Στατιστική και πιθανότητες για μηχανικούς

10

2

2.1 , , . 2.1.1 2.1.2 2.1.3 . 2.1.4 2.2 2.2.1 2.2.2 2.2.3 2.3 -2.4. 2.4.1 2.4.2 2.4.3 2.5 2.6 2.6.1 2.6.2 2.6.3 2.6.4 2.7

-

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11

. . , . . , , o OhmI=E/R.. R . , I. . . , . , . , . . .

, , , . .

. , .


12

2.1 , ,

, , , , .

2.1.1

. , . , , , , , , , , .

, . , , . , , .

. , ,

1. ,

2. ,

3. - ,

4. .


13

2.1.2

S.

2.1E1: . , D, , D .

S={D, D }. 2.2E2: . S ,

S={, , , , , , , }.

2.3E3: 1, 2 3. . , S

},,,,,,,{ 121321121321321321321321 AAAAAAAAAAAAAAAAAAAAAAAAS

2.4E4: . .

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

2.5E5: 10 . .

S={10, 11, 12, }.

. , (universal) S. o s, N(S).


14

2.6E6: .

S={x| x }.

2.7E7: 24-. 0C.S={x| m

15

2.1.3

E 1 2 S1 S2 . . S1S2 . ,

S=S1S2

{(s1,s2)| s1S1, s2S2, =1,2,,n1, =1,2,,n2}.

.

2.8 , , 2.1. 1={ } 2={ }

S1={ K, } S2={1, 2, 3, 4, 5, 6}. , S1 S2,

S=S1S2={ K, } {1, 2, 3, 4, 5, 6},S={(K,1), (,2), (,3), (,4), (,5),(,6), (,1), (,2), (,3), (,4), (,5), (,6)}.

1 2 3 4 5 6 1 2 3 4 5 6 {(K,1), (K,2), (K,3), (K,4), (K,5), (K,6), (,1), (,2), (,3), (,4) , (,5), (,6)}

2.1.


16

S ,

S={K1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6}.

,

S1S2Sn.

S, , . . .

2.9 ( 2.2) . , , 1={ 1}, 2={ 2} 3={ 3}, S1 ={K, }, S2={K, }, S3={K,} . , S ,

S=S1 S2 S3={K, } {K, } {K, },S={(,,),(,,),(,,),(,,), (,,), (,,), (,,) ,,)},

2.2, .

2.10 ( ) . 36 ,

S={(1,1),(1,2),(1,3),,(2,1),(2,2), (2,3),, (3,1), (3,2), (3,3),, (6,1), (6,2), (6,3), (6,4),(6,5), (6,6)}.

, 1 2 S1={1, 2, 3, 4, 5, 6} S2={1, 2, 3, 4, 5, 6} . , , , 1 2 S ,


17

S=S1 S2={ 1, 2, 3, 4, 5, 6} {1, 2, 3, 4, 5, 6}S={(1,1),(1,2),(1,3),,(2,1),(2,2), (2,3),, (3,1), (3,2), (3,3),, (6,1), (6,2), (6,3), (6,4),(6,5), (6,6)}.

, . , S, S1 S2 S3. , .

2.1.4

. .

, S .

2.11 . 2.1 2.8, , :

1: { } 1={D}.A2: { } 2= {, , , , }.3: { } 3= }AAA,AAA,AAA{ 321121321 .A4: { 5 } 4={1, 2, 3, 4, 5}.A5: { } 5={10}.A6: { } 6={x| 0

18

. . S . , .

S , . , S , . , . , Venn ( 19 ).

S , 2.2. , .

2.12 . ,

S={(,)

19

A , , .

h,

={(, ) >h}, 2.3.

2.2

2.2.1

. , , . , , , , , ..., .

: S S . A .

: S, , . .

max 2.3.

max

=h

S


20

. : , S, , =, . , . , .

.,}{,

,},{}{,,

SAAAAAAA

SSAAAAAASA

, , . .

AASAAAAAAASSA ,,}{,, ,

: S, , , =. , , 2.5.

A

S A

S

) , A ) ) , =

2.4. Venn

A

B

S


21

. ,

.

, , , , . , . 10 8 15 , 8 10 .

.},{},{}{,, AASAAAAAAASA

, ,

.

: S, , =. , , 2.5.

: S, - , =-. , , 2.5.

S

= = =-.

2.5. , .

S S

-.


22

, - , . , BA .

2.2.2

, . , , , , , . , , , .

Venn , , 2.6 ().

2.13 1 2 2.6. (1, 2, 3) , =1 , =0 , 8

S={(0,0,0), (0,0,1), (0,1,0),(0,0,1), (1,1,0), (1,0,1), (0,1,1), (1,1,1)}.

3 1 2, 1. , 3 , 2,

: . ,

BA ,

.


23

1={((0,0,0), (0,1,0), (1,0,0)}2={(0,0,1),(1,1,0), (1,0,1), (0,1,1), (1,1,1)}.

1 2 , 12= , .

2.2.3

, . ,

: ,

=, =.

: ,

()=(), ()=(.

: ,

()=()(), () =( ) ().

1

2

3

) ) Venn, 12=

2.6. 2.13.

S (0,0,1) B2 (0,1,1) (1,0,1) (1,1,1) (1,1,0)(0,1,0) (1,0,0)

(0,0,0) B1


24

. , .

, , ,

=, =2, ()()==, (+)(+)=++2++, (-) =, (-)+=.

. De Morgan ,

nn

nn

AAAAAAAA

AAAAAAAA

........

........

321321

321321 ,

. , AA , ,

nn AAAAAAAA ........ 321321 .

, ,

.

)()(

).()()(

ABA

ABAABAABA

BABABA

2.14

De Morgan: ,

BABABABA , . (2.1)


25

2.14 ( 2.6, ()). 1, 2, 3, .

, , ,

),( 213 AAA

, 3 1, 2 .

, , 3 2 1. , ,

).( 213 AAA

, , , De Morgan,

).()( 213213 AAAAAA

2.15 : , , . , , . () ,() ,() .:

() , . , , ,

{ }= )( H

() , , , , ,


26

{ }=))(())(())(( ,

{ }=

)()()( HHH ,

{ }=)()()( HHH .

() ,

H .

1.3 -

: S S*. :

S S* A A S* A S* A A1 S* A2 S* A1 A2 S*, A1 A2 S*,

() () ()

2.7. Venn, 2.16.

S

S

S

H

-()

-()-()


27

2.16 , S={,}. S, {}, {}, {,} . S* 22=4

S*={ { }, {}, {}, {,}}.

S , S* S .

2.4.

. , , , , . , ; . , , , ..

S . , . , S . , , , (1 6) 1/6.

, n ,

A1-A2 S*. S n , S* ,, 2n,

(S*)=2n


28

S={s1, s2, , sn},

si sj. si, P(si), 1/n. , r S ,

r ={ sr1, sr2, , srk },

/n.

.

2.17 , , ={2,4,6}, 5, ={1,2,3,4},

2

1

6

3 )S(N

)A(N)A(P ,

3

2

6

4 )S(N

)B(N)B(P .

2.18 , S 8 ,

S={, , , , , , , }.

,

={ , , },

3

: (S) , S, () , P(A) () (S),

N(S)

N(A)P(A) . (2.2)


29

8

3

)(

)()(

SN

ANAP .

(A) . , , , . . , . , , . , .(B) S ( ) . S , .

S , S . , () (S) . , , S . 2.19 . ;

, S , (S)= . , . ,


30

, ()=. , P(A) , .

, , S={1, 2}, 1 2 , , ={1}. , P(A) 1/2.

2.4.2

. n x , x n, x / n. , n x / n ,

n

xA

nlim)A(P (2.3)

. x x n , , .

. , fA, n xA/n,

n

xf AA .

fA :

0 fA 1. fA=1

.


31

5) fA P(A), n, . , , . , fA .

, (2.3). P(A) n.

2.20 P(); , , 100 , , . 45 , 45/100 P() . 50 100 . , . , , 1/2 .

2.4.3

, , .

fA=0 .

, U

fAU= fA + f. fA n

P(A) n. A

nfAP

lim)( .


32

, , . , , , P()=2/3 P()=1/3. , .

, . . , . , . , . , . , . , . o Kolmogorov o1933, . , , . ( Kolmogorov) .

2.5

S S* , S*={A1, A2, }. P S* S* P(), , .


33

. P() fA , fA . P() .

P(). . P().

S* ,

0 P() 1.

S , P(S)=1

. , S , S .

S, , S* , , =,

P( )=P( )+P().

. , S, S, . (S, S*, P) .


34

2.1:

, S*, ,

P()=0. (2.4)

: S* , =U. ,

P()=P(U)=P(A)+P() P()=0.

o . (2.1) . , P()=0, .

2.2:

A S*

)AP(1P(A) . (2.5)

: S , AAS

A , 1 2

1= P()+ P( A ). , .

P(), A , P( A ), .

2.3:

, S* , ,

P() P().


35

: -,=U(-),

P(B)=P[U(-)]=P(A)+P(B-A), P(B-A)0 P() P().

. , .

2.4:

S, S*,

P(-)=P()-P(). (2.6)

: -,= ()U(-),

P(B)=P[()U(-)]=P()+P(B-A),

.

2.4 , .

2.5:

, , S*,

)()()()()]([ BAPAPBAPAPBAP

:


36

)()()()(

)]()()([)(

)]()[()(

)]([)()]([

BAPAPBAPAP

BAPAPBAPAP

ABAPAP

BAPAPBAP

2.6:

, S*,

P(U)=P()+P()-P(). (2.7)

: , U, -, 2.8,

U= U(-),

3 ,

P(U)=P[U(-)]=P(A)+P(B-A)

2.4 ,

P(U)=P(A)+P(-)= P(A)+P(B) - P().

S A B

B-AAB

2.8 Venn ,


37

2.6 . III, = , III. , 2.8, . P(A) P(B), P() , .

2.7:

, , S*,

P(UU)=P(A)+P(B)+P()-P()-P()- P()+ P() (2.8)

: UU U, U(U), (2.7). .

. , , . 1, 2, , n n ,

)9.2()...()1(...

)()()()...(

211

32121

nn

rrji

ji

n

jiji

n

iin

AAAP

AAAPAAPAPAAAP

1, 2, , n n , , , , n, ,

)(....)()()()...( 32121 nn APAPAPAPAAAP

2.21 ( , ). ,


38

,

36% , 20% , 12% .

,() ;() ;() ;: .

12,0)(,36,0)(,20,0)( YMPMPYP .() -. , 2.4,

24,012,036,0)()()( YMPMPYMP

() . ,

YM ,

2.2 .

(0,8)

S M Y

M-Y (0,24)

S

MY

YM (0,56)

() () ()

2.9 Venn

S M

(

MY Y

(0,24) (0,12)

(0,20)

MY (0,44)


39

56,044,0112,020,036,01

)]()()([1)(1)(

YMPYPMPYMPYMP

()

)MY()YM( ., ( 2.4),

32,012,020,012,036,0

)()()()(

)()()]()[(

MYPYPYMPMP

MYPYMPMYYMP

2.22 , , . : 20% , 16% , 14% , 8% , 5% , 4 % , 2% , .

, :() ,() ,() . :

S

BA

A B

S B A

A-(B)

S

-

-

-

() , , () () , ,

2.10. 2.22, Venn.


40

={ },={ },={ }.

- : BA- : -()- :

)()()( ABBABA

() 2.2 ,

.65,035,01

)02,005,004,008,014,016,020,0(1)(1)(

BAPBAP

() ( ) . .... ( 2.5) , ,

09,002,005,008,020,0

)()()()()]([

BAPAPBAPAPBAP

() , Venn, -, -, -. , , ,

11,002,004,002,005,002,008,0

)()()()()()(

)()()()]()()[(

ABPBpBAPAPBAPBAP

ABPBAPBAPABBABAP

2.6

P() , P()=nA/nS, nA nS


41

. , nA nS. , .

, , , ; , ( , , Q , J , 10 ), .

, . , , , S . .

2.6.1

, 1, 2, , , S1, S2,, S. S , ( 2.1, ) ,

S=S1 S2 S

, S ,

S ={(s1,s2,, s)| s1S1, s2S2,,sS, =1,2,,n1, =1,2,,n2, , =1,2,,n}.

, n, ,

n=n1n2n, (2.10)


42

n1, n2,,n () S1, S2,, S .

2.23 , 5, 6 2 , ; : , . , n1=5 , n2=6 n3=2 . , . 2.10

n=n1n2n3=562=60 .

2.24 15 18 . , ;:

( ), ( 1) ( 2). , ,

n=n1n2=1518=270 .

2.6.2

, n . .

2.25 , , , ;


43

, , ( 1), ( 2) ( 3). (2.10) 2.11

n1n2n3 = 321 =6 .

2.26 n . ; n n 1,2,3,, n, .

1 2 3 . . . n

n ( 1), n . , (n-1) ( 2), (n-2) ( 3), (n-3) ( 4), . , , ,

1

2

3

4

5

6

2.11. , ,


44

n n ,

n (n-1) (n-2) (n-3) 2 1= n! (n ).

, . n , , n!. nPn (Permutations),

nPn = n!, (2.11)

n! n 1 n,

n! = n (n-1)(n-2)1. (2.12)

0! = 1.

2.27 0, 1, 2, 3, , 8, 9;: , (2.12),

P(10,10)=10!=109821=3.628.800.

, , 3.628.800. 1, 2, 3, , 9 . ,

P(9, 9)=9!=9821=362.880

, 3.628.800 362.880 = 3.265.920. , 1, 2, 3, , 9, .... , 9981=3.265.920.


45

n . r , 0 r n, . . P(n, r) n , r. n r 1, 2, 3, , r, . n , (n-1), (n-2) , . , ,

n (n-1)(n-2)(n- r+1)

. ,

)!(

!),(

rn

nrnP

. (2.13)

2.28 . ;: . , , .... , ,

8407654!3

!7

)!47(

!7)4,7( P .

2.6.3

r n . , ,, , n =3, , r=2. , , . , .


46

n r, n!/( n- r)!. C(n, r) r n, (, C(n, r) ). r r! . , , r n C(n, r) r!, ,

)!(

!!),(

rn

nrrnC

r n ,

!)!(

!),(

rrn

nrnC

. (2.14)

n r. 2.14 ,

!r

)rn()n)(n(n

!r)!rn(

!n

r

n)r,n(C

121

, n)( ,

rnrn

r

n

r

n

0

)( .

S, n . , n2 . . ,


47

, , , n.

n

n...

nnn

210

.

.2)11( nn

( ):

r

n

r

n

r

n

rn

n

r

n

1

1

1)2

)1

, (2.14).

2.29 10 . 5 .() ;() , ; () ; :() . , ,

25254321!5

109876!5

!5)!510(

!10)5,10(

C .

() , ,


48

240.30)!510(

!10)5,10( P .

, ,

120!5)5,5( P .,

30.240 =(252 )(120 ).

() , ,

56338

838 !)!(

!),(C .

,(56 )(120 )=6.720 .

2.30 80 20 . 10 , . ;: S. , C(100, 10). ; , 5 20 5 80 ; C(20, 5) C(80, 5). , , 5 5 10

0210

10

100

5

80

5

20

.

.

(A) r n ,


49

C(n, r). r n , rn .(B) . Stirling,

212 /nnne!n .

2.6.4

. .

n , n1 , n , , n1! n!. n1! n, n1 , n-n1 . , n1 . n , n1 ,

!

!),(

1n

nnnP .

, n , , , n1 , n2 , n3 , ..., n = n1+ n2 + n3 + + n , n

!!...!!

!),(

321 knnnn

nnnP . (2.15)

, . . , , , . ; , n =10, n1=4, n2=3, n3=1, n4=2. , (2.15),

600122134

10.

!!!!

! .


50

() n , n1=1, n2=1, n3=1, , n=1 n , (2.15), n!.() , , r n-r , (2.15):

),()!(!

!),( rnC

rnr

nnnP

.

() A n r. :

rn

n

r

n

2.31 12 . 12 , 5 7 . (2.15),

95040!7

!12

!1!1!1!1!1!7

!12)12,12( P .

2.7

, . , . , , , ,


51

.

, () (). . , . . . , .

. , , . , . , . , .

, .


52

-

1) ;

)()()()()()(

)()()()()()()(

CBBACBACBABABA

BBABACBACABA

(), (), ().()

).()()]([)]()[(][

)]([)]([)()(

CBACBABACBABA

CABCAACABA

() ,

)()()()()( BASBABBBABBA

() ,

}{}{)()()( CACBBACBBA

() ().() BABA .() De Morgan,

CBACBACBA )()( .

2) `. .


53

,

S={, , , ,.....}.

3) , b, c, d . : ={ }, ={b }() ,() BA BA .

() 4 4=24. 24 ,

S={abcd, abdc, adbc, adcb, acbd, acdb, dabc, dacb, dbac, dbca, dcab, dcba, bacd, badc. bcad, bcda, bdca, bdac, cbad, cbda, cdab, cdba, cabd, cadb}()

A={abcd, abdc, adbc, adcb, acbd, acdb},B={abcd, abdc, dbac, dbca, cbad, cbda},

BA ={ abcd, abdc},BA ={ abcd, abdc, adbc, adcb, acbd, acdb, dbac, dbca, cbad, cbda}.

4) (

54

() , , .() , , .

() S={3, 4, 5}. , :

S={(1,2,3), (1,2,4), (2,3,4), (1,3,4), (1,2,5), (1,3,5),,(3,4,5)}

, ,A={(1,2,3)},B={(1,2,4), (1,3,4), (2,3,4)}, ={(1,2,5), (1,3,5),, (3,4,5)}.

() S , ,

102

20

!3)!35(

!5)3,5( C

, ,

10

6)(

10

4)(

10

1)( PBPAP .

6) 4 . . 4 . ;

. . , (2.17)


55

N(S)=4 4 4 4=256, , ,

N()=4 3 2 1=24., 4 . 3 ( ). , 2 , .

, ,

256

24

)(

)( SN

ANp .

1) . ::, , . () ,() ,() .

={ },={ },={ }.

De Morgan , () { }= )( BA ,() { }= BA ,() { }= )( BA .


56

2) . , , . .

={ }={ }={ }={ }.

0, 1, 2 3 , ,

BAE

BABABAE

BAABBAE

BAE

3

2

1

0

)()()(

)()()(

)(

3) . . :() /4.() /4.() .

() , S={1, 2, 3, 4}.

, ={ 2, 3, 4}.

75,04

3

)(

)( SN

ANp

, () p=0,5,() p =0,6666.

1 2 3 4


57

4) 1 2 . . . () .() P(1)=0,2, P(2)=0,3 P(1 2)=0,1, .

() ={ }.

S1 ={ 21 , } S2 ={ 22 , },)},(),,(),,(),,{( 21122121 SSS .

S ( ) :

)()()()( 21212121 S ,

)()()( 212121 A )( 21 A .

() ,

4,01,03,02,0)()()()()( 212121 PPPPAP

1 2


58

1) Venn.

: BBABABAA ,)(),()(, .

2) 11 , 15 , 8 12 . :

).(,,,,,,, BABBAABAABABA

3) , , . ,

.

4) r. r.

5) ;

6) 20 . ;

7) 8 . . ;

S

A B


59

8) . . . ;

1) . . , . 100 :

1 2 3

18 17 50

2 3 10

1, . , , .

2) . . , , 0,3 0,5 . 0,2. . ( -) , .

3) . P()=0, 60. () { }. () ..


60

4) . , , 1, 2 3. . ,

1={ }, 2={ } 3={ }.

={ }

1, 2 3.

5) . , , . , , . Venn :

- , .

- , , .

- , .

6) . (0, 1). 0 1, s0=t2, s0 0 t 0.

1

2 3

4


61

1 0 s1=t3, s1 1 t 1. . (0, 0,5) (0,5, 1).

7) . 120 , . , RA + RB = 120 . () (RA, RB) RA RB .

, 80 120 ,() , , () ,

1={ 2 4 },

8) . a b c P(A)=0.30, P(B)=0.20 P(C)=0.15 . P(AB)=0.08, P(AC)=0.05, P(BC)=0.05 P(ABC)=0.03. :

- ,- ,- .

RA RB

120

6


62

3 , - Bayes,

3.1 3.2 3.3 BAYES3.4 3.4.1 3.4.2 3.5

-

. . , . , . , . , , , P(A)0, P(B)0, , , .


63

.

3.1

, P(A|) , . P(A|), P(A) , S.

Venn, P(A) , S. P(A|) , . . S .

.

3.1 S 36, S={(x1,x2)| x1=1,2,,6, x2=1,2,,6}.

S A B

AB

S (2,1) (3,2) (3,1) (4,3) (4,2) (4,1) (5,3) (5,2) (5,1) (6,5) (6,4) A (6,2) (6,1) B

) S ) 3.1

3.1. , , Venn.

(4,5) (5,4)

(3,6) (6,3)


64

, 9, ={(x1,x2)| x1+ x2=9}. , ={(x1,x2)| x1> x2}. , ( 3.1, ()) ,

36

15

)(

)()(,

36

4

)(

)()(

SN

BNBP

SN

ANAP .

, Venn, , , . , , ,

36

2

)(

)()(

SN

BANBAP

, , P(A|). , 15, . , , (6,3) (5,4) . -

15

2

)(

)()(

BN

BANBAP .

S,

)(

)(

)(

)()(

)(

)(

)()(

BP

BAP

SN

BNSN

BAN

BN

BANBAP

(3.1)

, .


66

() , ,() , .

={ }={ }={ }.

, P(A)=0,20, P(B)=0,16, P()=0,14, P(AB)=0,08, P(AB)=0,02, P(AB)=0,35.() ,

57,035,0

20,0

)(

)(

)(

))(())((

BAP

AP

BAP

BAAPBAAP

, , , P(A). (), , AB, , AB,

25,008,0

02,0

)(

)(

)(

)]()[(])()[(

BAP

BAP

BAP

BABAPBABAP

.

P(AB)=0,02, 0,25.

(3.2)

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

A)P(A)P(BB)P(A (3.3)


67

. , , , .

3.3 100 . Pendium (P) Celeron (C). () (). 3.1 .

3.1 () ()

Pentium (P) 30 10 40Celeron (C) 40 20 60 70 30 100

.() Celeron;() . Celeron;: . , . :

={ }, , ()=70,

C={ Celeron}, , (C )=60,

C={ Celeron}, , (C)=40.

() C 40 100,

40.0100

40

)(

)()(

SN

CKNCKP


68

, , .() E P(C|K). (100 ) (70 ),

P(C|K)=40/70=4/7.

,

7

4

7.0

4.0

)(

)()(

KP

CKPKCP

) . P(K)=70/100,

4.010

4

7

4

10

7)()()( KCPKPCKP .

3.4 100 20 . . ;: ,

={ }, =1,2,3.

To , 3.3, :

))A...AA(AP(A...))A(AP(A)A)P(AP(A

)A...AAP(A

1n321n213121

n321

. (3.4)

, , . .


69

, ,(21)={

}(312)={

}.

, :

.98

18

99

19

100

20 )AAP(A,)AP(A,)P(A 213121

, 1, 2 3, 123, ,

98

18

99

19

100

20 ))A(A)P(AA)P(AP(A)AAP(A 213121321 .

3.2

. , ( 3.3 ). .

, i .

: 1, 2, , S,

1. , iAj= ij,2. , SA

k

1ii

,

3. P(Ai)>0 i.


70

1={1,2,3}, 2={4}, 3={5,6} S={1,2,3,4,5,6}, 1={1,2,3,4} 2={3,4,5,6} .

S 1, 2, , S. 3.3 Venn =5, . ,

)(....)()( 21 BABABAB (3.5)

3.3 . .

: (3.5) ,

S A1 A2 A3 A4 A5

A1B A2B A3B A4B

Venn, A S. 3.3.

1, 2, , S, , ,

))P(AAP(B...)P(A)AP(B)P(A)AP(BP(B) 2211 . (3.6)


71

)](....)()[()( 21 BABABAPBP k .

2.6 ,

)(....)()()( 21 BAPBAPBAPBP k .

P()

)()()( iii APABPBAP ,

.

. , P(B), P(i) 3.6 .

3.5 ( 2.30) 20 80 . ,

1={ },2={ },={ }.

, 1 2 S. , ,

)()()()()( 2211 APABPAPABPBP .

2.30

100

80

100

80

99

79

100

20

99

80)( BP


72

.

3.6 . 50% 1, 2 3 25% . , 2 3 2% 6%, . ;

:

1={ },2={ },3={ },={ }.

,

.06,0)(,02,0)(,0,0)(

,25,0)(,25,0)(,50,0)(

321

321

ABPABPABP

APAPAP

. 1, 2, 3 . ,

)()()()()()()( 332211 APABPAPABPAPABPBP .

,

P(B)=0,50 0 + 0,25 0,02 + 0,25 0.06.=0,02

3.7 , 3.4.


73

3.7 0 1. 0 1. 0 1 . 60% 1 40% 0. H 0,10 1 0, 0,2 0 1. 0 1 :

{=0} {=1} 0 1 .

0,5 0,25 0,25

1 2 3

1,00 0,00 0,98 0,02 0,94 0,06

B B B P(B1) P(B2) P(B3) 0,000,5=0,00 0,020,25=0,05 0,060,25=0,015 P(B)= 0,00 + 0,05 + 0,015 = 0,02

3.4. 3.6

1

0

1

0

60% X=1

40% X=0

Y=1

Y=0

0,90

0,1

0,8

0,2

3.5


74

{=0} {=1}. 3.5. 1={=0}, 2={=1} ={=1}. 1 2 . ,

62,040,020,060,09,0

})0({})0{}1({})1({})1{}1({)1(

XPXYPXPXYPYP

={=0} 0,

38,040,080,060,01,0

})0({})0{}0({})1({})1{}0({)0(

XPXYPXPXYPYP

3.3 BAYES

3.6, . , ;

, 3.7,

P(A1B). . , 2 3 . , , 3 2 , 3 . , ,


75

75,002.0

06.025.0

)(

)()(

)(

)()(

00.002.0

00.05.0

)(

)()(

)(

)()(

3333

1111

BP

ABPAP

BP

BAPBAP

BP

ABPAP

BP

BAPBAP

.

, , .

, , P(A1B), , , P(A1), P(A1).

Bayes.

Bayes - ,

,...,2,1)(

)()(

)(

)()( i

BP

ABPAP

BP

BAPBAP iiii .

. , , Ai. , , Ai.

Bayes , .

3.8

Bayes 1, 2, , S, S, P(Ai )

,...,,i 21P(B)

)A)P(BP(A)BP(A iii (3.7)


76

3.7, 1. ;

}1{}1{ YX, , 1, 1. , 1={=0}, 2={=1} ={=1}, Bayes,

87,062,0

54,0

62,0

6,09,0

})1({

})1({)}1{}1({)}1{}1({

YP

XPXYPYXP

1, 0,60 . 1 1, (0,90) .

3.9 , 20% , .. leizer, .

90%, , 80%, .

() ;() ; ,

1={ , },2={ , },={ ,, },

,

,10,0)(,80,0)(,80,0)(,20,0)( 2121 ABPABPAPAP


77

.() P(B). 1 2 , , ,

24,080,010,020,080,0

)()()()()( 2211

APABPAPABPBP

, 24% .() , 2 , P(2). , Bayes,

3

1

24,0

80,010,0

)(

)()()( 222

BP

APABPBAP

, , . , , 11.

3.4

S, . , , =, , P(AB)=0. , , . , , AB , , P(AB)=1.

3.4.1

. ,


78

. , , ,

={ },={ },

, . . . 36 , 6 , 18 , , 3, P(A)=6/36=1/6, P()=18/36=1/2, P()=3/36=1/12. ,

6

1

2/1

12/1

)(

)()(

BP

BAPBAP .

, P(AB ) P(A). , ,

)(2

1

6/1

12/1

)(

)()( BP

AP

BAPABP .

. , P(AB)= P(A) P()= P().

, , P()

)()()()()( APBPBAPBPBAP

)()()()()( BPAPABPAPBAP .

, P() P() P()= P()P(). .

: , , P()0 P()0,

P(B)P(A)B)P(A ). (3.8)


79

() , P(AB)= P(A) P()= P(), , .() , , (3.8). () , P()0 P()0. .

3.10 . , , . ;

H , P(A)=1/=0. , S , . , P()=1/2, 2.19. P(A)=0.

3.8 P(A)=0P(A)P()= 0 1/2=0

, , P()=0P(). 3.8, , P(A)0 P()0.

3.11 , . ={5} ={4},


80

. 2.15.

36 , , 12 24 , P(A)=12/36=1/3, P()=24/36=2/3. , , A,, 8

9

2

3

2

3

1)()(

9

2

36

8)(

BPAP

BAP

.

, . , n 1, 2, , n , , .

, ,

).()()()(

),

,()()(

),()()(

PBPAPBAPP(B)P()P(B

)PAPAPBPAPBAP

(3.9)

3.4.2

.

: N A1 A2, An P(A)0 =1,2, , n, , , , n-1 n .


81

.

.

, , 3.7, 15 S . , , . , . ,

.15/2)()()(

,15/6)()(

,15/5)()(

BPAPBAP

ABPBP

BAPAP

, BA , BA .

, P() 0 P() 0, , , , . P() > 0 P() > 0, . , , . , , .

, , . , , .

S

A B

AB

3.7. Venn


82

, ( ):

1. S, , S .

2. S , . BA , . BA ,

3.5

. , , . , , , Bayes.

, . , Bayes, . , .

. , .


83

- 1) , , . . , . .

={ },={ },={ }.

..., :

14

8

8/11

2/1...

2

1

2

1

2

1)(

...)()()()(2

1)(

.....)()()(

.....)(

74

432

43211

43211

AP

APPBPAPAP

ABAPAPAP

ABAAA

1/8 1/2.

2) . ; ;

S={0, 1,2 ,3 ,4 ,5 ,6 ,7 ,8 ,9}, .

, .


84

100 ( ).

0, 0 0, 1 0, 2 0, 3 0, 4 0, 5 0, 6 0, 7 0, 8 0, 91, 0 1,42, 0 2,23, 0 3,84, 0 4,1 4,65, 06, 0 6,47, 08, 0 8,3 8,89, 0

:

={(2,2), (1,4), (4,1), (7,2), (2,7), (8,3), (3,8), (4,6), (6,4), (8,8), (6,9), (9,6) }.

100

12

)(

)()(

SN

ANAP .

5 . 40 , (3, 8), (4, 6) (8, 8) .

40

6)( AP .

3) ; 3 ; 12 ;


85

={ }. :

321 AAAA .

),,(.42,06

51)()()(1)(

)(1)(1)()(

321

3

321

321321321

AAAAPAPAPAP

AAAPAAAPAAAPAP

,

287,036

351)(

12

BP .

, 12 3.: , ... . ,

)...()()(...)()...( 321321321 APAPAPAAAPAAAP

.

4) 3. () 3;() 3;

10. 3;

3 : S={(1,2), (2,1), (3,3), (2,4), (4,2), (5,1), (1,5), (6,6), (6,3), (3,6), (5,4), (4,5)}.() , 3 ={(3,3)}.


86

12

1

)(

)()(

Sn

AnAP .

() { 3} , ={(3,), (3, 6). (6, 3)}. ,

12

3

)(

)()(

SN

BNBP .

() 10, 28,

S={(1, 4, 5), (1, 5, 4), (1, 3, 6), (1, 6, 3), (2, 4, 4), (2, 3, 5), (2, 5, 3), (2, 2, 6), (2, 6, 2), (3, 6, 1), (3, 1, 6), (3, 5, 2), (3, 2, 5), (3, 3, 4), (3, 4, 3), (4, 5, 1), (4, 1, 5), (4, 4, 2), (4, 2, 4), (4, 3, 3), (5, 1, 4), (5, 4, 1), (5, 3, 2), (5, 2, 3), (5, 4, 1), (5, 1, 4), (6, 3, 1), (6, 2, 1)}.

, , 3. , 27 ,

P({ 3})27

15 .

5) , P(A) P(AB) , , 3.2. , ;

:

() P(AB)=0P(A), .

() P(AB)=P(AB)/P(B)=P(A)/P(B)P(A), = P(B)1.

() P(AB)=P(AB)/P(B)=P(B)/P(B)=1P(A).

{AB} ,

P(AB)P(A), ,

P(AB)P(A).


87

, .

6) . p1 p2. . .

I II .

1

2

1

2

11

1

)(

)()(

)(

)()(

p

p

p

p

AP

BAPBP

AP

ABPABP

7) 6 . ;

1={ }={ -1, -2, , 1 }, =2, 3, 4, 5, 6. R 6 ,

654321 AAAAAAR ,

() () ()

A B B A

S S S


88

)()....()()()( 543216213121 AAAAAAPAAAPAAPAPRP

6

1)(.

.....6

4)(

6

5)(

1)(

543216

213

12

1

AAAAAAP

AAAP

AAP

AP

6654321 6

!6

6

1

6

2

6

3

6

4

6

5

6

6)()( AAAAAAPRP

8) . . . .

={ }={ }.

R ,

)( 3211 ABAAR

)()()()(

)()())(()(

2131211

32112211

BAAPABPAPAP

ABAPAPABAAPRP


89

3

2)(

4

2)(

5

3)(

5

2)(

213

12

1

1

BAAP

ABP

AP

AP

0

5

3

3

2

4

2

5

3

5

2)( RP

1) . () p

90

)( 321 AAAR

() . 1 2 3. ,

32

23212321

321321321

2222

)()()()()()(

)()()())(()(

ppppppppp

AAPAPAPAAPAPAP

AAAPAAPAPAAAPRP

() , 2 , 1 , . ,

1)(

)(

)(

))((

)(

))(()(

2

2

2

21

2

2121

ARP

ARP

ARP

ARAP

ARP

ARAPARAP

2) A . . , , . 0,04. , 0,80.

() ;


91

() , ;

. R () R ,

)( BAAR .

992,080,004,096,0

)()()()()())(()(

ABPAPAPBAPAPBAAPRP

() .

968,0932,0

196,0

)(

)()(

)(

)()(

RP

ARPAP

RP

RAPRAP

3) . , , , , . , , .

0,7

0,7

0,8 0,9

0,8

0,7 0,8

0,9

0,9

0,7

0,7

0,8

0,8

0,9

0,9


92

, . .

. , R ( ) i, Bi, i, i (i=1, 2, 3) , , ,

878,0128,0762,0512,19,08,07,09,08,07,039,08,07,03

)()(

)()(

)()()()(

)]()()[()(

)()()(

333222

333222111333222

333111222111

333222111

333222111

333222111

BABABAPBABAP

BABAPBABAP

BAPBAPBAPRP

BABABAPRP

BABABAR

, ,

8648,099,096,091,0)81,09,09,0)(64,08,08,0)(49,07,07,0(

)]()()([

)]()()()][()()([)(

)()()()]()()[()(

)()()(

2121

21212121

212121212121

212121

PPP

BBPBPBPAAPAPAPRP

PBBPAAPBBAAPRP

BBAAR

0,878 > 0,8648, .

4) . , . , , , .


93

, . , . , , , , 0,4, 0,2 0,6 . , , , 0,10. , , 0,30. :() .

() .() , , ;

,

={ , },={ , },={ , }.

() , BA . )( BAP , De Morgan ,

)(1)(1)( BAPBAPBAP ,

54,09,06,0)1,01(6,0)](1[8,0)()()( ABPABPAPBAP

46,054,01)( BAP .


94

() BA . ,

.

622,0378,0170,009,06,01

)()()(1

)(1)(1)(

BAPABPAP

BAPBAPBAP

()

BA . BA . ,

26,0622,0

162,0

)(

)(

)(

)]()[()]()[(

BAPBAP

BAPBABAPBABAP

.

5) . 1, 2, 3, 4 p . , =1, 2, 3, 4. E q, () R={ },() R,() , 4 ;

() R . ,

1

2

3

4

qA B


95

))(()( 4321 AAAR

, E q.() , ,

)]()()([)]()()([

))(()()))(()(()(

43432121

43214321

APPAPAAPAPAP

APAAPAAAPRP

))1()(2(][][

)]()()()()()([)]()()()([2

43432121

qpqppppqppqppppp

PPAPPPAPAPAPAPAP

() ,

pqq

qpqp

pqqp

qpqpp

qpqpp

qppqp

AP

AAP

APAAP

AAPAAP

RP

AAAAP

RP

AAAAP

RP

RAPRAP

1)1(

)(

))(((

))()((

)]([()(

)]()[()(

)(

))]()(()[(

)(

))]](()[([

)(

)()(

22

2

22

43

443

4321

4432144321

4321444

6) . , , .

, , 0,1, 0,2 0,1 . 0,02. , .


96

, , .() ; () ( ), ; () , 0,1, ();

:

1={ },2={ },3={ }.

, ,

1={ }=1,2={ }= 23 AA .

() 1 2. , , R,

21 EER . ,

648,09,08,09,0)3()2(9,0)32(90,0

)32()1()2()1()21()(

APAPAAP

AAPAPEPEPEEPRP

321 A,A,A .()

714,01,02,01,02,0

2,0

)32(

)2(

)32(

)]32(2[)322(

AAP

AP

AAP

AAAPAAAP

()


97

() P(A2)=0,1. ,

526,019,0

1,0

1,01,01,01,0

1,0)322( AAAP .

, , 0,714 0,526.

7) . 0,01 0,02 . , 0,02 0,08. , ;

={ },={ },

, 08,0)(,02,0)(,01,0)( KPPKP .,

K ={ } ={ }.

P({ }), ,

)(

)(

)(

)]()[(])()[(

KP

KP

KP

KKPKKP

)()()(

))(1)((

)()()(

)()(

KPKPP

KPP

KPKPP

KPP

P(K ) P(K) ,

04,002,0

0008,0

)(

)()(

0008,008,001,0)()()(

P

KPKP

KPKPKP


98

676,00284,0

0192,0

0016,002,001,0

96,002,0])()[(

KKP

8) . , , . 0,6, 0,4 0,3 . , , :() .() , .

, , , ,

, , P(A)=P(B)=p, P()=3p. , , , ,

P(A)+P(B)+P()=1 p+p+3p=1

p=0,20 P(A)=0,20, P(B)=0,20 P()=0,6. , . ,

={ },={ }.

,() ,

,62,042,012,008,06,07,020,06,020,04,0

)()()()()()()(

PKPBPBKPAPAKPKP

() , Bayes,


99

677,062,0

6,07,0

)(

)()()(

KP

PKPKP

.

, , 0,6 0,677, .

9) . , , . 0,10 0,02.() . () 90% , 70% , 40% , , , , . ,

:

A={ }, ={ }, R={ }.

() R ,

)()( BABAR .

, R

0,20

0,20

0,60

P()P(A)=0,60,20=0,12

A P(A)P(A)=0,40,20= 0,08

B P(B)P(B)=0,40,20=0,08

B P(B)P(B)=0,60,20=0,12

P()P()=0,30,60=0,18

P()P()=0,70,6= 0,42

0,6

0,4

0,4

0,6

0,3

0,7


100

.

)()()()()()()]()[()( BAPBPABPAPBAPBAPBABAPRP

8,02,01

1,0

02,01

)(

)(1)(1)(

AP

BAPABPABP

8,0)( BAP . , P(R) = 0,10,8+0,10,8=0,16.

() ,={ },1={ },2={ },3={ }.

1, 2, 3 ,

82,0)3()2(1)1(

16,0)2(

02,0)()3(

APAPAP

AP

BAPAP

,

S

A1A2

A2 A3


101

858,0008,0112,0738,002,040,016,070,082,090,0

)3()3()2()2()1()1()(

APAEPAPAEPAPAEPEP

, )3( EAP Bayes:

084,0142,0

012,0

)(1

02,060,0

)(

)3()3()3(

EPEP

APAEPEAP

. )(EP )(EP .

10) . , . . 80%, 70%, 50% . , , 50%, 80% . , .

, , ;

,

={ },={ },={ },={ }.

0={ },1={ },2={ },3={ }.

S,


102

28,0)()(

47,050,030,080,050,070,020,050,070,080,0

)]()()(

)]()()[()(

22,050,030,020,050,070,020,050,030,080,0

)()()()()()()()()(

)()()(

)]()()[()(

03,0)()()()()(

3

2

1

0

KBAPAP

KABPKABPKABP

KABKABKABPAP

KPAPBPKPAPBPKPAPBP

KABPKABPKABP

KABKABKABPAP

KPAPBPKABPAP

,

766,028,0376,011,0028,0147,080,022,050,003,00

)()()()()()()()()( 33221100

APAEPAPAEPAPAEPAPAEPEP

11) . 3.9 (), , , ;

3.9,

1={ , },2={ , },1 ={ },2 ={ },

,

136,0008,0128,08,01,01,02,08,08,0

)()()()()( 2221112121

APABBPAPABBPBBP


103

)()()( 1211121 ABPABPABBP , . , 13,6% . , P(212). Bayes

06,0136

8

136,0

008,0

136,0

8,01,01,0

)(

)()()(

21

2221212

BBP

APABBPBBAP

6% . ( 3.9) 33%

0,80 0,20 0,9 0,1 0,2 0,8 0,9 0,1 0,9 0,1 0,2 0,8 0,2 0,8

0,008 0,128


104

1) 4 6 .() , ;() . 4 . 4 5 ;

2) . (P(BA)>P(B)), ( P() > P() ).

3) . . . ;

4) 1, 2, 3, , . . .: 1+2+3++ = (+1)/2

5) 60% , 80% . .

6) 95%. 8% , :

. ;


105

7) 12 10 . , 80%.() , ; () , 11 . ; 8)

p)B(P,)A(P,,)BA(P 4070() p ;() p ;

9) T , . W: , ( ) R: , ( ) . W R .

10) . ( ) 0,25, ; 11) 940

514 68

112 246


106

, , .() ).(),(),(),( BAPBAPBAPBAP () ;12) () , . :() 3 ().() 3 (), ;

13) 15 . . 4 , ;

14) . . . ;

15) 50%. , 90% .

16) 10 m . : 5}{M}2{mP .17) 1/2 . ,

2/Ae .


107

1) . 3 4 . , 0,002 0,006. .

,() ,() ,() .

2) . , , . .

,() ;() , ;

3) . (0, ). [t1, t2]

[0, t0].

4) . 25 5 .() ( ), ;() ( ), ;

7) 1, 2, 3, , .

1 2 3


108

() , p.() :

, ,

.

.

5) . n , n1 , n2 , n3 n4 (n= n1+ n2+ n3+ n4). p1, p2, p3, p4 . , , . .

6) . , .

1 2

+1 +2 2

1 2

+1 +2 2


109

1. q1

2. q2

, ,

Q1,

Q2.() Q1, Q2 .

() Q1, Q2 .

7) . p ( ).

() , , =1, 2, 3, 4, 5.() , ;

1 2

1

2

3

4

2

4

1

3

5A B


110

8) . 1, 2 3. 50% . P(1)=0,9, P(2)=0,7 P(3)=0,6. () ;() . 2 3;

9) . .

. .;

10) . 20% 80% . 40% . ;

11) . () () 1:4 . () () 1/4 () () 1/3. (), ( () ).

12) . 10% . .

0,1

0,3

0,2 0,1

0,1


111

0,9 0,1 . () ,() , .

13) . , , , . 1/3, 1/6 1/2 . .

( ). P(X1 =1 X2 =1).

14) . 250 , 0,5 , . 10 . 85 % . , 75%, , 0,25.() ;() ,

15) . 1, 2, 3 (1, 2, 3, 4) . 3 2, 1, 0 , , , . . () 1 4; () 4; () 5 ;


112

16) . . , . , 0,80 . , . , 70%, 50%. 0,90. () ;() ;() ;

17) . 20 ( , 1/20 ).

4 100 ( 1/25). .() ;() , ;

20) . . . , 0,70 0,60

--


113

. , 0,30.

, ,

0,95 , 0,6 , 0,2 .

.

21) . 0,20. , , 0,05. , , 0,30.() ;() . ;

22) . , . , 0,10, 0,25, 0,40 , . 5 , .() ;() ;() . ; 22) . . , 1( 20 ) 2( 30 ), 50 . p=0,005 p=0,008 .


114

, . , , .

1, 2, .() 1, 2, .() 1, 2, , .() , ;


115

1) . Grenoble Turin . Turin Grenoble, Chambery . Turin Grenoble . , , Turin Grenoble, Turin Chambery, Chambery Grenoble .

Turin Grenoble, . , ( ) P(A)=0,6, P(B)=0,7, P()=0,4, P(/)=0,5, P(A/B)=0,4.() Grenoble ;() Turin Grenoble Chambery;() Grenoble ;.

() Turi Grenoble R ,

)( BAR .

, R, ,

81,014,035,06,04,05,07,05,07,06,0

)(}()()()(6,0

)()()()]([)(

BAPBPBPBPP

BAPBPAPBAPRP

TurinGrenoble

Chambery

A


116

() ,

35,05,07,0)()()( BPPBP

() , Grenoble .

2) . . , . , 0,2 0,80. , 0,10, 0,30 0,60.

, , . , , 60%,. .() ;() . ;

, ,

1=[ },2={ }.

1, 2 S1., S2

, ,

1={ }B2={ }3={ }.

, S S1 S2,


117

S=S1 S2={1, 2} {1, 2, 3}={(1,1), (1,2), (1,3), (2,1), (A2,B2), (A2, B3)}

6 S , Venn. , , .

02,01,02,0)1()1()11()11( BPAPBAPBAP

48,0)32(.24,0)22(,08,0)12(

12,0)31(,06,0)21(

BAPBAPBAP

BAPBAP

() , , ,

136,048,0024,0008,0112,0006,06,002,01

)32()32()22()22()12()12(

)31()31()21()21()11()11()(

BAPBAEPBAPBAEPBAPBAEP

BAPBAEPBAPBAEPBAPBAEPEP

() , , . 1 312111 BABABA . , ,

S 11 12 13

21 22 23


118

412.0136,0

056,0

136,0

12,00

136,0

06,06,0

136,0

02,01

)(

)31()31(

)(

)21()21(

)(

)11()11(

))31()21()11(])312111[(

EP

BAPBAEP

EP

BAPBAEP

EP

BAPBAEP

EBAPEBAPEBAPEBABABAP

312111 BA,BA,BA , ( 3.4)

3) . 1 2 3, . . 1 2 . 1, 2, 3 , ,

.1)12(,60,0)21(,30,0)2(,15,0)1( PPPP, 1 2, 3. , 1 2 , 3 0,25.

, 3.

1 2 , ,

21,21,12,21 .

32

1


119

, 3 ,

).21)213()21()213(

)12()123()21()213()3(

PPPPP

PPPPP

, :

.73,027,01)15,012,00(1)21(

,15,0115,0)12()1()21(

,12,040,030,0))21(1(30,0)21()2()12(

,0015,0))12(1(15,0)12()1()21(

P

PPP

PPPP

PPPP

, ,

153,015,003,073,0015,0112,025,0025,0)3( P

4) . , , . ( ). .

. 0,40 .

0,30. , , 0,10. , , 0,25. () .() , .


120

.

={ },={ },={ }.

, ,

.80,00,180,0)()()(0,00,080,0)()()(

12,060,020,0)()()(,08,040,020,0)()()(

XPPXPXPPXP

XPPXPXPPXP

,

475,025,070,030,0))((

,10,0))((

XEP

XEP

, ( 0,30) ( 0,700,25=0,175).() ,

X,X,X,X S,

05,00,00,080,00,012,010,0080,0475,0

)())(()())((

)())(()())(()(

XPXEPXPXEP

XPXEPXPXEPEP

E


121

() P(E) .

25,00,00,00,00,060,010,040,0475,0

)())(()())((

)())(()())(()(

XPXEPXPXEP

XPXEPXPXEPEP

.0,0({}))(0,0({}))(

60,0)()(40,0)()(

PXPPXP

XPXPXPXP

5) . , . 4 0,20. 0,60.() , ; () , ;

, , . ,

.....,4,06,01)(1)(

6,0)(,6,0)(,6,0)(,6,0)(

2,0)(,2,0)(,2,0)(

BAPBAP

BPBPABPBAP

PBPAP


122

() .. , , . , )AB(P 0,2. .

10,080,0

4,02,0

80,0

)()(

)(

)()( BAPBP

AP

ABPABP

.,)BA(P,,)B(P 1010

() ,

20,0

)(

)(

)()(

AP

AP

APAP

Venn . P() .

08,01,04,02,06,06,02,0

)()()()()()(

)()()]()[()(

ABPABPAPBAPABPAP

BAPBAPBABAPAP

40,020,0

08,0

)(

)()(

AP

APAP

6) . . 60%. , 0,10. ,


123

0,30. . ; . ,

1={ }2={ },

12={ },={ }=(12)(12).R={ }

,

03,030,010,0)12()1()21(

,30,0)12(,30,0)21(,10,0)2(,10,0)1(

AAPAPAAP

AAPAAPAPAP

12, . R ( ) . , , )AAB(P 21 0,60.

)21(

)12(

)21(

)21(

)21(

)]21()12[(

)21(

)]21()21[(

)]21)21[()]21)21[(

)]21)]21()21[[()21(

AAP

AAP

AAP

AAP

AAP

AAAAP

AAP

AAAAP

AAAAPAAAAP

AAAAAAPAABP


124

824,017,0

14,0

03,010,010,0

70,010,070,010,0

)21()2()1(

)21(1)(2(

)21()2()1(

)12(1)(1(

)21(

)21()2(

)21(

)12()1(

AAPAPAP

AAPAP

AAPAPAP

AAPAP

AAP

AAPAP

AAP

AAPAP

,

494,060,0824,060,0)21()( AABPRP

7) . 1 2, . , , 1 2. , 1, 2 ,

40,0)21(,50,0)12(,50,0)2(

,40,0)1(,40,0)2(,30,0)1(,60,0)(

AAAPAAAPAAP

AAPAPAPAP.

() , 1;() ; () 2, 1.

() , . , A S.

1

2


125

1 2 :

30,04,0)1(60,040,0

30,0)()1()()1()1(

AAP

APAAPAPAAPAP

15,0)1( AAP() , ,

12,050,040,06,0)12()1()()21( AAAPAAPAPAAAP .() 1. 2 Bayes ,

75,040,0

60,050,0

)2(

)()2()2(

AP

APAAPAAP

( ) P(A2)=0,75, 1,

3375,025,015,075,040,0)2()21()2()21(

)2())21(()2())21(()21(

AAPAAAPAAPAAAP

AAPAAAPAAPAAAPAAP

8) .

S , I, c, d.

R, S , I, d (). R,

R=d d

126

R=d+(I+S-c), I+S>d+c

,30,0})({,10,0})({,60,0})({ cdSIPSIPcdSIdP .

9) . . 0,02 ( 1). , , , , , ( 2). 20 . 0,08 ( ), , . , , , , . , F, ,, , . 1 , 2 ,() , F, M 1 , 2 ,() ,() , F M.


127

4 , 4.1 4.2 4.2.1 4.2.2 4.3

4.3.1 4.3.2 4.3.3 F(x)4.4 4.5 -

, . , . , , (1,6) (6,1) (5,2) (2,5) (3,4) (4,3). ,


128

.

4.1

. , . , , .., .

, x s S. , x=X(s) X S R, .

() s . X(s)= s, .() . , s S X(s),

S , , s S X(s)=x.

s X

X(s)=x

SRx

4.1


129

4.1. s , . s S . .() , , x , X(s)=x. , , , , ..., , x, y, z, .

4.1 . s . . , - (s)= ,- Y(s)= ,- Z(s)= . , s, (s) {0, 1, 2, 3}. , Y(s), Z(s) , Z s , .

Rx , , S. X(s)= s, S= Rx.

S. , , . S, , , RX. S R.

S s X(s) B, 4.2,


130

. , . , X(s) B , , , X(s) s . , S, RX.

4.2 , , . , ,

S={AAA, AAK, AKA, KAA, AKK, KAK, KKA, KKK}

. , S ()

0={ }={}1={ }={ AKK, KAK, KKA }2={ }={ AAK, AKA, KAA}3={ }={}

0, 1, 2, 3 x1=0, x2=1, x3=2, x4=3 , 4.3. T

X(s)

s

X

S RX

4.2. ,


131

={ }={sX(s)1}={s1,s2,s3,s4}S={(s)1}={x1, x2} RX

.

, , . , RX,

{X=x1}, {Xx1}, {x1 Xx2}

,

P(X=x1) P(Xx1) P(x1 Xx2).

RX

P(B)=P(A)

P(A) A S, A S.

RX,

{X=x1}, {Xx1}, {x1 Xx2}

s5=AAKs6=AKAs7=KAAs8=AAA

s1=KKKs2=KKAs3=KAKs4=AKK

X(s)

RX

x3=2

x4=3

x1=0

x2=1B

4.3 .

S


132

RX S. ,

P(X=x1), P(Xx1), P(x1Xx2),

S.

4.3 4.2, , ,

P({AAA})=1/8,P({AKK, KAK, KKA})=3/8,P({AAK, AKA, KAA})=3/8,P({KKK})=1/8.

A , {=0} ( RX ) {} ( S), {=1} { AKK, KAK, KKA}, {1} { , AKK, KAK, KKA} .... RX,

P(X=0)=P({KKK})=1/8,P(X=1)=P({AKK, KAK, KKA})=3/8,P(>1)=P({AAA, AAK, AKA, KAA})= 4/8.

,

P(1X2)=P({AKK, KAK, KKA , AAK, AKA, KAA})=6/8,P(X2)=P({AAK, AKA, KAA, AAA})=4/8,

.... .

( ) P(X=1)=3/8,

{ AKK, KAK, KKA}={sX(s)=1}

S, 3/8.


133

P(X=1), P(X2), ..., . RX , S . , , , , RX={0, 1, 2, 3}, (1/8, 3/8, 3/8, 1/8), S.

, : , .

4.2

4.2.1

4.4 48 . , . , 0, 1, 2, 3, 48. X

RX={0,1,2,,48}. , X=x=10.

X={ },X={ },X={ },

. ( RX) , . x1, x2, x3, , xn,, n .


134

X={ }

. .

.

. , .

, x1, x2, x3,,, xn, p(xi) ( 4.4)

RX={x1, x2, x3,,, xn,} xi p(xi)=P(X=xi). (xi , p(xi)), i=1, 2, 3, , n,, () .

f(xi)=P(X=xi)=p(xi)=pi.

,

.

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