?=c
M
# s?
B a?1
= ?
B aB?
M
# s?
1,
M
￥s?f
2,s?f
￥?é
3, ? ??
M
# s?f
4,èn ? ??
M
# s?
p
(1).
?s? (2),=[s?
(3),{
s? (4),+s?
5, ???
M
# s?f
6,èn ???
M
# s?
á
(1), (
s? ; (2),?
s? ; (3),·
s?
= a=?
M
# s?
1,=?
M
￥?l
2,=?
M
￥s?f
3,=? ? ??
M
# s?
p
4,=? ???
M
￥s?
á
5,H?s?
6,
M
￥? ??
7,
M
e?f
￥s?
1),B?
M
f
￥s?
2),=?
M
f
￥s?
? a?:k￥
T
(1) s?f
￥Vr
T )()( xXPxF ≤=
(2)9
T,
? ??
)()(
i
xx
xXPxF
i
==
∑
≤
???
dxxpxF )()(
∫
+∞
∞?
=
1
(3) ? X~ 5),(
2
σμN
σ
μ?
=
X
Y
jN(0,1)
(4) èn 7?
M
￥s?
ps?
á
(5) ?
s?à
q￥9
T
? X~ 5 ),(
2
σμN
)()()(
σ
μ
σ
μ?
Φ?
Φ=<≤
ab
bXaP
= a?? è5
1 a
!
M
X ￥s?f
1
>
≤≤
<
=
.1,1;10,
0,0
)(
2
x
xAx
x
xF
p
1 ￥′ A
2 X
)
2
1
,1(? # )2,
3
1
(
=￥à
q
3 X ￥à
q
áf
b
[3 ]
1?s?f
￥· ??? 1=x ?)μ
1)01()1( =+== FAF ' 1=A
2?s?f
￥?é?
4
1
)1()
2
1
())
2
1
,1(( ==?∈ FFXP
9
8
3
1
1)
3
1
()2())2,
3
1
((
2
=
=?=∈ FFXP
3 X￥à
q
áf
1
≤≤
><
==
.10,2;10,0
)(
)(
xx
xx
dx
xdF
xp
2 a
! O),2(~
2
σNX 3.0)42( =<< XP p )0( <XP
[3 ] y1
Φ?
Φ=<<=
σσ
2224
)42(3.0 XP
)0(
2
Φ
Φ=
σ
[
8.05.03.0
2
=+=
Φ
σ
2
?
^
2.0
2
1
2202
)0( =
Φ?=
Φ=
<
=<
σσσσ
X
PXP
3 aB
? a
? ] ?s-= B?
? ] ?s-B a
? ?×
?ê X
B?
??× b )5,55(~
2
1
NX )5,45(~
2
2
NX
1 V? ? |B? p ×
v? 50 Xà
q
2 V? ? |
? p
ì￥×
?l? 50 X￥à
q b
[3 ]
1
!
Yq V
U o ? |B? ×
v? 50 X p A
Yq V
U o ? |B?1a
? p
1
B
Yq V
U o ? |B?1B?
? p
2
B
5o O
21
,BB?=
21
BB ∪
3
1
)(,
3
2
)(
21
== BPBP
8413.0
5
5550
1)50()(
1
1
=
Φ?=>= XPBAP
1587.0
5
4550
1)50()(
2
2
=?
Φ?=>= XPBAP
? ?à
q
T¤
)()()()()(
2
2
1
1
BAPBPBAPBPAP +=
6138.01587.0
3
1
8413.0
3
2
=×+×=
2V? ? | 2 ?
??× v? 50 X￥à
q 6138.0=p l? 50 X￥ à
q
6138.011?=?= pq
! ? | 2?μ Y?v? 50 X5
),2(~ pBY
?
^
pà
q1
1492.0)6138.01()0(
2200
2
=?=== qpCYP
4 aJ aY a d |é?? ?
,
|￥
?
qGQ1 . , . , . ,
!
|
BQ,
k
p |
?9
￥à
qs? b
}3 ~ ¨ 9V
U |
?9
, 5 9￥ |′1 , , , .
¨ "V
U oJ
? p, #V
U oY
? p, $V
U o
? p, 5
1 9=
= 1 ? "? #? $
= . × . × . = . ,
1 9=
= 1 "? #? $
+ 1 ? " #? $
+ 1 ? "? # $
= . × . × . + . × . × . + . × . × . = . ,
1 9=
= 1 " #? $
+ 1 "? # $
+ 1 ? " # $
3
= . × . × . + . × . × . + . × . × . = . ,
1 9=
= 1 " # $
= . × . × . = . .
9
Q L . . . .
5 a
!
á
￥
l ?Z?1,V?
á
?
1° 5 qá
, ? Q
l? ??
1,5?
á
YV
l ?,?5?í YV, ?
á
￥Q
q1 0.05,
k p?
á
YV
l ?￥à
q,
}3 ~ ¨ XV
U 5qá
?￥Q
,5 X~B(5,0.05),??
á
YV
l ?￥à
q1
P(X≤1)=P(X=0)+P(X=1)
4150
)95.0()05.0(
1
5
)95.0()05.0(
0
5
××
+××
=
=0.9774,
6 a
z
k μ 10?ê 45,
5
μ A,B,C,D
1?s?ê4, ?o μB?s?
^?
?￥,
!
|
?5 (
1ê4s?,
k p?
3 10 ?5? zs 6 ?5￥à
q
^
$
}3 ~ ¨ XV
U 10?5 ?s￥5
"
,5 )
4
1
,10(~ BX,?
^?
3 10? 5? zs 6
?5￥à
q
^
4646
)
4
3
()
4
1
(
6
10
)
4
1
1()
4
1
(
6
10
)6( ××
=?××
==XP,
7 a
!
M
X μs?f
,
≥
<≤
<
=
1,1
10,
0,0
)(
3
x
xx
x
xF
k p,P(X≤?3),
)
2
1
( ≤XP
,)
2
1
3
1
( ≤< XP,
)
3
2
|
2
1
( ≤> XXP
,
}3 ~ P(X≤?3)=F(?3)=0,
8
1
)
2
1
()
2
1
()
2
1
(
3
===≤ FXP,
216
19
)
3
1
()
2
1
()
3
1
()
2
1
()
2
1
3
1
(
33
=?=?=≤< FFXP
,
4
)
3
2
(
)
3
2
2
1
(
)
3
2
(
)
3
2
,
2
1
(
)
3
2
|
2
1
(
≤
≤<
=
≤
≤>
=≤>
XP
XP
XP
XXP
XXP
64
37
)
3
2
(
)
2
1
()
3
2
(
)
3
2
(
)
2
1
()
3
2
(
3
33
=
=
=
F
FF
,
8 a
!
?
o￥×
X([ X9 )?V?
s?,X ~N(50,25),
(1). pV?
?? ? |Bo, ×
?@ 45 X￥à
q ;
(2).V?
?? ? |Bo, ×
o? 40 X? 60 X-W ￥à
q ; (3). ?V?
?? ?
|?o,
k p μ 2o??@ 45 X￥à
q ;
(4).V?
?? ? |Bo ×
?V 60 X￥à
q ;
}3 ~ (1),)1()
5
5045
()45(?=
=< ΦΦXP,
1587.08413.01)1(1 =?=?= Φ
(2),)
5
5040
()
5
5060
()6040(
=<< ΦΦXP
1)2(2)2()2(?== ΦΦΦ
=2×0.9772?1=0.9544,
(3).
! Y1 5o??×
?@ 45 X￥?
,
5 Y~B(5,0.1587),#
pà
q1
32
)413.0()1587.0(
2
5
)2(
==YP,
(4),)
5
5040
()60(1)60(
=≤?=> ΦXPXP,
0228.09772.91)2(1 =?=?= Φ
9 a
!
M
X μà
qs?
p,
X?3?2?1 0 1 2 3 4 5
p
k
0.08 0.02 0.03 0.17 0.15 0.05 0.20 0.16 0.14
k p Y=X
2
￥à
qs?
p,
}3 ~ Y￥ |′1 0,1,2,3,4,5, à
qs?
p1
P(Y=0)=P(X=0)=0.17,
P(Y=1)=P(X=?1)+P(X=1)=0.03+0.15=0.18,
5
P(Y=4)=P(X=?2)+P(X=2)=0.02+0.05=0.07,
P(Y=9)=P(X=?3)+P(X=3)=0.08+0.20=0.28,
P(Y=16)=P(X=4)=0.16,
P(Y=25)=P(X=5)=0.14,
'
Y 0 1 4 9 16 25
p
k
0.17 0.18 0.07 0.28 0.16 0.14
10 a
! ￥ ó
áf
1 ),( YX
≥≥
=
+?
e,0;0,0,
),(
)(
yxAe
yxf
yx
p
1"
A
2 ós?f
),( yxF
3 )1( >XP
4 )),(( DYXP ∈ ? }1,0,0:),{( ≤+≥≥= yxyxyxD
5 )(),( yfxf
YX
[3 ]
1? '
∫∫
+∞
∞?
+∞
∞?
== 1),( dxdyyxf
∫∫
+∞ +∞
+?
=?=
00
)(
11 AdxdyAe
yx
2?
H 0,0 ≥≥ yx
)1)(1(
),(),(
00
)( yx
xy
yx
xy
eedxdye
dxdyyxfyxF
+?
∞?∞?
==
=
∫∫
∫∫
? yx,1
e′
H 0),( =yxF
[
>>
=
e,0;0,0),1)(1(
),(
yxee
yxF
yx
3 )1(1)1(1)1(
X
FXPXP?=≤?=>
3679.0),1(1
1
==+∞?=
eF
6
4 {}
∫∫
=∈
D
dxdyyxfDYXP ),(),
∫∫
=?==
1
0
1
0
1
2642.021]][[ edyee
ydyx
(5)
<
≥
==
∫
∫
+∞
+?
∞+
∞?
0,0;0,
),()(
0
)(
x
xdye
dyyxfxf
yx
X
<
≥
=
.0,0;0,
x
xe
x
] ?
<
≥
=
0,0;0,
)(
y
ye
yf
y
Y
11 a
!=? ? ??
M
9, :
μà
qs?
p
9 :
p 9￥H?s?
p : ￥H?s?
p ;
}3 ~
9 : 1 9= J
1 := K
9
Q L
:
Q L
a
! X ￥à
q
á1
7
≤
>
=
.0,0;0,
)(
2
x
xxe
xf
x
x
λ
λ
Y
?V (
s? ),0( X
p
Y￥Hqà
q
á)( xyf
XY
￥ óà
q
á ),( YX
Y￥à
q
á b
<3 >
?X? Y
?V (
s?#
p￥),0( X Y￥Hqà
q
á
<<
=
e,0;0,
1
)(
xy
xxyf
XY
?Hq
áD ó
áf
-W￥1"á
ìμ
)()(),( xfxyfyxf
XXY
=
<<
=
e,0;0,
2
xye
xλ
λ
∫
+∞
∞?
= dxyxfyf
Y
),()(
?
H 0≤y 0),()( ==
∫
+∞
∞?
dxyxfyf
Y
?
H 0>y
y
y
x
Y
edxeyf
λλ
λλ
+∞
==
∫
2
)(
' Y￥à
q
áf
1
≤
>
=
.0,0;0,
)(
y
ye
yf
y
Y
λ
λ
8
M
# s?
B a?1
= ?
B aB?
M
# s?
1,
M
￥s?f
2,s?f
￥?é
3, ? ??
M
# s?f
4,èn ? ??
M
# s?
p
(1).
?s? (2),=[s?
(3),{
s? (4),+s?
5, ???
M
# s?f
6,èn ???
M
# s?
á
(1), (
s? ; (2),?
s? ; (3),·
s?
= a=?
M
# s?
1,=?
M
￥?l
2,=?
M
￥s?f
3,=? ? ??
M
# s?
p
4,=? ???
M
￥s?
á
5,H?s?
6,
M
￥? ??
7,
M
e?f
￥s?
1),B?
M
f
￥s?
2),=?
M
f
￥s?
? a?:k￥
T
(1) s?f
￥Vr
T )()( xXPxF ≤=
(2)9
T,
? ??
)()(
i
xx
xXPxF
i
==
∑
≤
???
dxxpxF )()(
∫
+∞
∞?
=
1
(3) ? X~ 5),(
2
σμN
σ
μ?
=
X
Y
jN(0,1)
(4) èn 7?
M
￥s?
ps?
á
(5) ?
s?à
q￥9
T
? X~ 5 ),(
2
σμN
)()()(
σ
μ
σ
μ?
Φ?
Φ=<≤
ab
bXaP
= a?? è5
1 a
!
M
X ￥s?f
1
>
≤≤
<
=
.1,1;10,
0,0
)(
2
x
xAx
x
xF
p
1 ￥′ A
2 X
)
2
1
,1(? # )2,
3
1
(
=￥à
q
3 X ￥à
q
áf
b
[3 ]
1?s?f
￥· ??? 1=x ?)μ
1)01()1( =+== FAF ' 1=A
2?s?f
￥?é?
4
1
)1()
2
1
())
2
1
,1(( ==?∈ FFXP
9
8
3
1
1)
3
1
()2())2,
3
1
((
2
=
=?=∈ FFXP
3 X￥à
q
áf
1
≤≤
><
==
.10,2;10,0
)(
)(
xx
xx
dx
xdF
xp
2 a
! O),2(~
2
σNX 3.0)42( =<< XP p )0( <XP
[3 ] y1
Φ?
Φ=<<=
σσ
2224
)42(3.0 XP
)0(
2
Φ
Φ=
σ
[
8.05.03.0
2
=+=
Φ
σ
2
?
^
2.0
2
1
2202
)0( =
Φ?=
Φ=
<
=<
σσσσ
X
PXP
3 aB
? a
? ] ?s-= B?
? ] ?s-B a
? ?×
?ê X
B?
??× b )5,55(~
2
1
NX )5,45(~
2
2
NX
1 V? ? |B? p ×
v? 50 Xà
q
2 V? ? |
? p
ì￥×
?l? 50 X￥à
q b
[3 ]
1
!
Yq V
U o ? |B? ×
v? 50 X p A
Yq V
U o ? |B?1a
? p
1
B
Yq V
U o ? |B?1B?
? p
2
B
5o O
21
,BB?=
21
BB ∪
3
1
)(,
3
2
)(
21
== BPBP
8413.0
5
5550
1)50()(
1
1
=
Φ?=>= XPBAP
1587.0
5
4550
1)50()(
2
2
=?
Φ?=>= XPBAP
? ?à
q
T¤
)()()()()(
2
2
1
1
BAPBPBAPBPAP +=
6138.01587.0
3
1
8413.0
3
2
=×+×=
2V? ? | 2 ?
??× v? 50 X￥à
q 6138.0=p l? 50 X￥ à
q
6138.011?=?= pq
! ? | 2?μ Y?v? 50 X5
),2(~ pBY
?
^
pà
q1
1492.0)6138.01()0(
2200
2
=?=== qpCYP
4 aJ aY a d |é?? ?
,
|￥
?
qGQ1 . , . , . ,
!
|
BQ,
k
p |
?9
￥à
qs? b
}3 ~ ¨ 9V
U |
?9
, 5 9￥ |′1 , , , .
¨ "V
U oJ
? p, #V
U oY
? p, $V
U o
? p, 5
1 9=
= 1 ? "? #? $
= . × . × . = . ,
1 9=
= 1 "? #? $
+ 1 ? " #? $
+ 1 ? "? # $
= . × . × . + . × . × . + . × . × . = . ,
1 9=
= 1 " #? $
+ 1 "? # $
+ 1 ? " # $
3
= . × . × . + . × . × . + . × . × . = . ,
1 9=
= 1 " # $
= . × . × . = . .
9
Q L . . . .
5 a
!
á
￥
l ?Z?1,V?
á
?
1° 5 qá
, ? Q
l? ??
1,5?
á
YV
l ?,?5?í YV, ?
á
￥Q
q1 0.05,
k p?
á
YV
l ?￥à
q,
}3 ~ ¨ XV
U 5qá
?￥Q
,5 X~B(5,0.05),??
á
YV
l ?￥à
q1
P(X≤1)=P(X=0)+P(X=1)
4150
)95.0()05.0(
1
5
)95.0()05.0(
0
5
××
+××
=
=0.9774,
6 a
z
k μ 10?ê 45,
5
μ A,B,C,D
1?s?ê4, ?o μB?s?
^?
?￥,
!
|
?5 (
1ê4s?,
k p?
3 10 ?5? zs 6 ?5￥à
q
^
$
}3 ~ ¨ XV
U 10?5 ?s￥5
"
,5 )
4
1
,10(~ BX,?
^?
3 10? 5? zs 6
?5￥à
q
^
4646
)
4
3
()
4
1
(
6
10
)
4
1
1()
4
1
(
6
10
)6( ××
=?××
==XP,
7 a
!
M
X μs?f
,
≥
<≤
<
=
1,1
10,
0,0
)(
3
x
xx
x
xF
k p,P(X≤?3),
)
2
1
( ≤XP
,)
2
1
3
1
( ≤< XP,
)
3
2
|
2
1
( ≤> XXP
,
}3 ~ P(X≤?3)=F(?3)=0,
8
1
)
2
1
()
2
1
()
2
1
(
3
===≤ FXP,
216
19
)
3
1
()
2
1
()
3
1
()
2
1
()
2
1
3
1
(
33
=?=?=≤< FFXP
,
4
)
3
2
(
)
3
2
2
1
(
)
3
2
(
)
3
2
,
2
1
(
)
3
2
|
2
1
(
≤
≤<
=
≤
≤>
=≤>
XP
XP
XP
XXP
XXP
64
37
)
3
2
(
)
2
1
()
3
2
(
)
3
2
(
)
2
1
()
3
2
(
3
33
=
=
=
F
FF
,
8 a
!
?
o￥×
X([ X9 )?V?
s?,X ~N(50,25),
(1). pV?
?? ? |Bo, ×
?@ 45 X￥à
q ;
(2).V?
?? ? |Bo, ×
o? 40 X? 60 X-W ￥à
q ; (3). ?V?
?? ?
|?o,
k p μ 2o??@ 45 X￥à
q ;
(4).V?
?? ? |Bo ×
?V 60 X￥à
q ;
}3 ~ (1),)1()
5
5045
()45(?=
=< ΦΦXP,
1587.08413.01)1(1 =?=?= Φ
(2),)
5
5040
()
5
5060
()6040(
=<< ΦΦXP
1)2(2)2()2(?== ΦΦΦ
=2×0.9772?1=0.9544,
(3).
! Y1 5o??×
?@ 45 X￥?
,
5 Y~B(5,0.1587),#
pà
q1
32
)413.0()1587.0(
2
5
)2(
==YP,
(4),)
5
5040
()60(1)60(
=≤?=> ΦXPXP,
0228.09772.91)2(1 =?=?= Φ
9 a
!
M
X μà
qs?
p,
X?3?2?1 0 1 2 3 4 5
p
k
0.08 0.02 0.03 0.17 0.15 0.05 0.20 0.16 0.14
k p Y=X
2
￥à
qs?
p,
}3 ~ Y￥ |′1 0,1,2,3,4,5, à
qs?
p1
P(Y=0)=P(X=0)=0.17,
P(Y=1)=P(X=?1)+P(X=1)=0.03+0.15=0.18,
5
P(Y=4)=P(X=?2)+P(X=2)=0.02+0.05=0.07,
P(Y=9)=P(X=?3)+P(X=3)=0.08+0.20=0.28,
P(Y=16)=P(X=4)=0.16,
P(Y=25)=P(X=5)=0.14,
'
Y 0 1 4 9 16 25
p
k
0.17 0.18 0.07 0.28 0.16 0.14
10 a
! ￥ ó
áf
1 ),( YX
≥≥
=
+?
e,0;0,0,
),(
)(
yxAe
yxf
yx
p
1"
A
2 ós?f
),( yxF
3 )1( >XP
4 )),(( DYXP ∈ ? }1,0,0:),{( ≤+≥≥= yxyxyxD
5 )(),( yfxf
YX
[3 ]
1? '
∫∫
+∞
∞?
+∞
∞?
== 1),( dxdyyxf
∫∫
+∞ +∞
+?
=?=
00
)(
11 AdxdyAe
yx
2?
H 0,0 ≥≥ yx
)1)(1(
),(),(
00
)( yx
xy
yx
xy
eedxdye
dxdyyxfyxF
+?
∞?∞?
==
=
∫∫
∫∫
? yx,1
e′
H 0),( =yxF
[
>>
=
e,0;0,0),1)(1(
),(
yxee
yxF
yx
3 )1(1)1(1)1(
X
FXPXP?=≤?=>
3679.0),1(1
1
==+∞?=
eF
6
4 {}
∫∫
=∈
D
dxdyyxfDYXP ),(),
∫∫
=?==
1
0
1
0
1
2642.021]][[ edyee
ydyx
(5)
<
≥
==
∫
∫
+∞
+?
∞+
∞?
0,0;0,
),()(
0
)(
x
xdye
dyyxfxf
yx
X
<
≥
=
.0,0;0,
x
xe
x
] ?
<
≥
=
0,0;0,
)(
y
ye
yf
y
Y
11 a
!=? ? ??
M
9, :
μà
qs?
p
9 :
p 9￥H?s?
p : ￥H?s?
p ;
}3 ~
9 : 1 9= J
1 := K
9
Q L
:
Q L
a
! X ￥à
q
á1
7
≤
>
=
.0,0;0,
)(
2
x
xxe
xf
x
x
λ
λ
Y
?V (
s? ),0( X
p
Y￥Hqà
q
á)( xyf
XY
￥ óà
q
á ),( YX
Y￥à
q
á b
<3 >
?X? Y
?V (
s?#
p￥),0( X Y￥Hqà
q
á
<<
=
e,0;0,
1
)(
xy
xxyf
XY
?Hq
áD ó
áf
-W￥1"á
ìμ
)()(),( xfxyfyxf
XXY
=
<<
=
e,0;0,
2
xye
xλ
λ
∫
+∞
∞?
= dxyxfyf
Y
),()(
?
H 0≤y 0),()( ==
∫
+∞
∞?
dxyxfyf
Y
?
H 0>y
y
y
x
Y
edxeyf
λλ
λλ
+∞
==
∫
2
)(
' Y￥à
q
áf
1
≤
>
=
.0,0;0,
)(
y
ye
yf
y
Y
λ
λ
8
