1
x5 a0
a1a3a2a3a4a3a5a3a6a3a7a3a8a3a9a3a10a12a11a12a13a12a14a16a15a18a17a12a19a12a20a12a21a12a22a3a23a12a24a12a25a12a14a16a15a18a26a12a27a12a13a12a28a12a29a12a30a3a31
p1; p2; ; pna15a32a1a33a2 p1; p2; ; pn a13a33a34a33a35a36a15a32a37a33a38a33a39a33a40a42a41a33a43a42a44a42a45a47a46
(1) a1a33a2 p1; p2; ; pn a13a33a48a33a49a33a34a33a35a51a50
(2) a52a33a53a33a54a33a55a33a56
a40a36a15a32a1a33a2 n a13a33a48a33a49a33a34a33a35a51a50
(3) H(p1; p2; p3) = H(p1; p2 + p3) + (o2 + p3)H( p2p
2 + p3; p3p
2 + p3
)a57
a52a33a58a33a59
a44a33a45a33a40a36a15a32a60a33a39a42a61a33a62
H(p1; ; pn) = c
nX
i=1
pilnpi; c > 0
a63 f(x) = lnx a2a33a64a16a65a66a34a33a35a33a67 H(p
1; ; pn) > cln
nX
i=1
p2i,
a1a33a68a33a69a33a39a33a40a33a11a33a70a51a46
(1) H(p1; ; pn) > 0a15a32a71 H(p1; ; pn) = 0 () P(X = c) = 1a57
(2) a53a33a54a33a72
a45a33a13a33a1a33a73a33a74
a57
(3) a75a33a76a33a77 ; a78a33a79
a15a32a80 H( ; ) = H( ) + H( )
a57
a17a33a48a33a49a33a22a33a23a33a24a33a25a33a14a36a15a32a10a42a81 H( ) =
Z 1
1
p(x)lnp(x)dxa57
a17a33a48a33a49a33a22a33a23a33a24a33a25a33a14a33a13a33a1a36a15a82a69a42a40a33a83a42a13a33a84a42a85a47a46
(1) a75 p(x) a86a16a87a66a52 [a; b] a64a36a15a32a39a33a88a33a89a33a28a33a29a33a13a33a1a42a73a42a74 a57
(2) a75 p(x) a86a16a87a66a52a33a90a33a91a33a92a33a93
a64a36a15a95a94a42a96a33a31 a15a32a39a33a97a33a98a33a28a33a29a33a13a33a1a33a73a42a74
a57
(3) a75 p(x) a86a16a87a66a52 [0;1) a64a36a15a32a88a33a99a33a31 a15a32a39a33a100a33a35a33a28a33a29a33a13a33a1a33a73a33a74 a57
1
2
a101a33a102a33a103 a104a33a105a33a106a33a107a33a108a33a109a33a105a33a106a33a110a33a111a33a112a42a113a33a114
x1 a104a33a105a33a106
a113a33a115 a116 X a31a33a117a33a118a33a119a33a99a33a23a33a24a33a25a33a14a36a15a95a27a42a13
a54a42a120
a28a42a29a42a30a33a31
P(X = k) = pk; k = 0; 1; 2; ;
a80a33a27a33a13a33a121a33a34a33a35a33a31
g(z) =
1X
k=0
pkzk:
a63a66a19 p
k > 0
a15a32a71
1X
k=0
pk = 1a15a32a122 g(z) a13a33a123a33a124a33a125a33a126 R > 1a57
a40a33a83a33a127a16a128a66a41a33a43a33a129a33a130a33a28a33a29a33a13a42a121a33a34a42a35
a57
(1) a131a33a132
a28a33a29
a116 X B(n; p)a15a32a80a33a27a33a13a33a121a33a34a33a35a33a31
G(z) =
1X
k=0
Cknpkqn kzk = (pz + q)n:
(2) Poisson a28a33a29
a116 X P( )a15a32a80a33a27a33a13a33a121a33a34a33a35a33a31
g(z) =
1X
k=0
k
k! e
zk = e (z 1):
(3) a41a33a133a33a28a33a29
a116 X
a134a33a135a33a136
a35a33a31 p a13a33a41a33a133a33a28a33a29a36a15a32a80a33a27a33a13a42a121a42a34a33a35a42a31
g(z) =
1X
k=0
pqk 1zk = pz1 qz:
a121a33a34a33a35a33a60a33a39a33a137a33a138a33a139a33a140a33a141a33a13a42a142a33a143a51a15a95a144a42a145a42a23a42a24a33a25a42a14a42a34a33a35a42a13a33a140a42a141a42a146a33a143a42a69
g(z) =
1X
k=0
pkzk = E(zX):
2
3
a121a33a34a33a35a33a69a33a39a33a40a33a11a33a70a51a46
(1) a121a33a34a33a35a33a147 a90
a9a33a10a33a148 X a13
a54a33a120
a28a33a29a36a15 p
k =
g(k)(0)
k! a57
(2) EX = g0(1); EX2 = g00(1) + g0(1)a57
(3) X a149 Y a150a33a151a33a78a33a79
a15a32a80
gX+Y (z) = gX(z)gY (z):
(4) a116 X1; X2; ; Xn a78a33a79a16a152
a28a33a29a36a15 Y
a149 X1; X2; a78a33a79
a15
Z =
YX
i=1
Xia15a32a80 Z a13a33a121a33a34a33a35a33a31
gZ(s) = gY (gX(s)):
(1)(2) a153 g(z) a154a33a155
a31 Taylor
a156
a35a33a157a33a60a33a67
a57
a40a33a83a42a158a42a159a16a160 (4)
a57
a63a18a121a42a34a42a35a42a13a42a140a42a141a3a137a3a138a3a142a3a143a161a15a162a39a3a163 X
1; X2; a150a42a151a42a78a42a79
a71a42a164a42a165 Y
a78a42a79
a15
a69
gZ(s) = EsZ
= Esx1+ +XY
= E[sX1sX2 sXY ]
= E[EsX1 EsXY ]
= EgX(s)Y
= gY (gX(s)):
(3) a166a33a167a33a52 (4) a87a66a168 P(Y = 2) = 1 a157a33a60 a57
3