13a2a239a2a160a2a37a39a38
a159
a72
a105a63a15a19a18a21a20
a234
a43
a22a22a17a1a13a2a239a19a160a2a37a2a15a19a89a2a101
a43a24a23a1a25
a38a66a55a2a70 X
(1); X(n)
a22a64a89a63a101
a201
a250a1a26a1a27
a148a39a38a25a236a2a130
X(1) = min X1; ; Xn;
7
Ch3 8
X(n) = max X1; ; Xn;
a77a2a117
FX(n)(x) = P(X(n) 6 x)
= P(X1 6 x; ; Xn 6 x)
= [F(x)]n:
a28a1a29 a15
a77
a148a48a95 X
(1)
a15a2a89a2a101
a79
a59a39a38
a55a2a70a2a6a2a150a2a15 ka22 a65
a77
a175a1a30a2a31a2a9a2a15 a28a24a29 a15a19a160a19a161a19a89a19a101
a79
a59a75a22a32a31
a43
a89a24a33a19a26a1a34a24a35
a6a1a36a39a22
a77
a148a2a118
FX(k)(x) =
NX
i=k
Cin(F(x))i(1 F(x))n i; k = 1; 2; ; n:
a130a2a89a2a33a2a245a2a89a1a37a2a67
a77
a175a1a38a19a119
a72
a9a19a15a1a39a19a68a19a67a75a74
X
ni = kCni pi(1 p)n i = n!(k 1)!(n k)!
Z p
0
tk 1(1 t)n kdt;
a172 a127a39a22 X
(k)
a15a2a89a2a101
a79
a59
a77
a175a1a40a1a41a2a115
FX(k)(x) = n!(k 1)!(n k)!
Z F(x)
0
tk 1(1 t)n kdt:
a116 F(x)
a61
a87a2a88
a79
a59 p(x) a109a39a22 F
X(k)
a65
a61
a16a2a192a2a15a2a87a2a88
a79
a59a23a74
p(k)(x) = n!(k 1)!(n k)!(F(x))k 1(1 F(x))n k p(x):
a222a2a223a2a232a2a106 X
(1); ; X(n)
a15a2a91a2a92a2a89a2a101a23a38a111a116 F(x)
a61
a87a2a88
a79
a59 p(x) a109a23a22a111a130
a42a2a76a2a87a2a88a2a152
a77
a175a2a148a2a118 X
(1); ; X(n)
a15a2a91a2a92a2a87a2a88a2a115
g(x1; ; xn) =
(
n!p(x1) p(xn); x1 < < xn;
0; a86a2a51,
8
x3 a0a2a1a2a3
a4a2a5a2a6a2a7a2a8a2a9a2a10a2a11a2a12a2a13a2a14a2a15a2a16a2a17a19a18a2a20a19a21a23a22a25a24a2a4a19a26a2a4a19a27a2a28a19a29a2a30a19a31a23a22a25a32a2a6a19a33
a10a2a11a2a34a2a35a2a36a2a37a2a15a2a16a2a17a2a18a19a20a19a21a39a38
a40a2a41 a42 X
1; ; Xn a43a2a44a2a45a2a46a48a47
( ;F; P) a31a2a15 n a28a2a34a2a35a2a36a2a37a39a22a50a49a2a51a2a52
a43
a16a2a17a2a18a2a20a2a15a2a53a2a54a2a55a2a56a19a57 n a28a2a58a2a59 x
1; ; xn a60a2a61
P(X1 6 x1; ; Xn 6 xn) = P(X1) P(Xn):
a58a63a62a63a31a39a22a64a34a63a35a2a36a2a37a19a15a2a18a2a20a19a21a2a65
a43
a13a19a14a2a15a2a18a19a20a2a21a23a22a66a31a2a67a19a68a2a69a2a70a19a13a2a14 fX
1 6
x1g; ;fXn 6 xng a16a2a17a2a18a2a20a39a38
a71a2a70a2a34a2a35a2a36a2a37a2a15a2a16a2a17a2a18a2a20
a61a2a72
a9a19a15a2a21a19a73a75a74
a53a63a54 X
1; ; Xn
a16a63a17a63a18a63a20a39a22 f
1(x); ; fn(x) a60a63a43
a6a63a76 Borel
a77a63a78a63a79
a59a39a22
a80 f
1(X1); ; fn(Xn)
a65
a43
a16a2a17a2a18a2a20a2a15a39a38
a55a81a70a81a82a81a83a81a84a81a34a81a35a81a36a81a37 X
1; ; Xn
a22a85a42a81a86a81a87a81a88
a79
a59a81a89a63a90 p
1(x1); ; pn(xn)
a22
a80
X1; ; Xna16a2a17a2a18a2a20
()
nY
i=1
pi(xi)
a43
(X1; ; Xn)a15a2a91a2a92a2a87a2a88,
a31a63a6a63a93a63a94a96a95a97a12a63a98a2a76a2a99a2a100a2a89a2a101a63a15a2a102a2a103a2a89a2a101a23a22a104a24a2a4a2a105a2a106a2a51a2a15a2a107a2a28a2a89a2a37a2a108a2a109a2a18
a20a39a38
a110 3.1 a42 (X; Y ) N(
1; 2; 21; 22;
a22a111a80 X; Y a16a2a17a2a18a2a20a2a15a2a112a2a26a19a113a19a14
a43
= 0a38
a114 X; Y a15a2a102a2a103a2a87a2a88
a79
a59a2a89a2a90a2a115
PX(x) = 1p2
1
exp (x 1)
2
2 21; P
Y (y) =
1p
2 2 exp
(y 2)2
2 22
,
a116 = 0 a109a39a22
a61
p(x; y) = pX(x)pY (y):
a77a2a117 X; Y a16a2a17a2a18a2a20a39a22a25a112a2a89a2a21a19a118a2a119a23a38
1
Ch3 2
a120 X; Y a16a2a17a2a18a2a20a23a22a122a121a123a70 p(x; y); p
X(x); pY (y) a60a2a43
a82a2a83
a79
a59a23a22a111a124a2a55a19a56a19a57
a15 (x; y)
a60a2a61
pX(x) pY (y) = p(x; y)
a125 a90a2a15a2a126 (X; Y ) = (
1; 2)
a22
a61
1
2 1 2 =
1
2 1 2p1 2,
a121a123a127a2a128a48a95 = 0a38a25a129a2a26a2a21a2a118a2a119a39a38
a27a2a8a2a130a2a131a2a12a2a87a2a88
a79
a59a2a6a2a28a2a21a19a73a75a74a25a53a2a54 p
1(x; y); p2(x; y) a60a2a43
(X; Y ) a15a2a87
a88
a79
a59a39a22a25a132a2a51a2a52
a60
a4 (x
0; y0) a133
a82a2a83a39a22a25a80a2a129
a61
p1(x0; y0) = p2(x0; y0):
x4 a134a2a135a2a136a2a137a2a138a2a139a2a140a2a141a2a142a2a143
a42 n
a144
a34a63a35a96a145a97a37 X = (X
1; ; Xn)
a15a63a91a63a92a63a89a63a101
a79
a59a2a115 F
X(x1; ; xn)
a22
a146
Y1 = g1(X1; ; Xn);
Y2 = g2(X1; ; Xn);
Ym = gm(X1; ; Xn);
a86a48a147 g
i(x1; ; xn); i = 1; ; m a60a2a43
n a76 Borel a77a2a78a2a79
a59a39a22a25a24a2a4a2a26a19a148a149a95 m
a144
a34a2a35a48a145a123a37 Y = (Y
1; ; Ym)
a15a2a91a2a92a2a89a2a101
a79
a59 F
Y (y1; ; ym)
a38
a6a81a150a81a15a81a151a81a152a81a53
a72
a74a85a153 C = f(x
1; ; xn)jgi(x1; ; xn) 6 yi; i = 1; ;g
a22
a80
FY (y1; ; ym) = P(g1(X1; ; Xn) 6 y1; ; gm(X1; ; Xn) 6 ym)
= P((X1; ; Xn) 2 C):
a116 X
a61
a91a2a92a2a87a2a88 p(x
1; ; xn)
a109a39a22
a61
FY (y1; ; ym) =
Z
Z
C
p(x1; ; xn)dx1 dxn:
2
Ch3 3
a72
a9a2a94a48a95a123a154a2a28a2a155a2a156a39a38
a110 4.1 a42 (X; Y ) a15a2a91a2a92a2a87a2a88a2a115 p(x; y)a22a25a148 Z = X + Y a15a2a89a2a101a39a38
a157 a158a2a159a2a160a2a161 Z a15a2a89a2a101
a79
a59
FZ(z) = P(X + Y 6 z)
=
ZZ
x+y6z
p(x; y)dxdy
=
Z 1
1
dx
Z z x
1
p(x; y)dy
=
Z 1
1
dx
Z z
1
p(x; u x)du
=
Z z
1
du
Z 1
1
p(x; u x)dx
a27a2a162a48a163 Z
a164
a61
a87a2a88
a79
a59
pZ(z) =
Z 1
1
p(x; z x)dx:
a121a123a55a2a49a2a21
a77a2a117
a22 p
Z (z) =
Z 1
1
p(z y; y)dya38
a110 4.2 a42 X; Y a16a2a17a2a18a2a20a23a22a111a132
a60a19a165a19a166
a89a19a101 N( ; 2)a22a111a148 X + Y a15a2a89a2a101
a87a2a88
a79
a59 g(z)a38
a157 a121a123a31a2a155a2a158a2a159a2a160a2a161
a77
a118
g(z) = 1p2 p2 exp (z 2 )
2
4 2
,
a27a2a162a48a163 X + Y N(2 ; 2 2)a38
a110 4.3 a42 X; Y a16a2a17a2a18a2a20a2a132
a60a2a165a2a166
a89a2a101 N(0; 1)a22a167a148 Z = pX2 + Y 2 a15
a89a2a101a2a87a2a88a39a38
a157 a130a2a168a2a169a2a170a2a36a2a171
(
x = r cos ;
y = r sin ; r > 0; 0 6 6 2,
3
Ch3 4
FZ(z) = P(pX2 + Y 2 6 z)
=
ZZ
pX2+Y 26z
1
2 exp
1
2(x
2 + y2) dxdy:
=
Z 2
0
Z z
0
1
2 e
r22 rdr
=
Z z
0
re r22 dr
a172 a127a39a22
pZ(z) =
(
ze z22 ; z > 0;
0; a86a2a51,
a27a2a28a2a89a2a101a2a49a2a115 Rayleigh a89a2a101a39a38
a31a63a9a63a15
a60a63a43
2 ! 1 a15a63a173a63a174a39a22
a72
a9a2a26a2a10a2a11 n ! n a15a63a173a63a174a39a22a176a175 n = 2 a115a63a155a39a38
a42 (X; Y )
a61
a91a81a92a81a87a81a88 p(x; y)a22a85a132a75a177a179a178 A
a180a181a77
a175a81a182a81a183a81a184a81a9a48a185a187a186a63a188 P((X; Y ) 2
A) = 1a189
a55a2a36a2a190 ( )
a189 (
u = f(x; y);
v = g(x; y);
a116 (x; y) 2 A a109a39a22 (u; v) a15a2a191a2a178a2a115 Ga22a25a132 ( ) a186a2a188a23a74
(1) A ( ) ! G
a43
a6a2a6a2a55a2a192a2a15
a189
(2) f; g a4 A a147
a61
a82a2a83a2a193a2a194a2a59
a189
(3) @(u; v)@(x; y) a4 A a147 a133a2a133a2a195
a115 0
a189
a80 U(= f(X; Y )); V (= g(X; Y ))
a164
a61
a91a2a92a2a87a2a88
a79
a59a23a74
q(u; v) =
8<
:
p(x(u; v); y(u; v))
@(x; y)
@(u; v); (u; v) 2 G;
0; a86a2a51,
a86a96a147a196a22 x(u; v); y(u; v)
a43
a121 ( )
a197a63a198
a15a63a199
a79
a59a23a22 a146 @(u; v)
@(x; y)
a115a63a16a63a192a63a15a63a200
a77a48a201a123a202
a203 a67a39a38
a110 4.4 a42 X; Y a16a2a17a2a18a2a20a23a22a204a132
a60a205a165a19a166
N(0; 1)a22 (R; )
a43
a184a2a9a2a31a2a34a19a35a19a206
(X; Y ) a16a2a192a2a15a2a168a2a207a39a208a25a168a2a209a23a22a167a210
a61
(
X = R cos ;
Y = R sin ; R > 0; 0 6 6 2,
4
Ch3 5
a148 (R; ) a15a2a91a2a92a2a87a2a88a39a38
a157 (X; Y ) a131 (R; ) a15a2a36a2a190 ( ) a115
8
<
:
R = pX2 + Y 2;
= arctan YX ;
a16a2a192a2a15a2a200
a77a48a201a123a202
a203 a67
@(u; v)
@(x; y)
= 1
r,
@(x; y)
@(u; v)
= r:
a70
a43
a22 (R; ) a15a2a91a2a92a2a87a2a88
a79
a59a2a115
q(r; ) = 12 e 12 (x2+y2)
@(x; y)
@(u; v)
= 12 e 12 r2 r; r > 0; 0 < < 2,
a211
a77
a175a81a106a75a95a196a22 R
a212
a43
a16a81a17a81a18a81a20a63a15a39a22 R
a165a81a166
Rayleigha89a81a101a213a22 a146
a165a81a166
(0; 2 )
a31a2a15a2a214a2a215a2a89a2a101a39a38
a27a205a28a205a155a205a156a205a15a205a216a205a54 a211
a77
a175a213a130a213a105a205a217a213a218a213a18a205a20a213a15a213a107a213a28a205a99a213a100a213a34a205a35a213a59a96a22
a164a213a219a213a220
a152a213a53
a72
a74a50a221a2a217a2a218a2a107a2a28a2a16a19a17a19a18a2a20a19a15 (0; 1) a15a2a214a2a215a2a89a2a101a2a15a2a34a2a35a2a59 U
1; U2
a22a25a222a2a223a2a126
X = ( 2lnU1)12 cos 2 U2;
Y = ( 2lnU1)12 sin 2 U2;
a80 X; Y
a43
a16a2a17a2a18a2a20a2a15 N(0; 1) a34a2a35a2a59a39a38
a159
a72
a105a2a224
a144
a99a2a100a2a89a2a101a2a15a2a225a19a21
a79
a59a19a15a2a89a19a101a23a38
a40a2a226 a42 X N( ; )a22 B
a43
a56a2a6 n
a227a2a228a2a229
a171a2a230a48a231a196a22a25a80
Y = BX N(B ; B B0):
a119a48a163a123a158a2a159a2a160a2a161 Y a15a2a89a2a101
a79
a59
a77
a118a39a38
a232a2a94a48a95a123a6a2a28a2a71a2a70a2a224
a144
a99a2a100a19a89a2a101a19a15a19a102a2a103a19a89a2a101
a79
a59a2a15
a198a19a233
a38
a40a2a226 a42 X N( ; )a22 a120
=
1 0
0 2
!;
5
Ch3 6
a80
X1 N( 1; 1); X2 N( 2; 2):
a86a48a147a196a22
1 a43
m a227a2a220
a231a196a22
2 a43
n m a227a2a220
a231a196a22 a146 X
1; X2; 1; 2 a43
a16a2a192a2a15 a203
a145a123a37a39a38
a121a97a31a63a9a63a15a63a107a63a28
a198a2a233a2a234a2a77
a175
a229
a95a123a6a2a150a2a173a2a174a2a15a2a224
a144
a99a2a100a2a89a2a101a2a15a2a102a2a103a2a89a2a101a2a12a23a38
a40a2a226 a42 X N( ; )a22a25a55 a32
a202
a89a2a235
=
11 12
21 22
!;
a80
X1 N( 1; 1); X2 N( 2; 2):
a86a48a147a196a22
1 a43
m a227a2a220
a231a196a22
2 a43
n m a227a2a220
a231a196a22 a146 X
1; X2; 1; 2 a43
a16a2a192a2a15 a203
a145a123a37a39a38a25a236a2a130a2a230a48a231a123a15a19a36a19a190
a234a19a77
a175a19a119a149a163a237a38
a55a63a70 n a28a63a18a63a20a96a238a97a89a19a101a2a15
a165a19a166
N(0; 1) a15a63a34a63a35a63a36a63a37a39a22a25a51a2a52a19a15a2a184 a220a19a212
a165a19a166
a15
a89a2a101a2a4a2a239a2a160a2a8a2a9a2a192a2a130a2a240a19a241a23a38
a72
a9a19a105a2a160a19a161a2a27a19a28a19a89a2a101a23a38
a42 X
1; ; Xn
a16a63a17a63a18a63a20a63a132a96a238a123a89a2a101a23a22
a165a2a166
N(0; 1)a22a242a148 Y = X21 + + X2n
a15a2a89a2a101a2a87a2a88
a79
a59a39a38
a157 a153 D
y = f(x1,; xn)jx21 + + x2n 6 yg
a22a25a80
P(Y 6 y) = P(X21 + X2n 6 y)
=
Z
Z
Dy
1
(p2 )n exp
x21 + + x2n
2
dx
1 dxn
a32
a202 n a144a2a243
a169a2a170a2a36a2a190
8
>>>
>>>
<
>>>
>>>
:
x1 = r cos 1;
x2 = r sin 1 cos 2;
xn 1 = r sin 1 sin 2 sin n 2 cos n 1;
xn = r sin 1 sin 2 sin n 2 sin n 1;
a86a48a147a237a22 r > 0; 0 6
1; 2; ; n 2 < ; 0 6 n 1 < 2
a38a244a70
a43
a31a2a9a2a15a19a245a19a89a149a177a246a178
a234
a171a2a115
D y = f(r; 1; ; n 1)j0 < r 6 py; 0 6 1; 2; ; n 2 < ; 0 6 n 1 < 2 g:
6
Ch3 7
a70
a43
a31a2a9a2a15a2a245a2a89a2a67a2a36a2a115
P(Y 6 y) =
Z
Z
D y
1
(p2 )n e
r22 rn 1f( 1; ; n 1)drd 1 d n 1
= c
Z py
0
rn 1e r22 dr
= c Ry0 sn2 1e s2 ds:
a70
a43
a22 Y
a61
a87a2a88
a79
a59
pY (y) = cy n2 1e y2 ;
a86a247a147 c a115a213a248a213a6a213a171a213a249a213a59a96a38
a201a251a250
a175a39a31a213a87a39a88a39a252 (n
2;
1
2)
a15a213a87a213a88
a79
a59a96a22
a77a213a117 Y
(n2; 12)a38
a31a2a9a2a118a2a131a2a15a2a89a2a101a2a49a2a115a254a253a123a121a246a88a2a115 n a15a2a255
a220
a89a2a101a39a22a25a153a1a0 2(n)a38
a222a2a223a39a22a25a94a48a95a123a6a2a28
a61
a71a19a255
a220
a89a19a101a2a15
a198a2a233
a38
a40a2a226 a42 X
1; ; Xn
a18a2a20a48a238a123a89a2a101 N(0; 1)a22a25a80
(1) X = 1n(X1 + + Xn) N(0; 1);
(2)
nX
i=1
(Xi X)2 2(n 1);
(3) X a252
nX
i=1
(Xi X)2 a16a2a17a2a18a2a20a39a38
x5 a2a1a3a1a4a1a5a2a137
a40a81a41 a42 X
1; ; Xn
a18a81a20a75a238a179a89a81a101 F( )a22a85a55a7a6a81a28 ! 2 a22a9a8 X
1(!); ; Xn(!)
a166a1a10
a131a1a11a1a12a1a13a23a74
X(1)(!) 6 X(2)(!) 6 X(n)(!);
a27a7a14a7a15a81a52a81a118a81a131 na28a7a16a81a15a81a34a81a35a81a36a63a37 X
(1)(!); ; X(n)(!)
a22 a49a81a51a81a52a81a115 X
1; ; Xn
a15a1a17a1a13a2a239a2a160a2a37a39a38
a159
a72
a105a63a15a19a18a21a20
a234
a43
a22a22a17a1a13a2a239a19a160a2a37a2a15a19a89a2a101
a43a24a23a1a25
a38a66a55a2a70 X
(1); X(n)
a22a64a89a63a101
a201
a250a1a26a1a27
a148a39a38a25a236a2a130
X(1) = min X1; ; Xn;
7
Ch3 8
X(n) = max X1; ; Xn;
a77a2a117
FX(n)(x) = P(X(n) 6 x)
= P(X1 6 x; ; Xn 6 x)
= [F(x)]n:
a28a1a29 a15
a77
a148a48a95 X
(1)
a15a2a89a2a101
a79
a59a39a38
a55a2a70a2a6a2a150a2a15 ka22 a65
a77
a175a1a30a2a31a2a9a2a15 a28a24a29 a15a19a160a19a161a19a89a19a101
a79
a59a75a22a32a31
a43
a89a24a33a19a26a1a34a24a35
a6a1a36a39a22
a77
a148a2a118
FX(k)(x) =
NX
i=k
Cin(F(x))i(1 F(x))n i; k = 1; 2; ; n:
a130a2a89a2a33a2a245a2a89a1a37a2a67
a77
a175a1a38a19a119
a72
a9a19a15a1a39a19a68a19a67a75a74
X
ni = kCni pi(1 p)n i = n!(k 1)!(n k)!
Z p
0
tk 1(1 t)n kdt;
a172 a127a39a22 X
(k)
a15a2a89a2a101
a79
a59
a77
a175a1a40a1a41a2a115
FX(k)(x) = n!(k 1)!(n k)!
Z F(x)
0
tk 1(1 t)n kdt:
a116 F(x)
a61
a87a2a88
a79
a59 p(x) a109a39a22 F
X(k)
a65
a61
a16a2a192a2a15a2a87a2a88
a79
a59a23a74
p(k)(x) = n!(k 1)!(n k)!(F(x))k 1(1 F(x))n k p(x):
a222a2a223a2a232a2a106 X
(1); ; X(n)
a15a2a91a2a92a2a89a2a101a23a38a111a116 F(x)
a61
a87a2a88
a79
a59 p(x) a109a23a22a111a130
a42a2a76a2a87a2a88a2a152
a77
a175a2a148a2a118 X
(1); ; X(n)
a15a2a91a2a92a2a87a2a88a2a115
g(x1; ; xn) =
(
n!p(x1) p(xn); x1 < < xn;
0; a86a2a51,
8
