Ch2 1
x4a0a2a1a2a3a2a4
a5a2a6a8a7 X(!)
a9a2a10a2a11a2a12a14a13 ( ;F; P)a15a2a16a2a17a2a18a2a19a2a20a2a21a23a22a25a24a27a26a23a28a23a29a23a30a32a31a33a17
a34a2a35
a21
a35a2a36a38a37
F(x) = P(X 6 x) = P(f!jX(!) 6 xg); 1 < x < 1
a39 X(!)
a16a2a10a2a11a2a40a2a41a2a29a2a30a43a42
a44a2a45a2a46a14a47
a24a48a40a2a41a2a29a2a30a23a49a2a50a23a51a23a52a2a53a23a54a25a55
(1) F(x)a56a2a57a2a58a2a59a61a60
(2) lim
x! 1
F(x) = 0; lim
x!+1
F(x) = 1;
(3) F(x)a62a2a63a2a64a43a24a48a65a2a66a2a67a2a68a23a69a2a70a25a60
(4)a28 F(x ) 6= F(x)a16a2a71 xa39a2a72a2a73 a71a43a24a48a26
a72a2a73
a71a23a74a2a50a23a75a23a30a2a18a61a42
(5)a76a2a77a2a78a2a19a2a20a2a21a2a22a2a16a2a40a2a41a2a29a23a30a23a9a2a79a23a80a2a29a23a30a61a42
a5a2a6a82a81a2a83
a19a2a20a2a21a2a22 Xa24a48a84a2a85a2a69a2a70a2a75a2a86a2a29a2a30 p(x)a87a2a88
F(x) = P(X 6 x)
Z x
1
p(t)dt
a26a2a28 Xa39 a63a2a64a2a78a2a19a2a20a2a21a2a22a43a24a90a89a23a91a2a16a61a24 p(x)a28
a39 X
a16a2a10a2a11a2a92a2a93a2a29a2a30a43a42
a81a2a83
a63a2a64a2a78a2a19a2a20a2a21a2a22 Xa24a48a94a2a95a2a96a2a97a2a17a2a18
a36 a
a16a2a10a2a11
a39 0
a42
a52a2a98a2a99
a47a101a100
a18a2a102a2a103a2a16a2a63a2a64a23a78a2a19a23a20a23a21a2a22a23a16a2a10a23a11a23a92a2a93a23a29a23a30a61a42
(1)a104a2a105a2a40a2a41
a28a2a10a2a11a2a92a2a93
a39
p(x) =
(; a106 a 6 x 6 b;
0; a107a2a108 ;
a16a2a19a2a20a2a21a2a22 Xa39a14a109 a13 [a; b]a15a2a16a2a104a2a105a2a40a2a41a43a24a48a110a23a111
X U[a; b]:
a44a2a45a2a46a14a47
a24 = 1b aa42a48a104a2a105a2a40a2a41a2a112a2a9a2a17a2a113a23a16
a100
a97a23a10a2a78a61a42
(2)a114a2a30a2a40a2a41
1
Ch2 2
a28a2a10a2a11a2a92a2a93
a39
p(x) =
(
e x; x > 0 ( > 0);
0; x < 0;
a16a2a19a2a20a2a21a2a22 Xa39a2a115a2a116a2a117 a30
a39
a16a2a114a2a30a2a40a2a41a43a42
a44a2a45a2a118a2a119
a24a48a114a2a30a2a40a2a41a23a50a2a84a23a52a23a53a2a54a25a55
P(X > a + bjX > a) = P(X > b):
a120
a112a2a9a2a121a61a24a27a122
a100
a97a23a40a23a41a123a17a23a124a25a24a125a114a23a30a23a40a23a41a23a49a123a50a23a126a23a110a23a127a23a53a25a42a125a128a130a129a131a16a23a9a25a24a125a114a23a30a23a40a23a41
a9a2a63a2a64a2a78a2a16a2a19a2a20a2a21a2a22a61a42
(3)a132a2a133a2a40a2a41
a84a2a85a2a19a2a20a2a21a2a22 Xa16a2a10a2a11a2a92a2a93a2a29a2a30
a39
p(x) = 1p2 e (x )
2
2 2 ( 1 < x < 1);
a107a14a134
a117
a30 a75a2a51
a39
a96a2a97
a35
a30a25a24a27a135
a117
a30 > 0a24a136a26a2a28 X a115a2a116a2a117 a30
a39 ;
a16a2a132
a133a2a40a2a41a43a24a48a110a2a111 X N( ; )a42
a132a2a133a2a40a2a41a2a16a14a137a101a138a2a84a2a52a61a55
O x
y
a44a2a45a2a46a14a47
a24a139a137a101a138a2a9a2a140
a83a23a141a23a142 x = a81
a28a2a16a43a42a90a143 P(X 6 a) = P(X >
+ a)a42
a44a2a45a2a119a14a144
a24a48a145 X N( ; )a24 Y N(0; 1)a24a48a26
P(a < X < b) = P(b < Y < a ):
a83
a9a2a70a2a146a2a147a2a16a2a148a2a149a43a24a150a74a23a151a23a152a2a153a23a154 N(0; 1)a16a2a40a2a41a2a155a2a75a43a42 N(0; 1)a16a2a40a2a41a2a29a2a30a2a110
a39 (x)
a42
2
Ch2 3
a132a2a133a2a40a2a41a2a9a2a156a2a146a14a134a101a91a2a103a2a157a2a158a2a16a2a40a23a41a61a24a48a159a2a158a23a160a2a22a23a161a2a162a61a24a48a163a23a164a2a114a23a165a2a166a23a9
a115a23a116
a132a2a133a2a40a2a41a43a42a48a167a2a168a2a169
a116a23a170a2a171
a15
a119a130a144a173a172a2a174
a17a2a71a61a42a175a49a23a176a2a177a23a178a2a70a23a167a23a168a2a169a23a179a2a180
a171
a42
(4)a181a2a182a2a40a2a41
a84a2a85a2a19a2a20a2a21a2a22 Xa16a2a10a2a11a2a92a2a93
a39
p(x) =
8<
:
( )x
1e x; x > 0 ( > 0; > 0);
0; x 6 0;
a26a2a28 Xa115a2a116a2a117 a30
a39 ( ; )
a16a2a181a2a182a2a40a2a41a43a24a48a110a2a111 X ( ; )a42
a181a2a182a2a40a2a41a2a9a2a17a123a18a23a159a23a183a123a16a23a40a23a41a123a184a25a24a185a114a23a30a23a40a123a41a23a9a23a94a123a16a23a17a23a18a123a186a23a184 ( = 1)a42a188a187
a189
a24a48a70a2a156a2a146a2a190a14a134a101a102a23a191a23a16
2
a40a2a41a2a184
a120
a9a2a94a2a16a2a17a2a18a2a186a2a184
= n
2; =
1
2
a42
x5a192a2a193a2a194a2a195
a3a2a4a2a196a2a197a2a198a2a199a23a0a2a1
a174
a179a2a152a2a180
a171
a16a14a200a101a201a2a9a43a24a139a202a173a153 X(!)a16a2a10a2a11a2a40a2a41a43a24a48a203
Y (!) = f(X(!))
a16a2a10a2a11a2a40a2a41a43a24a48a107a14a134a204a24 f( )a9a14a202a101a153a2a16a2a17
a34a2a35a2a36a2a35
a21a23a29a23a30a61a42
a70
a174
a179a2a74a2a180
a171 a100
a18a2a205a2a186a43a24a175a206a23a17a2a207a23a16a2a180
a171
a70a2a52a23a17a23a169a2a17a23a208a2a180
a171
a42
a209 5.1
a202a101a153 X N( ; )a24a48a203 Y = aX + ba16a2a40a2a41a2a29a2a30 (a > 0)a42
a210
FY (y) = P(Y 6 y) = P(aX + b 6 y)
= P(X 6 y ba )
= FX(y ba ):
a83
a9
F0Y (y) = F0X(y ba ) 1a
= 1p2 a exp (y a b)
2
2a2 2
,
a120
a112a2a9a2a121a61a24 Y N(a + b; a2 2)a24a175a155a2a132a2a133a2a40a2a41a2a211a2a212
a142
a53a23a21a23a213a23a214a23a215a23a216a23a9a23a132a2a133a23a40
a41a43a42a48a217a2a218a2a16a43a24a48a95 Y = X a24a48a50 Y N(0; 1)a42
3
Ch2 4
a209 5.2 X N(0; 1)
a24a48a203 Y = X2a16a2a40a2a41a43a42
a210
a106 y 6 0a148a43a24a48a219a2a220a2a50 FY (y) = 0a42a48a106 y > 0a148a43a24
FY (y) = P(Y 6 y) = P(X2 6 y)
= P( py 6 X 6 py)
= 1p2
Z py
py
e x22 dx
= 1p2
Z y
0
x 12 e x2 dx
a75a2a153a2a107a2a92a2a93a2a29a2a30
a39
pY (y) =
(
y 12 e y2 ; y > 0;
0; y 6 0;
a120
a112a2a9a2a121a43a24 Y 12; 12 a42
a209 5.3 a7 X U(0; 1)
a24a48a203 Y = 1(X)a16a2a40a2a41a43a42
a210
FY (y) = P(Y 6 y) = P( 1(X) 6 y)
= P(X 6 (y)) = (y):
a174a2a221a14a144 Y N(0; 1)
a42
4
x4a0a2a1a2a3a2a4
a5a2a6a8a7 X(!)
a9a2a10a2a11a2a12a14a13 ( ;F; P)a15a2a16a2a17a2a18a2a19a2a20a2a21a23a22a25a24a27a26a23a28a23a29a23a30a32a31a33a17
a34a2a35
a21
a35a2a36a38a37
F(x) = P(X 6 x) = P(f!jX(!) 6 xg); 1 < x < 1
a39 X(!)
a16a2a10a2a11a2a40a2a41a2a29a2a30a43a42
a44a2a45a2a46a14a47
a24a48a40a2a41a2a29a2a30a23a49a2a50a23a51a23a52a2a53a23a54a25a55
(1) F(x)a56a2a57a2a58a2a59a61a60
(2) lim
x! 1
F(x) = 0; lim
x!+1
F(x) = 1;
(3) F(x)a62a2a63a2a64a43a24a48a65a2a66a2a67a2a68a23a69a2a70a25a60
(4)a28 F(x ) 6= F(x)a16a2a71 xa39a2a72a2a73 a71a43a24a48a26
a72a2a73
a71a23a74a2a50a23a75a23a30a2a18a61a42
(5)a76a2a77a2a78a2a19a2a20a2a21a2a22a2a16a2a40a2a41a2a29a23a30a23a9a2a79a23a80a2a29a23a30a61a42
a5a2a6a82a81a2a83
a19a2a20a2a21a2a22 Xa24a48a84a2a85a2a69a2a70a2a75a2a86a2a29a2a30 p(x)a87a2a88
F(x) = P(X 6 x)
Z x
1
p(t)dt
a26a2a28 Xa39 a63a2a64a2a78a2a19a2a20a2a21a2a22a43a24a90a89a23a91a2a16a61a24 p(x)a28
a39 X
a16a2a10a2a11a2a92a2a93a2a29a2a30a43a42
a81a2a83
a63a2a64a2a78a2a19a2a20a2a21a2a22 Xa24a48a94a2a95a2a96a2a97a2a17a2a18
a36 a
a16a2a10a2a11
a39 0
a42
a52a2a98a2a99
a47a101a100
a18a2a102a2a103a2a16a2a63a2a64a23a78a2a19a23a20a23a21a2a22a23a16a2a10a23a11a23a92a2a93a23a29a23a30a61a42
(1)a104a2a105a2a40a2a41
a28a2a10a2a11a2a92a2a93
a39
p(x) =
(; a106 a 6 x 6 b;
0; a107a2a108 ;
a16a2a19a2a20a2a21a2a22 Xa39a14a109 a13 [a; b]a15a2a16a2a104a2a105a2a40a2a41a43a24a48a110a23a111
X U[a; b]:
a44a2a45a2a46a14a47
a24 = 1b aa42a48a104a2a105a2a40a2a41a2a112a2a9a2a17a2a113a23a16
a100
a97a23a10a2a78a61a42
(2)a114a2a30a2a40a2a41
1
Ch2 2
a28a2a10a2a11a2a92a2a93
a39
p(x) =
(
e x; x > 0 ( > 0);
0; x < 0;
a16a2a19a2a20a2a21a2a22 Xa39a2a115a2a116a2a117 a30
a39
a16a2a114a2a30a2a40a2a41a43a42
a44a2a45a2a118a2a119
a24a48a114a2a30a2a40a2a41a23a50a2a84a23a52a23a53a2a54a25a55
P(X > a + bjX > a) = P(X > b):
a120
a112a2a9a2a121a61a24a27a122
a100
a97a23a40a23a41a123a17a23a124a25a24a125a114a23a30a23a40a23a41a23a49a123a50a23a126a23a110a23a127a23a53a25a42a125a128a130a129a131a16a23a9a25a24a125a114a23a30a23a40a23a41
a9a2a63a2a64a2a78a2a16a2a19a2a20a2a21a2a22a61a42
(3)a132a2a133a2a40a2a41
a84a2a85a2a19a2a20a2a21a2a22 Xa16a2a10a2a11a2a92a2a93a2a29a2a30
a39
p(x) = 1p2 e (x )
2
2 2 ( 1 < x < 1);
a107a14a134
a117
a30 a75a2a51
a39
a96a2a97
a35
a30a25a24a27a135
a117
a30 > 0a24a136a26a2a28 X a115a2a116a2a117 a30
a39 ;
a16a2a132
a133a2a40a2a41a43a24a48a110a2a111 X N( ; )a42
a132a2a133a2a40a2a41a2a16a14a137a101a138a2a84a2a52a61a55
O x
y
a44a2a45a2a46a14a47
a24a139a137a101a138a2a9a2a140
a83a23a141a23a142 x = a81
a28a2a16a43a42a90a143 P(X 6 a) = P(X >
+ a)a42
a44a2a45a2a119a14a144
a24a48a145 X N( ; )a24 Y N(0; 1)a24a48a26
P(a < X < b) = P(b < Y < a ):
a83
a9a2a70a2a146a2a147a2a16a2a148a2a149a43a24a150a74a23a151a23a152a2a153a23a154 N(0; 1)a16a2a40a2a41a2a155a2a75a43a42 N(0; 1)a16a2a40a2a41a2a29a2a30a2a110
a39 (x)
a42
2
Ch2 3
a132a2a133a2a40a2a41a2a9a2a156a2a146a14a134a101a91a2a103a2a157a2a158a2a16a2a40a23a41a61a24a48a159a2a158a23a160a2a22a23a161a2a162a61a24a48a163a23a164a2a114a23a165a2a166a23a9
a115a23a116
a132a2a133a2a40a2a41a43a42a48a167a2a168a2a169
a116a23a170a2a171
a15
a119a130a144a173a172a2a174
a17a2a71a61a42a175a49a23a176a2a177a23a178a2a70a23a167a23a168a2a169a23a179a2a180
a171
a42
(4)a181a2a182a2a40a2a41
a84a2a85a2a19a2a20a2a21a2a22 Xa16a2a10a2a11a2a92a2a93
a39
p(x) =
8<
:
( )x
1e x; x > 0 ( > 0; > 0);
0; x 6 0;
a26a2a28 Xa115a2a116a2a117 a30
a39 ( ; )
a16a2a181a2a182a2a40a2a41a43a24a48a110a2a111 X ( ; )a42
a181a2a182a2a40a2a41a2a9a2a17a123a18a23a159a23a183a123a16a23a40a23a41a123a184a25a24a185a114a23a30a23a40a123a41a23a9a23a94a123a16a23a17a23a18a123a186a23a184 ( = 1)a42a188a187
a189
a24a48a70a2a156a2a146a2a190a14a134a101a102a23a191a23a16
2
a40a2a41a2a184
a120
a9a2a94a2a16a2a17a2a18a2a186a2a184
= n
2; =
1
2
a42
x5a192a2a193a2a194a2a195
a3a2a4a2a196a2a197a2a198a2a199a23a0a2a1
a174
a179a2a152a2a180
a171
a16a14a200a101a201a2a9a43a24a139a202a173a153 X(!)a16a2a10a2a11a2a40a2a41a43a24a48a203
Y (!) = f(X(!))
a16a2a10a2a11a2a40a2a41a43a24a48a107a14a134a204a24 f( )a9a14a202a101a153a2a16a2a17
a34a2a35a2a36a2a35
a21a23a29a23a30a61a42
a70
a174
a179a2a74a2a180
a171 a100
a18a2a205a2a186a43a24a175a206a23a17a2a207a23a16a2a180
a171
a70a2a52a23a17a23a169a2a17a23a208a2a180
a171
a42
a209 5.1
a202a101a153 X N( ; )a24a48a203 Y = aX + ba16a2a40a2a41a2a29a2a30 (a > 0)a42
a210
FY (y) = P(Y 6 y) = P(aX + b 6 y)
= P(X 6 y ba )
= FX(y ba ):
a83
a9
F0Y (y) = F0X(y ba ) 1a
= 1p2 a exp (y a b)
2
2a2 2
,
a120
a112a2a9a2a121a61a24 Y N(a + b; a2 2)a24a175a155a2a132a2a133a2a40a2a41a2a211a2a212
a142
a53a23a21a23a213a23a214a23a215a23a216a23a9a23a132a2a133a23a40
a41a43a42a48a217a2a218a2a16a43a24a48a95 Y = X a24a48a50 Y N(0; 1)a42
3
Ch2 4
a209 5.2 X N(0; 1)
a24a48a203 Y = X2a16a2a40a2a41a43a42
a210
a106 y 6 0a148a43a24a48a219a2a220a2a50 FY (y) = 0a42a48a106 y > 0a148a43a24
FY (y) = P(Y 6 y) = P(X2 6 y)
= P( py 6 X 6 py)
= 1p2
Z py
py
e x22 dx
= 1p2
Z y
0
x 12 e x2 dx
a75a2a153a2a107a2a92a2a93a2a29a2a30
a39
pY (y) =
(
y 12 e y2 ; y > 0;
0; y 6 0;
a120
a112a2a9a2a121a43a24 Y 12; 12 a42
a209 5.3 a7 X U(0; 1)
a24a48a203 Y = 1(X)a16a2a40a2a41a43a42
a210
FY (y) = P(Y 6 y) = P( 1(X) 6 y)
= P(X 6 (y)) = (y):
a174a2a221a14a144 Y N(0; 1)
a42
4
