Ch3 1
x6 a0a2a1a2a3a2a4
a5a2a6a2a7a2a8a2a9a2a10a2a11a2a12a2a13a15a14a17a16a19a18a2a20a22a21a24a23a19a25a19a26a19a27a2a28a19a12a19a13a15a14a29a16a2a11a19a30a19a31a22a21
a32 (X; Y )
a33
a9a2a10a2a26a2a27a2a28a2a12a2a13a15a14a17a16a22a34a24a35a19a36a2a37a19a38a19a18
P(X = xi; Y = yj) = pij; i; j = 1; 2; ;
a39a2a35a2a40a2a41a2a36a2a37a2a38a2a18
P(X = xi) =
X
j
pij = pi ; i = 1; 2; ;
P(Y = yj) =
X
i
pij = p j; j = 1; 2;,
a32 p
j > 0
a34a43a42a2a44a2a5 fY = y
jg a45a2a46
a11a2a30a2a31a2a47 fX = x
ig a45a2a46
a11a2a48a2a49a22a21a51a50a29a52a19a53
a48a2a49a2a54a2a55a2a56
P(X = xijY = yj) = P(X = xi; Y = yj)P(Y = y
j)
= pijp
j; i = 1; 2; ;
a57 a6a2a58a2a48a2a49a2a18a2a5 Y = y
j
a11a2a52a2a53a2a47a2a12a2a13a2a59a2a16 X a11a2a52a2a53a2a36a2a37a2a38a60a21
a61 6.1
a62a64a63a64a65a64a66a64a67a2a63a2a68
a34a70a69
a45a2a71a73a72a15a74a76a75
a11a2a48a19a49a2a18 p(0 < p < 1)a34
a63a64a68a64a66
a67a19a77a19a68a73a72a73a74a76a75a19a78a19a79
a18a81a80a82a21 X; Y a36a19a83a19a84a19a85a19a86
a62a19a79a19a87
a86a19a9
a79a81a68a88a72a73a74a89a75a81a90a19a63a81a68
a11
a79a2a91
a34a93a92 X; Y a11a2a94a2a95a2a36a2a37a2a96a2a52a2a53a2a36a2a37a22a21
a97 a98a2a99a2a100
P(X = m; Y = n) = p2qn 2; m = 1; ; n 1; n = 2; 3; ;
a101
a33
a40a2a41a2a36a2a37a2a38a2a18
P(X = m) =
1X
n=m+1
P(X = m; Y = n) = pqm 1 ; m = 1; 2; ;
P(Y = n) =
n 1X
m=1
P(X = m; Y = n) = (n 1)p2qn 2; n = 2; 3;,
1
Ch3 2
a102a2a103a2a11a2a52a2a53a2a36a2a37a2a18
P(X = mjY = n) = p
2qn 2
(n 1)p2qn 2 =
1
n 1;
P(Y = njX = m) = p
2qn 2
pqm 1 = pq
n m 1:
a5a19a25a19a104a19a105a19a28a19a11a19a30a19a31a106a21a89a32 (X; Y )
a33
a104a19a105a19a28a19a12a19a13a19a59a19a16a106a34a107a6
a33a81a108a19a109a81a110
a11 x; y
a111 a100
P(X = x) = P(Y = y) = 0;
a112a2a113a2a114a2a115 a52a2a53a2a48a2a49a2a54a2a55a19a116a19a117a2a118a19a119a2a120a22a21
a18a2a121a60a34a93a122a2a18a2a119a2a120a2a47a2a123a19a11a2a52a19a53a19a48a2a49a19a11a2a124a19a125a82a126
lim
"!0+
P(X 6 xjy " < Y 6 y + ")
= lim
"!0+
P(X 6 x; y " < Y 6 y + ")
P(y" < Y 6 y + ")
= lim
"!0+
F(x; y + ") F(x; y ")
FY (y + ") FY (y ")
=
@F(x; y)
@y
d
dyFY (y)
=
Z x
1
p(u; y)du
pY (y),
a57 a121a81a124a81a125a81a18a81a5 Y = y a47 X a11a81a52a81a53a81a36a81a37a81a127
a91
a34a129a128a131a130 P(X 6 xjY = y)
a132
FXjY (xjy)a21 a101 a33a2a133
a100
FXjY (xjy) =
Z x
1
p(u; y)
pY (y) du:
a134 a121a2a55a2a135a2a136a2a137a81a127
a91
a54a19a55a88a138a29a139a82a34
a133a141a140
a99 a57 p(x; y)
pY (y)
a18a2a52a2a53 Y = y a47 X a11a2a52a2a53
a136a2a137a2a127
a91
a34a93a128a2a130 p
XjY (xjy)
a34a93a142
pXjY (xjy) = p(x; y)p
Y (y)
:
2
Ch3 3
a61 6.1 a42a2a44a2a143a2a144a2a145
a72a29a146a19a147
a11a19a148a19a59a82a21a149a32
a109a19a62a19a150a19a151a81a146a19a147a19a77a19a152
a148a81a59a19a153a19a154a19a155a19a11
a90a15a156a17a33a2a62
a12a2a13a2a59a2a16a60a34a158a157a19a159a2a160
a91
a18 y a11a2a161
a91
a36a2a37a163a162
a108
a113a15a164a17a165 a28a2a11
a146a19a147
a34a24a35a19a160
a91
y a33
a113a15a164 a11a88a166a167a21a169a168a19a32
a108
a101 a143a81a144a19a145
a72
a12a81a13a19a170a19a171a81a11
a146a81a147
a34a169a35a19a160
a91 y a33a2a172a2a62
a12a2a13a19a59
a16 Y a11
a62
a58a64a173a60a34a175a174 Y a157a64a159 ( ; )a34a175a35
a72
a160
a91 ; a33
a50a129a176a64a177
a146a64a147
a11a2a178a2a179a2a52a2a53
a153a2a180
a151
a11a2a181a2a11a19a182
a91
a21a183a92a19a159a19a143a19a145
a72
a12a19a13a19a170a19a171a19a11
a146a81a147a19a77a19a152
a148a19a59a19a153a81a154
a90a73a156 X
a11a2a48
a49a2a136a2a137a2a127
a91
a21
a97 a50a17a184
a74
a52a2a53a2a185a2a186 X a11a2a52a2a53a2a48a2a49a2a136a2a137a2a18
pXjY (xjy) =
(
ye yx; x > 0;
x 6,
(X; Y ) a11a2a94a2a95a2a136a2a137a2a18 p(x; y) = pXjY (xjy)pY (y)a34 a101 a33 X a11a2a48a2a49a2a136a2a137a2a18
pX(x) =
Z 1
1
p(x; y)dy
=
Z 1
1
pXjY (xjy)pY (y)dy
=
Z 1
1
ye yx
( )y
1e ydy
=
( )
Z 1
1
y e( +x)udy
=
( )
( + 1)
( + x) +1
=
( + x) +1,
a142 X a11a2a136a2a137a2a127
a91
a18
pX(x) =
8
<
:
( + x) +1 ; x > 0
0; x 6 0:
a61 6.2 a32 (X; Y ) N(
1; 2; 21; 22; )
a34a93a92 p
YjX(yjx)
a21
a97a188a187a190a189a191a186a191a192 X N(
1; 21)
a34a194a193a191a195a191a52a191a53a191a48a191a49a191a54a64a55a64a119a191a120a64a196a64a197a64a185a191a186 P
YjX(yjx)
a33
a181a64a198a64a36a64a37 N(
2 +
2
1 (x 1); (1
2) 2
2)
a11a64a136a64a137a64a127
a91
a21a200a199
a133a2a33a2a201
a34a200a9a2a202a2a181
a198a2a36a2a37a2a11a2a52a2a53a2a136a2a137a2a203
a33
a181a19a198a2a36a19a37a22a21
3
Ch4 4
a204a15a205a17a206 a207a2a208a2a209a2a210a2a211a2a212a2a213a2a214a2a215
x1 a207a2a208a2a209a2a210a2a211a2a212a2a216a2a217a2a218
a108
a101
a62
a58a81a12a81a13a81a59a81a16a82a34a29a219a131a220a131a221a81a222a131a223
a77a131a62
a58
a91
a173a81a116a131a224a131a225a131a226a81a11a131a171a131a173a81a11a131a227a131a228a131a229
a230 a21a158a5a2a42a2a44a2a227a2a228a2a11
a90a19a231
a34 a113a19a232a19a232 a155a19a25a19a226a19a11a19a171a2a173a82a34a233a203a19a155a19a42a2a44
a77
a226a19a171a19a234a19a235a19a173a19a11a2a102
a103a2a11a2a48a2a49a60a21a93a18a2a121a22a34a24a219a19a220
a151a19a236a19a237a2a91a19a238a2a239
a222a22a21
a23a2a25a2a26a2a27a2a28a2a12a2a13a2a59a2a16a60a21
a240a2a241 a32a2a26a2a27a2a28a2a12a2a13a2a59a2a16 X a11a2a48a2a49a2a36a2a37a2a38a2a18
P(X = xk) = pk; k = 1; 2; ;
a39 a57
a87a2a91 X
k
xkpk =
X
k
xkP(X = xk)
a18a2a12a2a13a2a59a2a16 X a11
a91a2a238a2a239
a222a60a34a93a128a2a130 E(X)a21
a5a2a6a2a7a60a34a70a203a2a155a2a155a2a92a2a242a2a123a2a11a2a243
a91a2a244a19a108a2a245a19a246
a34a93a6a2a247a19a248 a114a19a249a2a250a19a251
a87a19a91
a113a19a252a2a253 a92
a87a2a79a2a254
a11a2a255a1a0a60a21a93a47a19a123a3a2a5a4a7a6a2a58a19a182 a115 a11a19a26a2a27a19a28a19a36a2a37a19a11
a239
a222a22a21
(1) a78a1a8
a36a2a37
a32 X a157a2a159a2a160
a91
a18 p a11
a78a1a8
a36a2a37a60a34a10a9a1a11a2a119a19a120
E(X) = 0 q + 1 p = p:
(2) a9a1a12a2a36a2a37
a32 X B(n; p)a34a93a142
P(X = k) = Cknpkqn k; k = 0; 1; ; n
a39
E(X) =
nX
k=0
kCknpkqn k
=
nX
k=1
nCn 1k 1pkqn k
= np
n 1X
k=0
Ckn 1pkqn 1 k
= np:
4
Ch4 5
a5
a151a64a236
a9a13a12a64a36a19a37a2a11
a90a2a231
a34a15a14
a77a3a16
a226a2a178a1a17a2a242
a133a2a33 n a58 a78a13a8
a36a64a37a64a11
a87
a11a2a36a19a37a22a34 a6
a7a1a18
a201a5a19a17a237
a9a1a12a2a36a2a37a2a11
a91a19a238a2a239
a222a2a199a3a20 a101 n a58
a78a1a8
a36a2a37a2a11
a91a2a238a2a239
a222a2a11
a87
a21
(3) Poisson a36a2a37
a32 X a157a2a159a2a160
a91
a18 a11 Poisson a36a2a37a60a34a93a39
E(X) =
1X
i=0
k e k
k
k! =
1X
k=1
e k
k 1
(k 1)! =
(4) a6a1a21a2a36a2a37
a32 X a157a2a159a2a160
a91
a18 p a11a1a6a1a21a2a36a2a37a60a34a93a39
E(X) =
1X
k=1
k pqk 1 = p 1(1 q)2 = 1p:
a118a2a47a2a116
a33
a104a2a105a2a28a2a12a2a13a2a59a2a16a19a11
a239
a222a22a21
a240a2a241 a32 X
a22
a100a2a136a2a137 p(x)a34a10a23a1a24
Z 1
1
jxjp(x)dx < 1a34a93a39 a57
Z 1
1
xp(x)dx
a18 X a11
a239
a222a60a34a93a128a2a130 E(X)a21
a118a2a47a2a116
a33
a6a2a58a2a182 a115 a11a2a104a2a105a19a28a2a12a19a13a19a59a2a16a19a11
a239
a222a22a21
(1) a228a1a25a2a36a2a37
a32 X a157a2a159 (a; b) a242a2a11a2a228a1a25a2a36a2a37a60a34a93a39
E(X) =
Z b
a
1
b axdx =
a + b
2,
a199
a133a2a33a2a201
a34a93a228a1a25a2a36a19a37a19a11
a239
a222a1a26a3a27
a33a5a28a19a156
a11
a72a7a8
a21
(2) a161 a91
a36a2a37
a32 X a157a2a159a2a160
a91
a18 a11a2a161
a91
a36a2a37a60a34a93a39
E(X) =
Z 1
0
x e xdx = 1,
(3) a181a2a198a2a36a2a37
5
Ch4 6
a32 X N( ; 2)a34a93a39
E(X) =
Z 1
1
x 1p2 e
(x )2
2 2 dx
= 1p2
Z 1
1
( t + )e t22 dt
=,
(4) a29a1a30
a36a2a37
a32 X ( ; )a34a93a39
E(X) =
Z 1
1
x
( )x
1ee xdx
= 1 ( )
Z 1
1
t e tdt
= ( + 1) ( )
=,
x2 a207a2a208a2a209a2a210a1a31a2a212a2a211a2a217a2a218a1a32a2a217a19a218a19a211a1a33a3a34
a240a36a35 a32 X = (X
1; X2 ; Xn)
a100a191a94a191a95a191a136a191a137 p
(x1; ; xn)
a34 Y = f(X
1; ; Xn)
a34
a39
E(Y ) =
Z
Z
Rn
f(x1; ; xn)p(x1; ; xn)dx1 dxn:
a6a2a58
a151
a197a2a11
a110a2a236
a5 a101 a34 a113a19a115 a92a5a4 Y a11a2a36a2a37a60a34a24a117a2a118a1a37a1a38 X a11a2a36a2a37
a133
a185a2a8
a92a5a4 Y a11
a239
a222a60a21a10a39 X a18a2a26a2a27a2a28a2a11
a90a2a231
a34a93a102a19a103a19a11a2a59a19a18
E(Y ) =
X
i1; ;in
f(x1i1; ; xnin)p(X1 = x1i1; ; Xn = xnin):
a61 2.1 a32 X U(0;
2)
a34a93a92 E(sin X)a21
6
Ch4 7
a97 E(sin X) = 2
Z
2
0
sin xdx = 2 a21
a61 2.2 a32 X; Y
a40a1a41
a164 a36a2a37 N(0; 1)a34a93a92 E(pX2 + Y 2)a21
a97 a50 X; Y
a40a1a41
a186a2a226a2a220a2a11a2a94a2a95a2a136a2a137
a133a19a33a3a42 a140
a136a2a137a19a11a3a43a1a44a22a34 a101
a33
E(pX2 + Y 2) =
ZZ
R2
p
x2 + y2 12 e x
2+y2
2 dxdy
=
Z 2
0
d
Z 1
0
1
2 re
r22 rdr
=
Z 1
0
p2t1
2 e tdt
= p2 (32) = p 2,
a91a2a238a2a239
a222a2a100a2a47a2a123a2a11a1a45a1a46a1a47a3a48a82a126
a91a2a238a2a239
a222a2a130a2a18
a62
a58a2a120
a147
a34
a33a3a49
a47a19a11a22a34a24a142
E(c0 + c1X1 + + cnXn) = c0 + c1E(X1) + + cnE(Xn):
a61 2.3 N a53a1a50a1a51
a72
a100 M a53
a33a2a79
a51a60a34a93a225a2a5a2a159
a72a53a52a3a54a56a55
a11a19a171 n
a79
a34a93a128
Xi =
(
1; a86 i a79
a171
a77
a181a1a51 ;
0; a86 i a79
a171
a77a2a79
a51 ;
i = 1; 2; ; n;
a57 Y =
nX
i=1
Xia34a93a142 Y a133a2a33
a171
a77
a11
a79
a51a1a58
a91
a34a158a92 E(Y )a21
a97 a5a2a6a2a7a60a34a10a59
a110a2a77 X1; ; Xn a100a2a102 a164 a11a2a36a2a37a60a34a93a174 E(X1) = MN a34 a112
E(Y ) =
nX
i=1
E(Xi) = nMN,
a6a2a7 Y a157a2a159a1a60a1a6a1a21a2a36a2a37a60a34a93a35
a239
a222a19a173a1a23a3a24 a115 a36a19a37a2a117a19a118a19a119a2a120 a252a1a61a3a62a3a63a1a64 a21
a61 2.4
a62
a58a1a65
a147a15a72a53a66
a100
a75
a242 1
a67 N a11 N a68a1a69a1a70
a34a158a8a2a100
a54a56a55
a11a1a71a19a55
a62
a68a2a62a1a68
a11a1a72a2a171a60a34a10a23a3a24a3a73a1a72
a77 r a68
a113a15a164 a11
a69a1a70
a34a93a92a3a72a2a171
a79a2a91 Sr a11 a239
a222a2a173a60a21
a97 a128 X
1; X2; ; Xr
a36a64a83a64a18a13a72
a77a2a62a1a68a1a74
a8a1a75
a75a1a76
a111a19a113 a102 a164 a11
a69a3a70
a153a2a154
a11
a79a2a91
a34a93a39 Y = X
1 + + Xr
a21
a98a2a99a2a100 X
1 = 1
a34 a112 E(X
1) = 1
a21
7
Ch4 8
X2 a157a2a159a2a160 a91
a18 p = N 1
N
a11a1a6a1a21a2a36a2a37a60a34 a112 E(X
2) =
N
N 1
a21
a164 a247a60a34 E(X
i) =
N
N 2; i = 1; 2
a21
a101
a33
a34 ES
r = E(X1 + + Xr) = N(
1
N r + 1 +
1
N )
a21
a5 N
a77a1a78
a229a2a11
a90a2a231
a34 a138a29a139
a78a3a79a19a150a1a80
a30a1a81a82a126 r = N; r = N
2
a21
r = N a90
a34 ES
N = N
1 + 1
2 + +
1
N
NlnN
a82
r = N2 a90
a34 ES
N
2
= N 1N
2 + 1
+ + 1N Nln2a21
a185a2a8a2a25a5a4 a34
a245a1a83a2a62a1a84
a11
a69a1a70
a154a2a155a19a11a2a227a19a228
a79a19a91a1a85
a64 a101
a68a2a91
a11
a62a3a84
a34a93a174
a245a3a83
a86a1a87 a11
a69a1a70
a34a93a154a19a155a2a11
a79a2a91a3a88
a18
a68a19a91
a11 lnN
a89
a34a93a199
a133a2a33a2a201
a34a91a90
a77a3a92
a123a22a34
a245a3a83a19a62
a68a1a93
a11
a69a1a70
a90a1a94a60a21
a239
a222a2a203a2a100a2a47a2a123a2a11a1a47a1a48a22a126
a23a1a24a2a12a2a13a2a59a2a16 X; Y a102a1a95
a40a1a41
a34a93a39 E(XY ) = E(X)E(Y )a21
a240a2a241 a57 EXk a18 X a11 k
a96a1a97
a34 E(X EX)k a18 X a11 k
a96a15a72a53a98a1a97
a21
a99 a101 X a11
a97
a34a93a100a2a47a2a123a2a11a1a100a3a101a22a126
a108a2a109a2a110
a11 > 1a34 a111 a100 EjXj > (EjXj) a21
a225a64a5a64a42a64a44 f(a) = E(X a)2a34 f0(a) = 2a 2EXa34 a185a64a186 f(a) a5 a = EX
a90
a171
a77a1a102
a230 a173a60a21
8
x6 a0a2a1a2a3a2a4
a5a2a6a2a7a2a8a2a9a2a10a2a11a2a12a2a13a15a14a17a16a19a18a2a20a22a21a24a23a19a25a19a26a19a27a2a28a19a12a19a13a15a14a29a16a2a11a19a30a19a31a22a21
a32 (X; Y )
a33
a9a2a10a2a26a2a27a2a28a2a12a2a13a15a14a17a16a22a34a24a35a19a36a2a37a19a38a19a18
P(X = xi; Y = yj) = pij; i; j = 1; 2; ;
a39a2a35a2a40a2a41a2a36a2a37a2a38a2a18
P(X = xi) =
X
j
pij = pi ; i = 1; 2; ;
P(Y = yj) =
X
i
pij = p j; j = 1; 2;,
a32 p
j > 0
a34a43a42a2a44a2a5 fY = y
jg a45a2a46
a11a2a30a2a31a2a47 fX = x
ig a45a2a46
a11a2a48a2a49a22a21a51a50a29a52a19a53
a48a2a49a2a54a2a55a2a56
P(X = xijY = yj) = P(X = xi; Y = yj)P(Y = y
j)
= pijp
j; i = 1; 2; ;
a57 a6a2a58a2a48a2a49a2a18a2a5 Y = y
j
a11a2a52a2a53a2a47a2a12a2a13a2a59a2a16 X a11a2a52a2a53a2a36a2a37a2a38a60a21
a61 6.1
a62a64a63a64a65a64a66a64a67a2a63a2a68
a34a70a69
a45a2a71a73a72a15a74a76a75
a11a2a48a19a49a2a18 p(0 < p < 1)a34
a63a64a68a64a66
a67a19a77a19a68a73a72a73a74a76a75a19a78a19a79
a18a81a80a82a21 X; Y a36a19a83a19a84a19a85a19a86
a62a19a79a19a87
a86a19a9
a79a81a68a88a72a73a74a89a75a81a90a19a63a81a68
a11
a79a2a91
a34a93a92 X; Y a11a2a94a2a95a2a36a2a37a2a96a2a52a2a53a2a36a2a37a22a21
a97 a98a2a99a2a100
P(X = m; Y = n) = p2qn 2; m = 1; ; n 1; n = 2; 3; ;
a101
a33
a40a2a41a2a36a2a37a2a38a2a18
P(X = m) =
1X
n=m+1
P(X = m; Y = n) = pqm 1 ; m = 1; 2; ;
P(Y = n) =
n 1X
m=1
P(X = m; Y = n) = (n 1)p2qn 2; n = 2; 3;,
1
Ch3 2
a102a2a103a2a11a2a52a2a53a2a36a2a37a2a18
P(X = mjY = n) = p
2qn 2
(n 1)p2qn 2 =
1
n 1;
P(Y = njX = m) = p
2qn 2
pqm 1 = pq
n m 1:
a5a19a25a19a104a19a105a19a28a19a11a19a30a19a31a106a21a89a32 (X; Y )
a33
a104a19a105a19a28a19a12a19a13a19a59a19a16a106a34a107a6
a33a81a108a19a109a81a110
a11 x; y
a111 a100
P(X = x) = P(Y = y) = 0;
a112a2a113a2a114a2a115 a52a2a53a2a48a2a49a2a54a2a55a19a116a19a117a2a118a19a119a2a120a22a21
a18a2a121a60a34a93a122a2a18a2a119a2a120a2a47a2a123a19a11a2a52a19a53a19a48a2a49a19a11a2a124a19a125a82a126
lim
"!0+
P(X 6 xjy " < Y 6 y + ")
= lim
"!0+
P(X 6 x; y " < Y 6 y + ")
P(y" < Y 6 y + ")
= lim
"!0+
F(x; y + ") F(x; y ")
FY (y + ") FY (y ")
=
@F(x; y)
@y
d
dyFY (y)
=
Z x
1
p(u; y)du
pY (y),
a57 a121a81a124a81a125a81a18a81a5 Y = y a47 X a11a81a52a81a53a81a36a81a37a81a127
a91
a34a129a128a131a130 P(X 6 xjY = y)
a132
FXjY (xjy)a21 a101 a33a2a133
a100
FXjY (xjy) =
Z x
1
p(u; y)
pY (y) du:
a134 a121a2a55a2a135a2a136a2a137a81a127
a91
a54a19a55a88a138a29a139a82a34
a133a141a140
a99 a57 p(x; y)
pY (y)
a18a2a52a2a53 Y = y a47 X a11a2a52a2a53
a136a2a137a2a127
a91
a34a93a128a2a130 p
XjY (xjy)
a34a93a142
pXjY (xjy) = p(x; y)p
Y (y)
:
2
Ch3 3
a61 6.1 a42a2a44a2a143a2a144a2a145
a72a29a146a19a147
a11a19a148a19a59a82a21a149a32
a109a19a62a19a150a19a151a81a146a19a147a19a77a19a152
a148a81a59a19a153a19a154a19a155a19a11
a90a15a156a17a33a2a62
a12a2a13a2a59a2a16a60a34a158a157a19a159a2a160
a91
a18 y a11a2a161
a91
a36a2a37a163a162
a108
a113a15a164a17a165 a28a2a11
a146a19a147
a34a24a35a19a160
a91
y a33
a113a15a164 a11a88a166a167a21a169a168a19a32
a108
a101 a143a81a144a19a145
a72
a12a81a13a19a170a19a171a81a11
a146a81a147
a34a169a35a19a160
a91 y a33a2a172a2a62
a12a2a13a19a59
a16 Y a11
a62
a58a64a173a60a34a175a174 Y a157a64a159 ( ; )a34a175a35
a72
a160
a91 ; a33
a50a129a176a64a177
a146a64a147
a11a2a178a2a179a2a52a2a53
a153a2a180
a151
a11a2a181a2a11a19a182
a91
a21a183a92a19a159a19a143a19a145
a72
a12a19a13a19a170a19a171a19a11
a146a81a147a19a77a19a152
a148a19a59a19a153a81a154
a90a73a156 X
a11a2a48
a49a2a136a2a137a2a127
a91
a21
a97 a50a17a184
a74
a52a2a53a2a185a2a186 X a11a2a52a2a53a2a48a2a49a2a136a2a137a2a18
pXjY (xjy) =
(
ye yx; x > 0;
x 6,
(X; Y ) a11a2a94a2a95a2a136a2a137a2a18 p(x; y) = pXjY (xjy)pY (y)a34 a101 a33 X a11a2a48a2a49a2a136a2a137a2a18
pX(x) =
Z 1
1
p(x; y)dy
=
Z 1
1
pXjY (xjy)pY (y)dy
=
Z 1
1
ye yx
( )y
1e ydy
=
( )
Z 1
1
y e( +x)udy
=
( )
( + 1)
( + x) +1
=
( + x) +1,
a142 X a11a2a136a2a137a2a127
a91
a18
pX(x) =
8
<
:
( + x) +1 ; x > 0
0; x 6 0:
a61 6.2 a32 (X; Y ) N(
1; 2; 21; 22; )
a34a93a92 p
YjX(yjx)
a21
a97a188a187a190a189a191a186a191a192 X N(
1; 21)
a34a194a193a191a195a191a52a191a53a191a48a191a49a191a54a64a55a64a119a191a120a64a196a64a197a64a185a191a186 P
YjX(yjx)
a33
a181a64a198a64a36a64a37 N(
2 +
2
1 (x 1); (1
2) 2
2)
a11a64a136a64a137a64a127
a91
a21a200a199
a133a2a33a2a201
a34a200a9a2a202a2a181
a198a2a36a2a37a2a11a2a52a2a53a2a136a2a137a2a203
a33
a181a19a198a2a36a19a37a22a21
3
Ch4 4
a204a15a205a17a206 a207a2a208a2a209a2a210a2a211a2a212a2a213a2a214a2a215
x1 a207a2a208a2a209a2a210a2a211a2a212a2a216a2a217a2a218
a108
a101
a62
a58a81a12a81a13a81a59a81a16a82a34a29a219a131a220a131a221a81a222a131a223
a77a131a62
a58
a91
a173a81a116a131a224a131a225a131a226a81a11a131a171a131a173a81a11a131a227a131a228a131a229
a230 a21a158a5a2a42a2a44a2a227a2a228a2a11
a90a19a231
a34 a113a19a232a19a232 a155a19a25a19a226a19a11a19a171a2a173a82a34a233a203a19a155a19a42a2a44
a77
a226a19a171a19a234a19a235a19a173a19a11a2a102
a103a2a11a2a48a2a49a60a21a93a18a2a121a22a34a24a219a19a220
a151a19a236a19a237a2a91a19a238a2a239
a222a22a21
a23a2a25a2a26a2a27a2a28a2a12a2a13a2a59a2a16a60a21
a240a2a241 a32a2a26a2a27a2a28a2a12a2a13a2a59a2a16 X a11a2a48a2a49a2a36a2a37a2a38a2a18
P(X = xk) = pk; k = 1; 2; ;
a39 a57
a87a2a91 X
k
xkpk =
X
k
xkP(X = xk)
a18a2a12a2a13a2a59a2a16 X a11
a91a2a238a2a239
a222a60a34a93a128a2a130 E(X)a21
a5a2a6a2a7a60a34a70a203a2a155a2a155a2a92a2a242a2a123a2a11a2a243
a91a2a244a19a108a2a245a19a246
a34a93a6a2a247a19a248 a114a19a249a2a250a19a251
a87a19a91
a113a19a252a2a253 a92
a87a2a79a2a254
a11a2a255a1a0a60a21a93a47a19a123a3a2a5a4a7a6a2a58a19a182 a115 a11a19a26a2a27a19a28a19a36a2a37a19a11
a239
a222a22a21
(1) a78a1a8
a36a2a37
a32 X a157a2a159a2a160
a91
a18 p a11
a78a1a8
a36a2a37a60a34a10a9a1a11a2a119a19a120
E(X) = 0 q + 1 p = p:
(2) a9a1a12a2a36a2a37
a32 X B(n; p)a34a93a142
P(X = k) = Cknpkqn k; k = 0; 1; ; n
a39
E(X) =
nX
k=0
kCknpkqn k
=
nX
k=1
nCn 1k 1pkqn k
= np
n 1X
k=0
Ckn 1pkqn 1 k
= np:
4
Ch4 5
a5
a151a64a236
a9a13a12a64a36a19a37a2a11
a90a2a231
a34a15a14
a77a3a16
a226a2a178a1a17a2a242
a133a2a33 n a58 a78a13a8
a36a64a37a64a11
a87
a11a2a36a19a37a22a34 a6
a7a1a18
a201a5a19a17a237
a9a1a12a2a36a2a37a2a11
a91a19a238a2a239
a222a2a199a3a20 a101 n a58
a78a1a8
a36a2a37a2a11
a91a2a238a2a239
a222a2a11
a87
a21
(3) Poisson a36a2a37
a32 X a157a2a159a2a160
a91
a18 a11 Poisson a36a2a37a60a34a93a39
E(X) =
1X
i=0
k e k
k
k! =
1X
k=1
e k
k 1
(k 1)! =
(4) a6a1a21a2a36a2a37
a32 X a157a2a159a2a160
a91
a18 p a11a1a6a1a21a2a36a2a37a60a34a93a39
E(X) =
1X
k=1
k pqk 1 = p 1(1 q)2 = 1p:
a118a2a47a2a116
a33
a104a2a105a2a28a2a12a2a13a2a59a2a16a19a11
a239
a222a22a21
a240a2a241 a32 X
a22
a100a2a136a2a137 p(x)a34a10a23a1a24
Z 1
1
jxjp(x)dx < 1a34a93a39 a57
Z 1
1
xp(x)dx
a18 X a11
a239
a222a60a34a93a128a2a130 E(X)a21
a118a2a47a2a116
a33
a6a2a58a2a182 a115 a11a2a104a2a105a19a28a2a12a19a13a19a59a2a16a19a11
a239
a222a22a21
(1) a228a1a25a2a36a2a37
a32 X a157a2a159 (a; b) a242a2a11a2a228a1a25a2a36a2a37a60a34a93a39
E(X) =
Z b
a
1
b axdx =
a + b
2,
a199
a133a2a33a2a201
a34a93a228a1a25a2a36a19a37a19a11
a239
a222a1a26a3a27
a33a5a28a19a156
a11
a72a7a8
a21
(2) a161 a91
a36a2a37
a32 X a157a2a159a2a160
a91
a18 a11a2a161
a91
a36a2a37a60a34a93a39
E(X) =
Z 1
0
x e xdx = 1,
(3) a181a2a198a2a36a2a37
5
Ch4 6
a32 X N( ; 2)a34a93a39
E(X) =
Z 1
1
x 1p2 e
(x )2
2 2 dx
= 1p2
Z 1
1
( t + )e t22 dt
=,
(4) a29a1a30
a36a2a37
a32 X ( ; )a34a93a39
E(X) =
Z 1
1
x
( )x
1ee xdx
= 1 ( )
Z 1
1
t e tdt
= ( + 1) ( )
=,
x2 a207a2a208a2a209a2a210a1a31a2a212a2a211a2a217a2a218a1a32a2a217a19a218a19a211a1a33a3a34
a240a36a35 a32 X = (X
1; X2 ; Xn)
a100a191a94a191a95a191a136a191a137 p
(x1; ; xn)
a34 Y = f(X
1; ; Xn)
a34
a39
E(Y ) =
Z
Z
Rn
f(x1; ; xn)p(x1; ; xn)dx1 dxn:
a6a2a58
a151
a197a2a11
a110a2a236
a5 a101 a34 a113a19a115 a92a5a4 Y a11a2a36a2a37a60a34a24a117a2a118a1a37a1a38 X a11a2a36a2a37
a133
a185a2a8
a92a5a4 Y a11
a239
a222a60a21a10a39 X a18a2a26a2a27a2a28a2a11
a90a2a231
a34a93a102a19a103a19a11a2a59a19a18
E(Y ) =
X
i1; ;in
f(x1i1; ; xnin)p(X1 = x1i1; ; Xn = xnin):
a61 2.1 a32 X U(0;
2)
a34a93a92 E(sin X)a21
6
Ch4 7
a97 E(sin X) = 2
Z
2
0
sin xdx = 2 a21
a61 2.2 a32 X; Y
a40a1a41
a164 a36a2a37 N(0; 1)a34a93a92 E(pX2 + Y 2)a21
a97 a50 X; Y
a40a1a41
a186a2a226a2a220a2a11a2a94a2a95a2a136a2a137
a133a19a33a3a42 a140
a136a2a137a19a11a3a43a1a44a22a34 a101
a33
E(pX2 + Y 2) =
ZZ
R2
p
x2 + y2 12 e x
2+y2
2 dxdy
=
Z 2
0
d
Z 1
0
1
2 re
r22 rdr
=
Z 1
0
p2t1
2 e tdt
= p2 (32) = p 2,
a91a2a238a2a239
a222a2a100a2a47a2a123a2a11a1a45a1a46a1a47a3a48a82a126
a91a2a238a2a239
a222a2a130a2a18
a62
a58a2a120
a147
a34
a33a3a49
a47a19a11a22a34a24a142
E(c0 + c1X1 + + cnXn) = c0 + c1E(X1) + + cnE(Xn):
a61 2.3 N a53a1a50a1a51
a72
a100 M a53
a33a2a79
a51a60a34a93a225a2a5a2a159
a72a53a52a3a54a56a55
a11a19a171 n
a79
a34a93a128
Xi =
(
1; a86 i a79
a171
a77
a181a1a51 ;
0; a86 i a79
a171
a77a2a79
a51 ;
i = 1; 2; ; n;
a57 Y =
nX
i=1
Xia34a93a142 Y a133a2a33
a171
a77
a11
a79
a51a1a58
a91
a34a158a92 E(Y )a21
a97 a5a2a6a2a7a60a34a10a59
a110a2a77 X1; ; Xn a100a2a102 a164 a11a2a36a2a37a60a34a93a174 E(X1) = MN a34 a112
E(Y ) =
nX
i=1
E(Xi) = nMN,
a6a2a7 Y a157a2a159a1a60a1a6a1a21a2a36a2a37a60a34a93a35
a239
a222a19a173a1a23a3a24 a115 a36a19a37a2a117a19a118a19a119a2a120 a252a1a61a3a62a3a63a1a64 a21
a61 2.4
a62
a58a1a65
a147a15a72a53a66
a100
a75
a242 1
a67 N a11 N a68a1a69a1a70
a34a158a8a2a100
a54a56a55
a11a1a71a19a55
a62
a68a2a62a1a68
a11a1a72a2a171a60a34a10a23a3a24a3a73a1a72
a77 r a68
a113a15a164 a11
a69a1a70
a34a93a92a3a72a2a171
a79a2a91 Sr a11 a239
a222a2a173a60a21
a97 a128 X
1; X2; ; Xr
a36a64a83a64a18a13a72
a77a2a62a1a68a1a74
a8a1a75
a75a1a76
a111a19a113 a102 a164 a11
a69a3a70
a153a2a154
a11
a79a2a91
a34a93a39 Y = X
1 + + Xr
a21
a98a2a99a2a100 X
1 = 1
a34 a112 E(X
1) = 1
a21
7
Ch4 8
X2 a157a2a159a2a160 a91
a18 p = N 1
N
a11a1a6a1a21a2a36a2a37a60a34 a112 E(X
2) =
N
N 1
a21
a164 a247a60a34 E(X
i) =
N
N 2; i = 1; 2
a21
a101
a33
a34 ES
r = E(X1 + + Xr) = N(
1
N r + 1 +
1
N )
a21
a5 N
a77a1a78
a229a2a11
a90a2a231
a34 a138a29a139
a78a3a79a19a150a1a80
a30a1a81a82a126 r = N; r = N
2
a21
r = N a90
a34 ES
N = N
1 + 1
2 + +
1
N
NlnN
a82
r = N2 a90
a34 ES
N
2
= N 1N
2 + 1
+ + 1N Nln2a21
a185a2a8a2a25a5a4 a34
a245a1a83a2a62a1a84
a11
a69a1a70
a154a2a155a19a11a2a227a19a228
a79a19a91a1a85
a64 a101
a68a2a91
a11
a62a3a84
a34a93a174
a245a3a83
a86a1a87 a11
a69a1a70
a34a93a154a19a155a2a11
a79a2a91a3a88
a18
a68a19a91
a11 lnN
a89
a34a93a199
a133a2a33a2a201
a34a91a90
a77a3a92
a123a22a34
a245a3a83a19a62
a68a1a93
a11
a69a1a70
a90a1a94a60a21
a239
a222a2a203a2a100a2a47a2a123a2a11a1a47a1a48a22a126
a23a1a24a2a12a2a13a2a59a2a16 X; Y a102a1a95
a40a1a41
a34a93a39 E(XY ) = E(X)E(Y )a21
a240a2a241 a57 EXk a18 X a11 k
a96a1a97
a34 E(X EX)k a18 X a11 k
a96a15a72a53a98a1a97
a21
a99 a101 X a11
a97
a34a93a100a2a47a2a123a2a11a1a100a3a101a22a126
a108a2a109a2a110
a11 > 1a34 a111 a100 EjXj > (EjXj) a21
a225a64a5a64a42a64a44 f(a) = E(X a)2a34 f0(a) = 2a 2EXa34 a185a64a186 f(a) a5 a = EX
a90
a171
a77a1a102
a230 a173a60a21
8
