!"#
!"#$%&'(
!"#)$%&'(*+,- /
0#$ 12*3"45*
)678$231239:2)312*3912*3:;'<
2312*=>3912*3=12>3
2?3*:>/.@'A12*>3912*312>3
( ) ( )
i i i i
i
x p pxΡ Χ = = ⇒ Ε =Χ
∑
( )
X
f x
( ) ( )
.
X
x f x dx
+∞
−∞
⇒ Ε =Χ
∫
!"#$$
)$BCDEF%GHI
!"#)$B'*
$
!"J)$
)678$23K239LM2)3K2*39
23*:>/.@'AK2*=>39K2*3=K2>3
2?3K2=*39K2*3:N;'<
!/O$
( ) ( )
( )
2
D
Χ = Ε Χ − Ε Χ
( )
( )
( )
( )
( )
( )
+∞
Χ
−∞
Χ = Ε Χ − Ε Χ Ε Χ =
Ε Χ =
∑
∫
2
2 2 2
2 2
( ) với . , nếu X rời rạc ;
. , nếu X liên tục.
i i
i
D x p
x f x dx
2
. ( )C D Χ
#
!"#$$
( ) ( )
D
σ
Χ = Χ
!"#$$
P
QRS*2"*TNU'8N83
!"#$%&'(*V
!"#)$%&'(*+,-:
,
)QWS*2-WS-X"*3
!"#$
!"J$FY*+5
( )
i i
x pΡ Χ = =
( )
X
f x
( ) ( )
⇒ Χ = =
0 0
neáu
X X
Mod x f x Max f x
( ) ( )
1/ 2, 1/ 2Med m m X mΧ = ⇔ Ρ Χ < ≤ Ρ > ≤
( )
1
( )
2
m
X X
MedX m F m f x dx
−∞
= ⇔ = =
∫
⇒ Χ = =
0 0
neáu
i i i
Mod x p Max p
QR-W
!"#?$QR-W8@P&Z
*<N'<$
9L$-R-W<
912*3$-R-W[-
?CO'< O'<48<UT2UW-G%3
\7S+$
( )
k
X a
Ε −
( )
[ ]
[ ]
cos , 0, / 2
~
0, 0, / 2
X
x x
f x
x
π
π
∈
Χ =
∉
( ) ( )
/2
0
. .cos 1
2
X
x f x dx x xdx
π
π
+∞
−∞
Ε Χ = = = −
∫ ∫
%
!"#$$
!"#$$
&
QRS*9L
QWS*9-
\7S+)$R*,4&@[@<U'8'
( )
( )
2
2
/2
2
0
cos 1 3
2
X
D X x xdx
π
π
π
Ε
= − − = −
÷
∫
1 44 2 4 43
( )
0
cos 1/ 2
sin 1/ 2 / 6
m m
X
f x dx xdx
m m
π
−∞
⇔ = =
⇔ = ⇔ =
∫ ∫
2 1 1
1 2 1 1
m m m k
m m m k
p pq pq pq pq pq
− − −
Χ − +
Ρ
QRS*9
QWS*9-
( )
1
2
1
1 1
( ) . .
1
k
k
E X kp q p
p
q
∞
−
=
= = =
−
∑
( )
2
2
2 1
1
2
3 2 2 2
1
( )
1 1 1 1
.
(1 )
k
k
D X k pq
p
q q q
p
q p p p p
+∞
−
=
Ε Χ
= −
÷
+ +
= − = − =
÷
−
∑
1 4 2 43
( )
( )
2
2 1
1 1/ 2
1 1/ 2
m
m m
p q q
p q q q
−
− −
+ + + ≤
⇔
+ + + + ≥
'
!"#$$
!"#$$
(
( )
1
1
1
1
1
. 1/ 2
1 1/ 2
1
2
1
1 1 1/ 2
. 1/ 2
2
1
ln ln 2 , 1 ln ln 2
ln 2 ln 2
1
ln ln
m
m
m
m m
m
q
p
q
q
q
q q
q
p
q
m q m q
m
q q
−
−
−
−
≤
≥
− ≤
−
⇔ ⇔ ⇔
− − ≥
≤
≥
−
⇔ ≤ − − ≥ −
− −
⇔ ≤ ≤ +
!"#$$
)
\7S+$R*,4&@[@<U'8'$
*)]^
BL:?L:L:
( )
( )
( )
( )
2
2
2 2 2
2.0, 4 5.0,3 7.0,3 4, 4
2 .0.4 5 .0,3 7 .0,3 4, 4D
Ε Χ
Ε Χ = + + =
Χ = + + −
1 4 4 44 2 4 4 4 43
( )
( ) 2,107D X
σ
Χ = =
X = 2 ; Med X = 5Mod
S_-A74`a1G
*
Qbc'<2c3$GdQRSWGe2edd3
*
F.@$QRSWGf
)L:?
]L:
^L:
H$4RPYa.@SgO
PYh&$GdG\
i i
x n
( )
( )
4, 4
2,107
x
x n
σ σ
=→ Ε Χ =
=→ Χ =
+
!"#$$
S_-A74`aQG$\RQRSW GK
*,SgOi, /0/1
.@'<O$
)ML:?Q=
]ML:Q=
^ML:Q=
PYh&$
GCIj6GX\Hk
( )
( )
4, 4
2,107
x
x n
σ σ
=→ Ε Χ =
=→ Χ =
$
!"#$$
!"#$$
\7S+?$
6_a]RUaUl[<:m8%
*n'<o-[
p
CqA712*3:K2*3
%&$% *'<o-RUaUlT
*r
p
[
p
@
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
1 2 5
1 5 1
1 2 5 1
5
5D D D D D
Χ = Χ + Χ + + Χ
Ε Χ = Ε Χ + + Ε Χ = Ε Χ
⇒ Χ = Χ + Χ + + Χ = Χ
( ) ( )
1 1
7 35
,
2 12
D⇒ Ε Χ = Χ =
1
1 2 6
1 1 1
6 6 6
X
X
P
?$ -
6V@V$
)6V@+$
\7S+?$
R
65-P @'>9'*
( )
Y
ϕ
= Χ
( ) ( )
( ) .
ii i i
i
x p E Y x p
ϕ
Ρ Χ = = ⇒ =
∑
( )
cos , 0,
2
0, 0,
2
π
π
∈
Χ =
∉
:
X
x x
f x
x
( ) ( ) ( ) ( )
.
X X
f x Y f x dxx
ϕ
+∞
−∞
Χ ⇒ Ε =
∫
:
#
!"#$$
!"#$$
( )
( )
( )
( )
( )
( )
2
/2
/2
0
0
3
/2
/2
2
0
0
2
2
2
sin 1
cos
2 2
sin 1
cos
sin
sin
3 3
1 1 1
3 4 12
x
Y xdx
x
Y x
x
x dx
D Y Y E Y
π
π
π
π
Ε = = =
Ε = = =
= Ε − = − =
∫
∫
]$ -
6V@V$
\7S+]$
)6V@+$2*:>3+,--[
p
r
p
d2U:A3
\7S+])$
( )
,Y
ϕ
Ζ = Χ
( )
( )
( )
,
,
.,
i j ij
ij
i j
i j
x
x Y y p
py
ϕ
Ρ Χ = = =
⇒ Ε Ζ =
∑
( )
,
i j ij
i j
Y x y pΕ Χ =
∑
( ) ( ) ( )
2
, . ,
Z
R
f x y dxdyx y
ϕ
⇒ Ε Ζ =
∫∫
14 2 43
( )
8 , neáu 0 1, ( 5.1)
,
0 , neáu traùi laïi.
xy x y hình
f x y
Ω
≤ ≤ ≤
=
6 447 4 48
%
!"#$$
23&
A
L*
!"#$$
&
↑
→
Ω
!"#$$
'
( ) ( )
( ) ( )
( )
( )
( )
( )
( ) ( )
2
2
2
2
2 2
2 2
1
0 0
1
0 0
. , .8
, .8
.
.
.
,
,
,.
.
y
R
y
R
R
R
f x y dxdy dy xydx
f x y dxdy dy xydx
f x y dxdy
f x y dxdy
f x y dx
x x
Y y y
Y y
X
dy
x
X Y xy
Ε = =
Ε = =
Ε =
Ε
Ε
Χ
=
=
∫∫ ∫ ∫
∫∫ ∫ ∫
∫∫
∫∫
∫∫
s$t
p
W
$12*:>39212*3:12>33
)C
p
@@'2R3$
!"#s$R2*:>391u2*f12*332>X12>33v
!"Js$R2*:>3912*>3X12*312>3
678$23*:>r
p
[
p
@5R2*:>39L
2)3R2*:*39K2*3
( )
( ) ( )
1 1 1 1
1 1 1
(3) cov , cov ,
(4) cov , cov ,
m n m n
i j i j
i j i j
m m m
i k i i k
i k i i k
Y Y
D X
= = = =
= = = ≠
Χ = Χ
÷
Χ Χ = Χ + Χ
÷
∑ ∑ ∑∑
∑ ∑ ∑ ∑
(
!"#$$
C
p
'<h
456'#,
,789:;<
=
>?
=
7#8
78
@6,
=
A;B
=
!AC!D<
=
E
"F:,G9H9:GIJ
=
E
?Q[
p
h,
( )
( ) ( )
cov ,
.
XY
Y
R
Y
σ σ
Χ
=
Χ
0
Y
R
Χ
⇒ =
1, ,
1 , , :
XY
XY
R Y
R a b c a bY c
≤ ∀Χ
= ⇔ ∃ Χ + =
XY
R
XY
R
( )
( ) ( )
( ) ( )
cov , , cov ,
,
cov , ,cov ,
Y
D Y
Y Y Y
Χ Χ Χ
Χ =
÷
÷
Χ
)
!"#$$
!"#$$
+
*
\7S+s$R4Y
,@'w4;
65-
p
'<h)4Y$
%&$
1 2 1 2
, , , ; , , ,
m n
Y Y YΧ Χ Χ
( ) ( ) ( )
1 2 3
cov , ;cov , ;cov ,
i j i j i j
p Y Y p Y pΧ Χ = = Χ =
( ) ( )
1 2 1 2
à
m n
U v V Y Y Y= Χ + Χ + + Χ = + + +
( )
( )
( ) ( ) ( )
( )
( )
( ) ( )
( ) ( )
3
1 1 1 1
1
1 1 1
2
3
1 2
cov , cov , . cov , . .
cov , cov , ( 1).
( 1).
cov ,
. .
.
1 . 1
m n m n
i i i j
i j i j
m n m
i k i i k
i k i i k
UV
U V Y Y m n p
D U X D m m m p
D V n n n p
U V
m n p
R
U V
m m m p n n n p
σ σ
= = = =
= = = ≠
= Χ = Χ =
÷
= Χ = Χ + Χ Χ = + −
÷
= + −
= =
+ − + −
∑ ∑ ∑ ∑
∑ ∑ ∑ ∑
]S_-A74`a
8/K"L,MNLOPDQ
O
;REJS,
OTO0
OTO0
OTO0
OTO0
O0LU
OV
( )
y Y=→ Ε
i i ij
x y p
( )
x X=→ Ε
( )
x n X
σ σ
=→
( )
y n Y
σ σ
=→
XY
r R=→
( )
xy XY=→ Ε
∑
#$
!"#$$
!"#$$
#
43xRQG$QeK1k1%xIF
U,SgOi$GCIj6xkGx9
.@SgO$
PYh&$
GCIj6Gf\Hk
GCIj6Gf\Hk
GCIj6Gf\Hk
GCIj6Gf\Hk
GCIj6Gf\Hk
GCIj6GfGyQ
( )
x X=→ Ε
( )
x n X
σ σ
=→
, ;
i j ij
x y p M +
( )
=→ Ε>y Y
( )
σ σ
=→>y n Y
=→>>
XY
r R
( )
=→ Ε
∑
> xy XY
TWX'#,U"SAY9:IDS ? "QAA,
:
&
$ $9 $9#
# $9 $9%
#
!"#$$
##
)
!"#$$
#
z&N4&'$
$ $9
$ & $9#
# $9
# & $9%
i
x
j
y
ij
p
3Z DSA>"[\9I,
!"#$$
#%
( )
1, 4x X=→ Ε =
( )
0,9165x n X
σ σ
=→ =
( )
4, 2y Y=→ Ε =
( )
0,9798y n Y
σ σ
=→ =
0,0891
XY
r R=→ = −
( )
5,8xy XY=→ Ε =
∑