最新-浙大概率论与数理统计课件概率论1-PPT文档资料_图文



4/21/2019

1


2


1.1 1.2 1.3 1.4 1.5 1.6


2.1 2.2 2.3 2.4 2.5



3.1

3.2

3.3

3

3.4




4.1 4.2 4.3 4.4




5.1 5.2


6.1 6.2

4




7.1 7.2 7.3




8.1 8.2 8.3 8.4 8.5 8.6 8.7


9.1 9.2 9.3 9.4
5




10.1 10.2 10.3


11.1 11.2 11.3


12.1 12.2 12.3 12.4

6


7



8

1








----



----

----

9



1. 2. 3.






10

2

()

EE S={e}




Se
S={}
S={0,1,2,...}

xy S={(x,y)|T0yxT1}
x S={ x|axb }

11

()
SAEA AA 89S{0,1,2,...}
A{10}{10,11,12,...} S AA
SS S

12

()



S

1 A B A B

B A

2

AB



AB BA


A={}B={} BA

A={10}B={5} BA

A={}B={}
BA
13



AB AB

S

AB

A B { x |x A x B } A B

AB AB,AB,AB

S

A B { x |x A x B } A B A B

n
AiA1, A2,An
i1
n
AiA1, A2 ,An
i1

AB= AB

S

AB

14

A B A B { x |x A x B }

S AB

A A , A A A A S , A A B B S , A ,B
S

""""

AA

n

n

Ai Ai A1 A2

n

n

An Ai AiA1A2 An

i1

i1

i1

i1

A={ }B={ }

A B {}

A B {}

A BAB {}

A BAB{}

15

3

()



fn (A)

nA n

n A --A()n-- f n ( A ) An



""n
n""1n

17"fn(A ) 15 1"7 88 % 15 A={}
fn ( A )
# A
16


1

2
18

**

1 0fn(A)1

2 fn(S)1

k

k

3 A1,A2,... ,Ak fn( Ai) fn(Ai)

i1

i1

f n ( A ) np

19

()

1f n ( A )pAP(A)=p
2
1 0P(A)1

2 P(S) 1

k

k

3 A 1 ,A 2 , ... , A k P ( A i)P (A i)

i 1

i 1

P(A)A
20


1 P(A)1P(A)

P(A)0A P(A)1AS

A AS P(A)P(A)1 P()0
2 A B P ( B A ) P ( B ) P ( A ) P ( B ) P ( A )

BA AB P (B )P (A )P (A B )

P ( B ) P ( A ) P ( A B ) P ( B A ) 0P(B)P(A)

3 P ( A B ) P ( A ) P ( B ) P ( A B )

AB A(B A B ) P ( A B ) P ( A ) P ( B A B ) B A B 2 P ( B A B ) P ( B ) P ( A B )

P ( A B ) P ( A ) P ( B ) P ( A B )

#3

n

n

P( Ai) P(Ai) P(AiAj)

i1

i1

1ijn



P(AiAjAk)(1)n1P(A1A2An)

1ijkn

21

4
E
1. S() 2. ()
PA A S
()
22

181813 48 A={ }P(A)



S={1,2,...,8} A={1,2,3}



P

A



3 8

23

2 A={}P(A)
P(A)C 3 1C 5 1/C 8 21 25 853.6%

3ND n
Ak{k}P(Ak) P (A k ) C D k C N n k D /C N n ,k 0 ,1 , ,n

L>mL<0

C

L m



0



24

4nN(nN)


A{ n }P(A)

......n





12

N

12

N





12

......



N

12

N

n2N2nN

NnA



C

n N

n!

P(A)CN nn!/Nn

n64N365 P (A ) 1 C N nn !/N n 0 .9 9 7
64 99.7%
25

55 2
55
77 A={ 2 }

PAC77555!3.7%
26

6: ()ababn



n

A k { k }k1,2,...,n P ( A k ) 1

n ... n

---- a n1,2,...,n

, , ,, 12 k n

......
n

... n

P(Ak)a((aabb)!1)!aa b

----------k

27

2


, ,,, 12 k n

C

a n

C

a 1 n 1



A k P(Ak)Cna 11/Cnaaa b

3



k

k



ab



S{ ...n }A
4 P(Ak)anaab

k



{ ...a

} a
0



k



S{,}A k {} P(Ak)12 28

71212 ?
12 212/712 =0.000 000 3.
" "()
29

5

90%95%

A={} B={}

P(A)=90% P(B)=85.5%

P(B|A)=95%

1. P(A)=0.90 1A 2. P(B|A)=0.95 1B 3. P(B|A)P(A)P(A|S)
SP(A)

P(B|A)=xP(A):P(AB)=1:x x P ( A B )
P(A)

A

S B



P(B| A) P(AB)
P(A)

P(A) 0

P(B|A)

P(B|A)1P(B|A)

P ( B C |A ) P ( B |A ) P ( C |A ) P ( B C |A )

B C P (B |A ) P (C |A )


P ( A B ) P ( A ) P ( B |A ) P ( B ) P ( A |B )
P ( A B C ) P ( A ) P ( B |A ) P ( C |A B ) P ( A 1 A 2 A n ) P ( A 1 ) P ( A 2 | A 1 ) P ( A 3 | A 1 A 2 ) P ( A n | A 1 A n 1 )
31

70%

30%80%

20%

A={}

AB A B

B={}

P(B)=0.3P(A|B)=0.2 P(A| B) 0

P(A)P(ABAB)P(AB)P(AB)



P (B )P (A |B ) P (B )P (A |B ) 0 .3 0 .2 0 .7 0 6 %
A B ,AA B ,
P (A )P (A B )P (B )P (AB )0.30.26% 32

360% 80% 90%



P ( A2 | A1)

Ai={ i }i=1,2,3

1 P ( A2 | A1) 1 0 .8 0 .2

A={ } AA 1 A 1A 2 A 1A 2A 3

P (A ) P (A 1 ) P (A 1 A 2 ) P (A 1 A 2 A 3 )

P ( A 1 ) P ( A 1 ) P ( A 2 |A 1 ) P ( A 1 ) P ( A 2 |A 1 ) P ( A 3 |A 1 A 2 ) 0 . 6 0 0 . 4 0 . 8 0 . 4 0 . 2 0 . 9 0 . 9 9 2

P ( A ) 1 P ( A ) 1 P ( A 1 A 2 A 3 ) 1 P ( A 1 ) P ( A 2 |A 1 ) P ( A 3 |A 1 A 2 )
1 0 .4 0 .2 0 .1 0 .9 9 2
33

522(1)2

""


Ai={i}i=1,2 B={2}

A1A2 A1A2


P(B)P(A 1A 2 A 1A 2) P(A1A2)P(A1A2)

P (A 1 )P (A 2 |A 1 ) P (A 1 )P (A 2 |A 1 )



1 2

P (B )1 21 21 21 2C 2 1(1 2)1(1 2)11 2
P (B ) 5 2 2 6 2 5 6 1 5 2 2 6 2 5 6 1 C 2 1 6 C 2 1 6/C 5 2 2 2 5 6 1
34

Bayes
SEB1,B2,...,Bn E
(i) B 1 B 2 B n S
( ii)B iB j ,i j,i,j 1 ,2 ,,n
B1,B2,...,BnS,

B1

S B1,B2,...,Bn



B2

Bn

35

ESAEB1,B2,...,BnS P(Bi)>0i=1,2,...,n

P(A) n P(Bj)P(A|Bj)

j1

A

B


i

A

B

j

B1


S

i j

A

A A S A B 1 A B 2 A B n

n

n

B2

Bn P(A) P(ABj) P(Bj)P(A| Bj)

j1

j1

P(Bi

|

A)



P(Bi A) P(A)



P(Bi | A)

P(Bi)P(A| Bi)
n

P(Bj)P(A| Bj)

Bayes

j1

36

*
P(Bj)=pj, P(A|Bj)=qj, j=1,2,...,n n
pj1
j1

P1 B1 q1

S

P. 2

B2
.

q2

A

..

..

qn

Pn Bn

PA n PBjPA|Bj

j1

37

80% 20% 90%(1) (2) A={}B={}
ABAB P ( A ) 0 . 8 0 , P ( B |A ) 0 . 2 0 , P ( B |A ) 0 . 9 0
1P (B )P (A BA B )P(AB)P(AB)
P ( A ) P ( B | A ) P (A )P ( B | A )
0 .8 0 .2 0 .2 0 .9 3 4 %
2 P (A |B )P (A B ) P (A B ) 1 68
P (B ) P (A B ) P (A B ) 3 41 7
38

5%

5%A={} C={}

P (A |C )5 % ,P (A |C )5 % , P(C)=0.005

P(C|A)

P(C| A) P(AC) P(A)

P(C)P(C)=0.8 P(C|A)=0.987




P (C )P (A |C )

0.087

P (C )P (A |C )P (C )P (A |C )

100 8.7

39

6

10822

,1Ai={i}i=1,2


P(A2|

A1)7 9P(A2)180

P(A2| A1)180P(A2) A1A2

A2A1

AB P(A )0,P(B)0 P(B|A)=P(B) P(AB)=P(A)*P(B) P(A|B)=P(A)AB
40

A,B A,B A,B A,B
PABPAPB
PABPAABPAPABPA1PBPAPB
A1,A2, ,Ann2kn,
k
PAA i1 i2 Aik PAij j1
A1,A2, ,An

1
2
41

0.80.7
A={},B={} C={}
C A B P ( C ) P ( A ) P ( B ) P ( A B )
AB P ( C ) 0 .7 0 .8 0 .5 6 0 .9 4
42

4() p

A i i ,i1 ,2,3,4 A
A A 1A 2A 3 A 4
A 1 ,A 2 ,A 3 ,A 4
P ( A ) P ( A 1 ) P ( A 2 A 3 A 4 ) p ( p 2 p p 3 )
P ( A ) P ( A 1 A 2 A 3A 1 A 4 ) p 3 p 2 p 5

2

3

1

4

43

p,p1 2,


A i i P A i p ,i 1 , 2 ,, 5
A
1

P A P A 1 A 2 A 1 A 2 A 3 A 1 A 2 A 3 p 2 2 p 2 1 p p 1
2

P A PA 1A 2A 3 A 4 A 5

p3C 3 11pp3C 4 21p2p3p 2

P 2P 13 P 2P 1 22P 1



p2



p1,

p



1 2

p2



p1,

p



1 2

44



1. S e

A S

2. A B;A B

A B;A B;A

3.

fn A

nA n

















0 P





AB

A 1; P S 1 PA B



P

A

P

B

1 P A 1 P A

2 A B P A P B

3 P A B =P A P B P AB

4.

P B |A

P AB P A



P AB

P AP B |A

B1, B 2 , , B n S

P ( A )

n
P (B j)P ( A | B j), P (Bi | A )

P (Bi)P (A | Bi)
n

j1

P (B j)P (A | B j)
j1

45

5.

1

1."AA=", 2. "ABAB=,AB"

3 . A B ,A B A B A B A B , " A ,B "
" A ,B A B A B " " A ,B A B "

4. A={}B={} AB={1}P(A)=0.7,P(B)=0.8, "P(AB)=0.7+0.8=1.5"
5.

6 . 1 0 , 1 9

A , A , S,SA ,A ,

S , A , P A 1 2

46

7.

8.ABP(AB)=P(B|A)
9 . A B ,P A 0 , P B |A P B P B |A P B |A 1 P B |A
10.AB nA1,A2,...,An
11.ABP(A)0,P(B)0,ABAB

12.ABP(A)=a,P(B)=b,

(1) AB,P(AB)

(2) AB, P(AB)

47

13.A1,A2,...,An
14.A,B,CABP(A)0, P(B)0 P(C|A)+P(C|B)
15.A,B,C,P(C)0,
P(AB|C)=P(A|C)+P(B|C)P(AB|C)

48



49

1

*
----...

----......

*

s e

x X=f(e)SX

* X



*



50

2

()

X x1
P p1

x2 ...

xi

...

p2 ...

pi

...


pi 0, pi 1 i1

S{ X=x1X=x2...X=xn... }





1P(S)P(Xxi)pi

i1

i1

# 1

2
51

3 p0<p<1X X


Ai={i}P(Ai)=pi=1,2,3 A1,A2,A3

P (X0)P (A 1)pP (X 1 ) P (A 1 A 2 ) (1 p )p
P (X 2 ) P (A 1 A 2 A 3 ) ( 1 p ) 2 p

P (X 3 ) P (A 1 A 2 A 3 ) ( 1 p )3

X0,X1,X2

X0

1

2

3

X3 S

p

p p(1-p) (1-p)2p (1-p)3

52

p0<p<1 X X
Ai={i}i=1,2,... A1,A2,...
P ( X k ) P ( A 1 A 2 A k 1 A k ) ( 1 p ) k 1 p , k 1 , 2 ,
Xp
53



01(p)

X

0

p

q



1 p

(p+q=1)


* nE A , A p(A)=p,0<p<1,En n





54


1. n
P 12

2.nA={1}


A,A,

PA 1 6

3.52nA={}


A , A , PA 1 2



55

AnX
P ( X k ) C n k p k ( 1 p ) n k k 0 1 n
Xp X b(np)
n
1(p q )n C n kp kq n k q 1 p k 0
Ai={ iA }n=3
P (X 0 ) P (A 1 A 2 A 3 ) ( 1 p )3
P ( X 1 ) P ( A 1 A 2 A 3A 1 A 2 A 3A 1 A 2 A 3 ) C 3 1 p 1 ( 1 p ) 3 1
P ( X 2 ) P ( A 1 A 2 A 3 A 1 A 2 A 3 A 1 A 2 A 3 ) C 3 2 p 2 ( 1 p ) 3 2
P (X 3 ) P (A 1 A 2A 3 )p 3
P ( X k ) C n k p k ( 1 p ) n k ,k 0 , 1 ,2 ,,n
56

80
0.01
420 380
57

X " 20 "
Aii1,2,3,4 " i 20
" 80

P A 1 A 2 A 3 A 4 P A 1 P X 2
Xb20,0.01,

1

1

PX21PXk1 C 2 k00.01k0.9920k0.0169

k0

k0

P A 1 A 2 A 3 A 4 0 .0 1 6 9

Y 80

,Yb80,0.01,

80

3

P Y 4 1 C 8 k 00 .0 1 k0 .9 9 8 0 k 0 .0 0 8 7 k 0

58

3 p0<p<1 Y (1)Y (2)2
Y b(3, p)
1 P ( Y k ) C 3 k p k ( 1 p ) 3 k ,k 0 , 1 ,2 ,3
2 P (Y 2 ) C 3 2p 2 (1 p )
59

np 0<p<1X(1) X (2)
n X b(n,p)
1 P ( X k ) C n k p k ( 1 p ) n k k 0 , 1 , ,n 2 P ( X 1 ) 1 P ( X 0 ) 1 ( 1 p ) n
limP(X1)1 n
pp0,n "" ""
60

10 255 p

L(p)



XY2

Xb(10,p)Yb(5,p)

P(A |1 X 2)

{X=i}{Y=j}A={} P(Y 0 |1 X 2)

P(Y 0)
L(P ) P (X0 )P (A |X0 ) P ( 1 X 2 ) P ( A |1 X 2 )

P (X 2 )P (A |X 2 ) L(P)=P(A) ( 1 p ) 1 0 [ 1 0 p ( 1 p ) 9 4 5 p 2 ( 1 p ) 8 ] ( 1 p ) 5

61

(Poisson) X
P (Xk)e k k0 ,1 ,2 ,,0
k!
X X ~()

X () 4.5
(1) (2) P(Xk)e4.54.5k k0,1,2,
k!
1 P ( X 2 ) 1 P ( X 0 ) P ( X 1 ) 1 e 4 . 5 ( 1 4 . 5 ) 0 . 9 3 8 9

2 P (X2|X2 )P (X2 ) 0 .1 1 9 8

P (X2 )

62


n10, p0.1,
Cnkpk1pnk ek k , np
63

3
X , x , P ( X x ) x
X , x , F (x ) P (X x ) X
F(x) X

x
F(x)

1) 0F(x)1 2 ) F ( x ) F ( ) 0 F ( ) 1

0 P ( x 1 X x 2 ) F ( x 2 ) F ( x 1 )

64



X p

0 q

1 p

X F x P X 1



0 x0
F(x)PXxq 0x1
1 x1
P(X1)p

F x
1
q

0

1

x

P ( X 1 ) p x 1 F ( x ) 1

65

4

X F ( x ) , f ( x ) , x ,



x

F(x) f (t)dt

X

f ( x ) X

66

f (x) 1 1) f (x)0

y f (x)

+
2) f(x)dx1
3) x1 x2(x2x1)

Px1Xx2

Px1Xx2

x2f(t)dt P(Xa)0
x1

4 ) f( x ) x F '( x ) f( x )

f(x)

x1 x2

f( x ) F '( x ) l i m F ( x x ) F ( x ) l i m P ( x X x x )

x 0 x

x 0

x



P ( x X x x ) f( x )x

f( x ) X x 67

c 0 x 1
X f (x) 2 9 3 x 6
0

(1)c

(2) X

(3) P(X k) 2k



3

1

1


f (t)dt

c

12 dt

6
dt



2



c



c1



0

93

3

3

2 F (x)P Xx

0

x0





x1 dt
03 11 dt 03 11 dt

x 2 dt

0 3

39

1

0 x1 1 x3 3 x6
x6

0





x 1

3 3



(

2

x



3)

/

9

1

3 P (X k)2 F (k) k 4 .5

3

x0 0 x1 1 x3 3 x6 x6
68





1

X f (x) ba

x (a, b)

0

X(a,b) XU(a,b)

acclb

0

c l 1

l

P (cXcl)

dt

---- c

c ba ba

F

(

x

)





x b



a a

1

f x

F x

1

xa a xb xb

1 ba

0

a

b

x

0a

b

x
69

(-1,2)XX P(X 0) 1010 0

X(-1,2) f (x) 13, 1 x 2
0,
P(X 0) 2 , 10Y0
3


Y b(10, 2 ) 3

P(Y2)C120232138

70



X

ex
f (x)

x0

0 x 0

>0X
X EP()

1ex x0

F(x)

0

x0

X

P (Xt0t|Xt0)

P( X t0 t) P( X t0 )

1F(t0t) et P(X t)

1F(t0)

71

tNt
t PoissonT
1 T 2 10
8
1 P N t k e tt k /k ! ,k 0 , 1 ,2 ,
F T t P T t 1 P T t
t0 F Tt0
t 0 F T t 1 P N t 0 1 e t
2 P T 1 8 |T 1 0 P P T T 1 1 0 8 e 8 P T 8
72


X , 2 f(x)X21e(x22)2 x
, 2 (Gauss)
X N(,2)



f (x)dx 1

tx

+



f (x)dx

1

t2



e 2dt



2



I

t2
e 2 dt



1

t2
e 2 dt

2

I2

(x 2 y2 )

2

r2



e 2 dxdy d re 2 dr I 2 f(x)dx1

0

0



73

X ~ N ( , 2 )

1 f (x) x

2

fmax f ( )

1 2

3 lim f (x) 0 x

( )

(

) f x

5

5

f x
0.798

0.399 0.266

0.5
1.0 1.5

0



1

x

0



x

74

X
X

75

Z ~ N ( 0 1 ) Z

Z x

1

x2
e2

Z (x)

2
x1

t2
e2dt

2

y ( x)

y

(x) ( x)

xx1

x 0

x

x

X~N(,2) b P(aXb)

P 1(ae X (x2 2)2b d)x(b )(a )

a 2

x t


b

P(aXb)

a



1

t2
e 2 dt

2

76

X ~N(,2)



P(X)P(X) (1)(1)2(1)10.6826

P (X 2 ) 2 (2 ) 1 0 .9 5 4 4
P (X 3) 2(3 ) 1 0 .9 9 7 4

3 2

68.26%



2 3

95.44%

99.74%

77

() X(cm)~N(,2)
(1) =100 =297.8cm (2) =100 90%(97,103)

(1)P(X97.8)(97.8100) 1(1.1)
2

===10.86430.1357

( 2 ) P 9 7 X 1 0 3 9 0 %

(1 0 3 1 0 0 ) (9 7 1 0 0 ) 2 ( 3 ) 1 9 0 %

( 3 ) 0.95 3 1.645





1.8237
78

X(cm )N(169.7,4.12) (1) 175cm(2) 5, 175cm 175cm

(1) P(X 175) 1(175169.7) 1(1.293)
4.1

10.90150.0985
( 2 ) 5 Y 1 7 5 c m Y b (5 ,p ), p 0 .0 9 8 5
P ( Y 1 ) 1 P ( Y 0 ) 1 ( 1 p ) 5 0 .4 0 4 5
P (Y 1 ) C 5 1 p 1 (1 p )4 0 .3 2 5 3
79

5

X

Y=g(X)Y
X X N(,2)

Y

X

X -1 0 1 pi 0.2 0.5 0.3

Y=X2Y

Y0,1

P(Y 0) P(X0)0.5

P(Y 1) P (X1 ) (X 1 ) P (X 1 ) P (X 1 ) 0 .5

(Y=0)(X=0)

(Y=1)(X=1)(X=-1)

80

X

f

X

(

x)





x 8

,

0 x4

Y=X2

0,

X,Y FX(x)FY(y)

F Y ( y ) P Y y P X 2 y P y X y

y0 F Y(y)0 ; y16 F Y(y)1

f(x)d

x
f(t)dt f(x)

dx a

0y16 ,

d

u(x)
f(t)dt f(u(x))u'(x)

dx a

F Y(y)P0Xy FX (

y
y ) f X (t)dt

1 fY(y)2 y

fX(

y),

0y1621y

y 1, 8 16

0 y 16



0,





0,



Y(0,16)

81

XY=g(X)Y
1. YY
y1,y2, yj, ,Yyj
(XD),P(Yyi)P(XD);
2. Y Y
F Y(y)P(Yy) Yy (XD ),
F Y(y)P(XD ) Y fY(y)

82

X -1 0

1

p

1

1

3

3

1 3

Y=2X,Z=X2,Y,Z

Y-2,0,2 Z0,1 (Y=-2)(X=-1)... (Z=1)(X=1)(X=-1)



Y p

-2
1

0
1

3

3

2
1 3

Z

0

1

p

1

2

3

3

83

X f(x) x YX2 Y fY(y)

Y F Y (y )

y0 FY(y) P(Y y) P(X2 y)

y
f (t)dt
y

y

y

0 f(t)dt0 f(t)dt

fY(y)FY'(y) [f (

y) f (

y)] 1 , 2y

y0



0,

y0

84

XfX ( x ) , x g '( x ) 0 ( g '( x ) 0 )

Y g ( X ) Y

fY(y) fX(h(y0 )),h'(y),

y


m in (g ( ),g ( )) m a x (g ( ),g ( ))

h (y ) x y g (x )

y y=g(x)

y
g'(x)0 , gx , h(y),y

h'(y)0

0

x

y F Y ( y ) P ( Y y ) P ( g ( X ) y ) P ( X ) 0

y FY (y) 1

y FY(y)P(Yy)P(g(X)y)

h( y)

P(Xh(y))

fX (t)dt

f Y ( y ) fX ( h ( y ) ) h '( y ) fX ( h ( y ) ) h '( y )

g '( x ) 0

85

X f X (x), {x f (x) 0} (a, b) , a x bg '(x) 0 (g '(x) 0) Y g ( X )Y

f

Y

(

y)





fX

(h(

y)) 0,



h

'(

y)

,

y


min(g (a), g (b)) max( g(a), g(b))

h(y) x y g(x)

86

X ~ N (,2 ) Y X Y fY ( y )


y

g(x)



xg

'(x)



fY(y)fX(y)



1


1

2

0
y2
e2

xh(y)y
Y~N(0,1)

X ~ N (,2 ) Y a X b Y ~ N ( a b , a 2 2 )

X f(x) 8 x, 0x4 YX3 fY(y)
0,



yg(x)x3x

1
y3



h( y)

g'(x)3x2

0fY(y)13y23

1
fX(y3)

fY

(

y)



1 24

1
y3

,

0 y 64

0 ,

87

X Fx X

1 Fx

2 YFX, Y U0,1

1 ,X fx 0 ex,,x x 0 0Fx10ex,,xx00

2YFX 1eX,X0
0 ,X0

0Y1

F Y y Y ,

y 0 F Y y P Y y 0 y 1 F Y y P Y y 1

0 y 1 F Y y P 1 e X yPeX1y

PX1ln

1y



1e1ln1y y

0, y0

FYyy, 0y1, YU0,1

1, y1

88

2

1.

2."n"

3.Ap,0<p<1n AX,X
P X k C n kp k1 k n k,k 0 ,1 , ,n
4.

5.

6.AP(A)=0,A

P(A)=1

7.X0X

0

8.X(a,b)X(a,b)

(a1,b1)(b1-a1)/(b-a),

89

9.XN(,2)Xf(x)x=X



!
2019/4/21


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