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(

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)21()21(

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))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

, :

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

P

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

4) . , , . ( ). .

. 0,40 .

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


120

.

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

, ,

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XPPXPXPPXP

XPPXPXPPXP

,

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

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, ( 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

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.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

)()(

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)()( BAPBP

AP

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() ,

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)()(

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Venn . P() .

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

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

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ABPABPAPBAPABPAP

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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(

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AAPAAPAPAP

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124

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70,010,070,010,0

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)21()2()1(

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,

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,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() , ,

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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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