Bài 1.

 !!"#$
% !%!%"&
!%'!
Bài 2.
()*+**, %-.-/%0%12*3%45678*9:-9;%0%
%<%0*%0%4;=>?,-6


%
4
5
@ABCDE)44;=>?FG/%H8*9,I=*%1

B
"&#
%0*/>%H%0%0%-J**K%0%L
MNBOPQR
MNB"OPQS
MNB&OP%QTU
MNB#OMNBOP4QVP5QT$VT$TU
Bài 3
6

=
&W66IAX

=
"#W66IA%Y
6Z[\C%9*,,IAB

b.
c. MN\
≤
]O'MN\c]O
Bài 07.
M0%[?9%HCD=*/d*"
%

e+*>f
y
P=Q%4K
Bài 8
Phác họa hàm phân phối của biến ngẫu nhiên trong bài 3
+ trường hợp 1 với n = 4
%d%0%+, %5d%?9DCDBJ=%0%0*/>"&#IA%0%
0%-J**,g
$


$
"

$
&

$
#

$
&

F
y
=≤=
ONQP
Nd_%GJ*8_*-JO
$

V
$
"

$
&
@
ONQP
δδ
+≤=+ YP
F
y
MNd_%G-JO
$
&
n,I=?9D*i%D.IE
$
&
*KL6e).=*KL
EMNBO
$
&
'


QPxu
V
$
"
QPxu
V
$
&
QPxu
V
$
#
QPxu
V
$
&
QPxu
V
$
"
QPxu
V
$

QPxu
+ trường hợp với n = 5
,*3,*/,p #
QPx
F

q*g%?9
F
Y
(
y
)
=
{
0n ế u y<0
y
2
nế u 0≤ y ≤1
1n ế u y>1
Bài 10.
Phác hoạ hàm phân phối của biến ngẫu nhiên Z trong ví dụ 5. Chỉ ra dạng của Z
p*/=r\%H)**s*/d)*8*/=r**?*5dt=*?9
uIA*-9
λ
v
F
X
(x)=
{
0,∧x<0
1−e
−λx
,∧x ≥0
M0%dK4K+*>
Bài 11
MP1Q







=== qpCxP
$
w
w
U
6&#&
U
w
6
U

6
yy6w
yU
QwP
−
=












=== qpCxP
&
&
&&
U
6#
U
w
6
U

6
y&y6
yU
QP
−
=










=== qpCxP
$
U
w
6
U

6
yy6&
yU
Q&P
&
&
U
=












=== qpCxP
1−e
−λx

6
U

6
ywy6
yU
QP
w
ww
U
=












=== qpCxP
&#
U
w
6
U


"

6
yy6U
yU
QUPQP
−
=












===== qpCxPxP
&
w
w
U
6"&
"

6
"

6
y$y6"
yU
Q$PQ"P
−
=












===== qpCxPxP
&
&
&&
U
6w"U
"

6
"

6

yU
Q#P
−
=












=== qpCxP
e+*>
@AxTctT
#&


6

x
6
yUy6
yU
QUP
U













=== qpCxP
#x


6

x
6
y"y6$
yU
Q$P
"$
"$"
U
=










=== qpCxP
&
##
###
U
6x#


6

x
6
y#y6#
yU
Q#P
−
=












=== qpCxP

$"
$"$
U
6"$U"


6

x
6
y$y6"
yU
Q"P
−
=












=== qpCxP
U
U
UU
U



6

x
6
yy6U
yU
QP
−
=












−
O

#
$
'
"
$
 

&
MN|z|
&
#
≥
OMN'bzb
&
#
−
OVMN
&
#
bzbO
fN
&
#
−
O'fN'OVfNO'fN
&
#



"
Bài 13:
F
X
(
x
)
=
{
1
3
+
2
3
(
x+1
)
2
n ế u−1 ≤ x ≤ 0
0 n ế u x <−1
•
A=
{
X >
1
3
}
~MN!O

1
3
•
C=
{
∣
X−
1
3
∣
<1
}
=
{
−2
3
< X <
4
3
}
~MNqO
P
[
(
−2
3
;
4
3
)

¿
=
2
3
0
¿
−F
X
¿
P
[
[
−1; 0
)
]
=F
X
¿
Bµi 14:
a. BiÕn ngÉu nhiªn x lµ BNN liªn tôc
( )





>
≤≤−
=



{N
&
 −
=−=






=






<
ρρ
X
[ ]
{ }
[ ]
#

 ==≤
ρρ
X
$

{ }
[ ]
$
&

$
&

#


#

=+=+






<≤=






≤≤
ρρρ
XX



=






>
XX
FFX
ρρρ
[ ]
 =≥X
ρ
[ ]
( )
[ ] [ ][ ]
( )
( )
#
&
{{ =−+=+=<
−
Xx
FFX
ρρρ
Bài 15:
P Q

  P  O
Y
k P k Y k P k k> ⇔ < ≤ + = +

P Q P Q
Y Y
F k F k= + −

 P Q P Q P Q
k k k k
k k k k
− − − −
= − + − − = − +
Bài 17:
•C%9D•=5%?9
QPrF
R





≥−
<
−


"
"
"

R
F
'
QP
−
σ
R
F
V
Q"P
σ
R
F
'
QP
σ
R
F

Q"P
σ
R
F
'
QP
−
σ
R
F
P'

"Tx−
Bài 18.
lCDuIA*-9
λ
*%*)0%-J*%HC
u



=
−

QP
x
X
e
xf
λ
λ
C

≥
Cb
@=*%?9u
( ) ( ) ( )
( )
xxt
x
t
x

x
X
e
xF
λ
nC
≥
nCb
@A0*/>4c1=D4,
‚MN
dX ≤
O
d
X
edF
λ
−
−=QP
MN
( )
dkXkd +≤≤
O
dkkdkddk
XX
eeeekdFdkF
QPQP
QPQPQQPP
+−−−+−
−=−−−=−+
λλλλ

≤
14O'f
X
P14Q}P'5
kd
λ
−
Q5
kd
λ
−
67;=%G4,%H,p*ƒ*3%*„1d.1%L
%+0%-J*6
MN
( )
dkXkd +≤≤
O
 4 14 P1VQ4
q0%0*/>0%-J**K%0%L1%L%+0%-J*
MNblb4Of
l
P4Q}f
l
PQ
MN4blb14Of
l
P14Q}f
l
P4Q
MN14blbP1VQ4Of


P Q fx x dx
+∞
=
∫
⇔


−∞
∫
V


P QCx x dx−
∫
V


∞
∫

⇔


P QCx x dx− ≤
∫

⇔
qP
"

 Cb
f
x
PQ&
"
}"
&
C
≤

≤

  Cc
6MN

"
b
≤
&
#
Of
x
P
&
#
Q}f
x
P

"

PQ
5  
 PV%Q5 }   
P Q5
 
c
a
c
x c x a
a
neu x a
< −



≤ <


−

− ≤ <


≥

b.kf

PQ
f


}P%'Q
"
O
c
a
PV%Q
"
'
c
a
P%'Q
"
f

PQ
P Q
a
x
a
t c dt
c
−
−
∫

a
c
−
P*'%Q
"

a
c
P'%Q
"
@=?9
f

PQ
( ) ( )
( ) ( )
" "
" "
 %  %  5 
'%  5 
c c
a a
a a
a c
c c

+ − − <




− − ≥


c6MN||bO


i)
0 ≤ F
X
(
x
∣
A
)
≤ 1
j*Dr
A
{ X ≤ x }¿
¿
P¿
F
X
(
x
∣
A
)
=¿
6qG%g
F
X
(
x
∣
A
)

P%Q
ii)
A
{X ≤ x }¿
¿
¿
P[ A]
P ¿
¿
lim
n→ ∞
¿
€d
X ≤ x
D1
x→ ∞
*k
X ≤ ∞
6er=*,,dIAl*d)1
l
A
{X ≤ x}¿
¿
¿
P[ A]
¿
A
S¿
¿
¿

A
{X ≤ a }¿
¿
A
{X ≤ b }¿
¿
P ¿
P ¿
¿
@kbD
{X ≤ a}
*%d%H
{X ≤ b}
A
{ X≤a}¿
¿
A
{ X ≤ b }¿
¿
A
{a< X <b }¿
¿
P ¿
P ¿
P ¿
¿
A
{X ≤ a} ¿
¿
A

A
)
=lim
h → 0
F
X
(
b+h
∣
A
)
=F
X
(
b
¿
A
)
€d
+¿
h →0=¿ b+h → b
¿
vi)
P
[
a< X ≤ b
]
=F
X
(

a
∣
A
)
+P
[
a< X ≤ b
]
=F
X
(
b
∣
A
)