Bài tập xác xuất thống kê phần 1

Bài 1.

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Bài 2.
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B
"&#
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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
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b1 P1=D4,Q
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1=D4,6
Bài 5
a.
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PQ

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
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y
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l:*?9*K?%%HL%d
δ
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F
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n,I=?9D*i%D.IE
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Bài 9.

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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

knkk
n
qpC
−
1W
t'
@AU
l:*TUctwTU
U
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U
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U
w
6
U

6
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yU
QUP
−
=







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

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

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

x$
U
w
6
U

6
y$y6"
yU
Q"P
$"
$"$
U
=
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

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
=== qpCxP
&x
U
w

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

6
yy6U
yU
QP
U
UU
U
=
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

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
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
=== qpCxP
@AT"ctT"
&
U
UU
U
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"

6

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


6
ywy6
yU
QwPQP
−
=




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
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



===== qpCxPxP
&
"$
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U
6&wx
"
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6
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yU
Q$PQ"P
−
=

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===== qpCxPxP
&
&
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U
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6
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6

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

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

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

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

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

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

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

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




=== qpCxP
$
w
ww
U
6w"


6

x
6
ywy6
yU
QP
−
=











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


=== qpCxP
e+*>
Bài 12:
@kzCD?9r*/D1d.N'{OD
MNzOMNz'OMNzVO
MNzcOMNbzbOfNO'fNO
'

"


"

MN|z|b

&
OMN

&
−
bzb

&
OfN

&
O}fN

&

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

O


U
'V'
w
U
 

#
MNzbO
MN

&
bzb

"
OfN

"
O'fN

&
O

&
#
'
#
$



"
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

•
B=
{
∣
X
∣
≥ 1
}
=
{
(
X ≤ 1
)
∪
(
X ≥−1
)
}

~MN•O
X ≤−1 }
¿
{X ≥1
¿
P ¿

F
X
(−1)


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

]

¿ P
[
(
−2
3
; 0
]
]

¿ F
X
(0)−F
X
(
−2
3
)=
16
27
•
D=
{
X <0
}
~MN€O
−¿
−¿
−1

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


#

#

nn
nF
x
b.
[ ]

"

==






−<
φρρ
X
$

#

"

&

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

&
{
#


#

=












=






<≤
ρρ
X

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

[ ]
( )
U
w

"

QP{
"

"

=+






−=+∞+











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

P Q
 P Q
Y
n
y
F y
y y
−
<

=

− ≥

•
{ } { }
[ ]
{ }
   " PQ "
Y Y
k P Y P P≤ ≤ ⇔ < ≤ = +

 P"Q P Q  "
k
Y Y
F F
−
= + − − = −
•
{ } { }

  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





≥−
<
−


"
"
"

khir
r
e
r
σ

k
MN
σσ
"
≤≤
R
O
N
σσ
"≤≤ R
ON•
σ
O
∪
N
σσ
"≤≤ R
O
MN
σσ
"≤≤ R
O
QP
σ

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'

e
r
""
"T#
σ
−
Q}P'
e
r
""
"T
σ
−
Q
ee
""T −−
−
MN•c
σ
&
O'MN•
σ
&≤
O'
Q&P
σ
F
R
'
e

"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
XX
x
XX
ee
x
edtedxxfdxxfdxxfdxxfxF
λλλλ
λ
−−−−
∞−∞−
−=−−=−===+==
∫∫∫ ∫∫


QPQP



@=0*/>%H?9, %IC*
( )



−
=
−



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

7=MN
( )
dkXkd +≤≤
O
( )
∫∫
+
+−−−−
+
−=
+
−==
dk
kd
dkkdxx
dk
kd
X
ee
kd
dk
edxedxxf
QP
QP
QP
QP
QP
λλλλ
λ
MNlc14O'MNl

≤
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

l
P14Q}f
l
PP1VQ4Q
MNlcP1VQ4O}f
l
PP1VQ4Q
Bài 19:
6…4i*‚%J*
P Q fx x dx
∞
−∞
=
∫
(_*10%
P Qfx x dx
∞
−∞
∫


P Qfx x dx
−∞
∫
V


P Qfx x dx
∫
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