Ι∆ΑΚΤΙΚΕΣ ΗΜΕΙΩΣΕΙΣ - users.uoa.grusers.uoa.gr/~npapadat/BOOKS/PithStatOpen.pdf · ΕΙΣΑΓΩΓΗ ΣΤΙΣ ΠΙΘΑΝΟΤΗΤΕΣ ΚΑΙ ΤΗ ΣΤΑΤΙΣΤΙΚΗ

( )

. , . , . .

2003

v

1

1. 1 2. 1 3. 10 4. , 13 5. 18 6. 19 7. 28 8. 39 9. 43 . 1 48 2

1. H 57 2. 61 3. 65 4. 68 . 2 75 3

1. 79 2. BERNOULLI 79 3. PASCAL 85 4. 93 5. POISSON 97 . 3 103

ii

4

1. 107 2. ERLANG 110 3. 117 4.

POISSON

126 5. 132

. 4 135

5 ,

1. 141 2. 145 3. 147 . 5 154

1 6

1. 157 2. 158 3.

161 7

1. 179 2. 179 3. 187 4. 194 5.

196 6. 200 . 7 201

iii

2 8

1. 213 2. () 215 3. 218 4. 221 5. 222 . 8 227 9

1. 229 2.

231 3. 237 . 9 245 10

1. 253 2. 256 3.

265

. 10 276 283 289 297

. , ( ) . : , , , . , : () , () , , . 1, 1-5 ( ), , (, , ) . 2-4 , 5 ( ) ( , , ), . ( ) , ( 6 7) ( 8-10). , , , . ( ) 8 10 , ( , , , , , ...), . , , ,

vi

, . , () ( 8-10). , , . , . , ( ) , . , , , , . , , , , , . ( ) . , 2003

. , . , ..

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

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

..: )( , )( .

2

)

..: 6,5,4,3,2,1 .

) )(

..: ...,,,, .

) ()

).6,6(...,),2,6(),1,6(............

),6,2(...,),2,2(),1,2(),6,1(...,),2,1(1),(1,:..

)

..: v...,,2,1,0 .

) ( )

..: ...,2,1,0 .

) .

..: .

: .

() . A , A . . BA . BA , . BA .

BA AB . BA = .

.

3

2.1. (..) ( ) ( ) . .

, , . . . . : }...,,,{ 21 N = : ...},,{ 21 =

. ( ) ( ( ) ). , :

) },{ =

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

) ...},,,{ =

) )}6,6(...,),2,1(),1,1{(=

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

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

) ),0[}0:{ +== tt .

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

...},,,{ 321 =

. , () .. ),0[ = 0

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

4

.. ( ).

2.2. . ( ). .

}{ = ,

, . .

, , . . () ( )

ABA = :{ }B ,

. , AAA ...,,, 21

jAAAA = :{21 L }...,,2,1 =j ,

AAA ...,,, 21 .

, ...,...,,, 21 AAA

jAAAA = :{21 LL ...},2,1=j ,

...,...,,, 21 AAA .

() ( )

A = :{ }B ,

. , AAA ...,,, 21

LL 2121

jA = :{ }...,,2,1 =j ,

5

AAA ...,,, 21 . ,

...,...,,, 21 AAA

LLLL 2121

jA = :{ }...,2,1=j ,

...,...,,, 21 AAA .

, = , ( ) . ( )

}:{ AA = ,

. A . ( )

ABA = :{ }B ,

. BABA = .

(). Venn ( ) , . . Venn 2.1-2.4 BA , BA , A = BA .

, , . 1 2 .

1 2 , 21 ,

1

2 ,

}`,:),{ 22112121 ( = .

6

...,,, 21 :

}...,,,:)...,,,{( 22112121 = L .

=== L21 .

A B

2.1: BA 2.2: BA

A

2.3: A 2.4: BA

2.1. () () .

},{ = ,

( ).

}{= }{=

. () () 2 . 2 . 2

7

.

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

2 },{ =

. 2 ,

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

0, 1 2 , .

2.2. () .

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

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

5,4,3,2,1 6

,

}1{1 = , }2,1{2 =B , }3,2,1{3 =B , }4,3,2,1{4 =B ,

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

5,4,3,2,1 6

.

11 AB = , 212 AAB = , 3213 AAAB = , 43214 AAAAB = ,

543215 AAAAAB = , B =6 .

2.3. 3 .

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

. 0 , 1 , 2 3 0, 1, 2 3 , ,

:

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

8

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

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

0321 A == .

() 3

0)},,{( A == .

2.4. , .

() , , ,

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

1

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

() , , :

}0{= .

, .

() , ,

}0:{2 tRt

9

1 .

, . 2 .

2.5. () , )( )( .

, ()

...},,,,{ = .

.

) 4

}{A =

) 4

...},,{ =

) 4

},,,{ = .

2.6. () , 0 1. ,

}111,110,101,011,100,010,001,000{1 = ,

( .. 1 .. 2.3).

() ... 4: , , AB . ,

),(),,(),,(),,(),,(),,(),,(),,(),,(),,{(2 = ,

)},(),,(),,(),,(),,(),,( ,

.. ),(

.

10

3.

De Moivre (1711). O , ( , ...) : , (). 1/2. . () . , ( ). Laplace (1812). , () . ( , ), () ( ). , )(AP ,

NANAP )()( = (3.1)

)(AN )(NN

. )(AP

( ) (3.1)

11

() : 0)( AP A ,

() : 1)( =P ,

() : )()()( BPAPBAP += (

) B .

(3.1) : 0)( AN )()()( BNANBAN +=

, .

)()()()( 2121 APAPAPAAAP +++= LL (3.2)

( , ) AAA ...,,, 21 . (3.1)

1)( AP .

0)( =P .

, = L21 AA = L21 ii A

)(/)()( iii NANAP = , ...,,2,1=i ,

)()()()( 21 APAPAPAP L= . (3.3)

( ) : ( ) () .

)()()(

AAP = , (3.4)

)( )( ( )

. (3.4), , (3.1). 3.1. jA j ,

.2,1,0=j )( jAP , .2,1,0=j

12

()

12

},{ = .

, , :

21})({})({ == PP .

, 2

)},(),,(),,(),,{(2 = ,

},{ = .

(3.3) 4 :

41

21

21})({})({)}),({( === PPP ,

41

21

21})({})({)}),({( === PPP ,

41

21

21})({})({)}),({( === PPP ,

41

21

21})({})({)}),({( === PPP .

, (3.1)

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

41)( 0 =AP , 2

1)( 1 =AP , 41)( 2 =AP .

3.2. r , r < .

.

=)( .

, , ,

2/r (. 3.1). r .

13

))(( rr .

))(()( rr = .

A B

r

r/2

.

r/2

ra

3.1

, (3.4),

=

==

r

r

rr

AP 11))((

)()()( .

, = ,

2

1)(

=

rAP .

4. ,

( , ) , , NANAP /)()( = , )(AN

)(NN .

. ( ), , :

14

. () 1

1 2 2

, 1 2

21 + .

( ). () 1

1

2 2 , 1 2

21 .

...,,, 21

().

v ...,,, 21 .

, 3=v ),,( 321 . 1 21 = (

) , 2 32 = ( )

3 23 = ( ) , 12321 =

21, 3 :

15

- }...,,,{ 21 = .

- )...,,,( 21 r

r ...,,2,1= . ( )

}...,,,{ 21 r , r ...,,2,1= .

. , , . ( ) . .

4.1. () , )( ,

)!(!)1()2)(1()(

=+= L , (4.1)

1 )1(321! = L ( 1)( 0 =v )1!0 =

()

,

)!(!!

!)(

==

. (4.2)

. () )...,,,( 21

}...,,,{ 21 = , 1

, , 2 ,

1 ,

1 . 121 ...,,, ,

, 1

16

, 1)1( +=

. , , (4.1).

() }...,,,{ 21

! , ! . ! (4.1) (4.2).

4.2.

= L . (4.3)

. )...,,,( 21

}...,,,{ 21 = i

. , , (4.3).

4.3.

)!1(!)!1(

!)1()1(1

+

=++

=

+

L . (4.4)

. }...,,,{21 iii

}...,,,{ 21 =

iii ...,,, 21 .

. iii L211

}...,,,{21 iii }...,,,{ 21 jjj

11 ij = , ...,122 += ij , )1( += ij ,

11 21 +

17

4.1. () . }...,,,{ 21

}...,,,{ 21 ccc .

,

,

, . j j

j ...,,2,1= =+++ L21 ,

!!!!

21 L

,

1

1

. 2

1

2

1

.

, 1 , -

=+++ )( 121 L ,

, ,

11

2

1

1

LL

)!(!

)!()!(!

)!()!(!

!

1

11

212

1

11

=

L

LL

.

() . }...,,,{ 21 ccc .

, }...,,,{

21 iii ccc ,

18

. , ( )

,

. ,

+ 1

,

. 5. . ( ) , , . ( ) , , . Von Mises . ( ) A . ( )

)(n . ,

n )( .

19

, Von Mises, :

nAP

)(lim)(

= .

(5.1)

, ,

() : 0)( AP A

() : 1)( =P

() : )()()( BPAPBAP += (

) .

. (5.1) . 6. , , . .

6.1. () ( ). A () )(AP

():

() ,

0)( AP A ,

() ,

1)( =P ,

() ,

20

LLL ++= )()()( 2121 APAPAAAP L++ )( AP

( ) Ai , ...,...,,2,1 i = .

6.1.

() : )()()( BPAPBAP += (

) BA , ,

)()()()( 2121 APAPAPAAAP +++= LL ,

( ) Ai ,

i ...,,2,1= .

() () )(AP

A . )(AP , A .

() . .

6.1. . }...,,,{ 21 =

=)( iii = }...,,,{ 21 .

)(AP

:

ii pP =})({ , Ni ...,,2,1= .

, }{}{}{21 iii A = L ,

,

})({})({})({)(21 iii PPPAP +++= L

iii pppAP +++= L21)( .

21

,

Nppp +++= L21)( ,

121 =+++ Nppp L .

, , . . ,

NPp ii

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

)(AP

NANAP )()( = ,

.

6.2. .

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

6)( == .

() , ( ), :

61})({ == jPp j , 6,5,4,3,2,1=j .

H

6)()( ANAP = ,

. , 5, }6,5{=A 2)( =AN ,

31)( =AP .

22

() , ( ) ,

cjjPp j == })({ , 6,5,4,3,2,1=j ,

c . 1654321 =+++++ pppppp ,

1)654321( =+++++c 21/1=c .

jjjA = }...,,,{ 21

21)( 21

jjjAP

+++=

L.

5, }6,5{=A

2111

2165)( =+=AP .

6.3. ABOBA ,,,

40% 14%, 42% 4%, . , . ,

%8282.042.040.0})({}))({}),({ ==+=+= OPAPOAP .

, , 18% ,

%1818.004.014.0})({})({}),({ ==+=+= ABPBPABBP .

(), () () .

6.1. () , ,

0)( =P . (6.1)

() Ai , i ...,,2,1= ( )

,

23

)()()()( 2121 APAAPAAAP +++= LL (6.2)

() A , ,

)(1)( APAP = . (6.3)

() A,B ,

)()()( ABPAPBAP = (6.4)

AB ,

)()()( BPAPBAP = . (6.5)

() BA, ,

)()()()( ABPBPAPBAP += (6.6)

)()()(1)( ABPBPAPBAP += . (6.7)

. () =iA , ...,2,1=i , = LL AAA 21

()

LLLL ++++== )()()()()( 2121 APAPAPAAAPP

LL ++++= )()()( PPP .

, () 0)( P .

,

0)()( =+++ LL PP ,

, 0)( =P .

() =iA , ...,2,1 ++= i .

() (6.1)

)()( 12121 LLL = + AAAAPAAAP

)()()()()()()( 21121 APAAPAPAPAPAP +++=+++++= + LLL .

() A ( ), = AA , AA = . (6.2) 2= () 1)()()( ==+ PAPAP ,

(6.3).

24

() BA == ABBA = :

=== ABBABABA )()()(

AABBABABA === )()()( .

, (6.2) 2= ,

)()()()()]()[()( ABPBAPBAPBAPBABAPAP +=+==

)()()()( ABPAPBAPBAP == .

AB BAB =

)()()( BPAPBAP = .

() BABA = , = BBA )( ,

BABBA = )( . (6.2),

)()(])[()( BPBAPBBAPBAP +==

(6.4) (6.6). )( = BABA ,

(6.3) (6.7).

6.2. () AAA v ...,,, 21 . ,

.6.1, ,,

(. 11):

i) )()()()()()()()( PBPAPBAP +++= ,

ii) )()()()()()()(1)( +++= .

6.1:

25

6.2. AAP ),( , ]1,0[ :

1)(0 AP A (6.8)

:

)()( PAP BA , BA . (6.9)

. , () ,

0)( AP , 0)( AP A

(6.3), )(1)( APAP = , (6.8).

, () , AB ,

0)( ABP ,

(6.5),

)()()( APBPABP = ,

BA , (6.9).

6.1 , .

6.3. . ,

)(BP .

B . )(BP

)(BP . ,

8 (. 2.3) B

81)( =BP

(6.3)

87

811)(1)( === BPBP .

26

)(BP

21, AA 3A

2,1 3 , .

87

81

83

83)()()()()( 321321 =++=++== APAPAPAAAPBP .

6.4. . . 365 , 366 . . , , 365 . , . )...,,,( 21 iii 365

}365,...,2,1{ , ri r ,

r ,...,2,1= . , , 365)( = .

. A . )(AP

)(AP . , A

)...,,,( 21 iii 365 }365,...,2,1{ ( )

)365()( = . (6.1),

AP

365)365(

)( =

(6.3) :

APAP

365)365(

1)(1)( == .

23= , 2/15073.0)( >=P .

6.5. 10 0 9 . . 3 () .

27

5. 5 BA }5,4,3,2,1,0{

}4,3,2,1,0{ . AB (6.5)

)()()( BPAPBAP = .

3

310)( =N , 10 }9...,,2,1,0{

3 , 36)( = , 6 }5,4,3,2,1,0{ 3

. 35)( =

091.0105

106)( 3

3

3

3== BAP .

6.6. (). 0 1 ( ).

0 1, . BA 0 1 ( ) (6.7),

)()()(1)( ABPBPAPBAP += .

39)( =AN ,

9 }9...,,2,1{ 3 ,

39)( =BN , 9

}9...,,3,2,0{ 3 AB 38)( =ABN , 8 }9...,,3,2{ 3

.

054.0108

10921)( 3

3

3

3=+=BAP .

6.7. 2,1,0 3

50%, 30%, 10% 10%, . 5

28

. 2,1 3.

1, 2 3. )( BAP 6.2 (ii)

)(1))(()( BAPBAPBAP ==

)()()()()()()(1 ABPBPAPABPPBPAP +++= .

.

()( PAP = 1 5 )

55 )7.0())}1{(1( == P .

5)9.0()()( == PBP . )(ABP :

()( PABP = 1 2 5 )

555 )6.0()1.05.0())}3,0{(( =+== P .

, 5)6.0()( =AP 5)8.0()( = .

,

()( PABP = 1 2 3 5 )

55 )5.0())}0{(( == P .

,

5555555 )5.0()8.0()6.0()6.0()9.0()9.0()7.0(1)( +++=

%29.101029.0 == . 7. , , () . , )(AP

A . A . , , ( )

29

AB= | :

() . , )|( ABP ,

( ), , , )(AP

)(ABP .

.

7.1. 5 1 5. 1 2 3, 4 5 .

() . :

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

: }2,1{=A . , ,

52)( =AP ,

53)( =AP .

() , , . O )(BP

. , )(BP ,

. )|( ABP ,

. , )|( ABP

)(AP )(ABP ,

() , ( ) .

() , , . 20)5()( 2 == NN

:

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

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

30

To ( ), , 8)( =AN :

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

o ( ), , 8)( =BN :

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

, ,

52

208)()( ===

NANAP ,

() . , .

)}1,2(),2,1{(=AB

2)( =ABN .

41

82

)()()|( ===

ANABNABP .

,

NABNABP )()( = ,

NANAP )()( =

)()()|(

APABPABP = .

, , ( )

52

208)()( ===

NBNBP .

31

, , , (. 7.3).

.

7.1. () ( ) A 0)( >AP . ,

, )|( ABP , B , :

)()()|(

APABPABP = , B . (7.1)

0)( =AP , )|( ABP . B

)|( ABP .

)|( ABP , B ,

, () :

0)|( ABP B ,

() :

1)|( =AP ,

() :

LLLL ++++= )|()|()|()|( 2121 ABPABPABPABBBP

( ) ,...,...,2 ,1 , = iBi

. ()

)|()|()|()|( 2121 ABPABPABPABBBP +++= LL

( ) ,...,2 ,1 , = iBi .

. (7.1)

)|()()( = . (7.2)

.

32

7.1. ( ). Ai

,...,2 ,1=i , 0)( 121 >AAAP L .

)|()|()|()()( 12121312121 = AAAAPAAAPAAPAPAAAP LLL . (7.3)

.

121221121 AAAAAAAAA LLL ,

)()()()( 121221121 APAAPAAAPAAAP LLL

0)( 121 >AAAP L ,

0)( 1 >AP , 0)(...,,0)( 12121 >> AAAPAAP L .

(7.3) (). (7.1)

)()(

)|(1

2112 AP

AAPAAP = ,

)()(

)|(21

321213 AAP

AAAPAAAP = ,,

)()(

)|(121

121121

=

AAAP

AAAAPAAAAP

L

LL

)(

)()(

)()(

)()()(

121

21

21

321

1

21121

=

AAAP

AAAPAAP

AAAPAPAAP

APAAAPL

LLL

)|()|()|()( 121213121 = AAAAPAAAPAAPAP LL .

7.2. 1 r . . . jA j

...,,2,1=j . AAA L21

, (7.3),

)|()|()()( 12112121 = AAAAPAAPAPAAAP LLL

)()(

11

11

r

r

r

r

=++

= L .

33

Lotto 49= 6= . r . 6=r , 6

810700000007.013998816

1 ==p .

. :

}...,,,{ 21 AAA Ai , i ...,,2,1= ,

, = ji AA , ji , ,

AAA = L21 , .

7.2. ( , ...). }...,,,{ 21 AAA 0)( >AP ,

...,,2,1= ,

=

=

1)|()()( ABPAPBP . (7.4)

.

BABABABAAAB === LL 2121 )( ,

BA = , ...,,2,1=

ji , == BAA jiji )( (. 7.2).

BA1 BA2 BA3 BAvL

1A 2A 3A L vA

7.2

, ,

34

)()()()( 21 BAPBAPBAPBP +++= L .

0)( >AP , (7.2),

)|()()( ABPAPBAP = , ...,,2,1= ,

)|()()|()()|()()( 2211 ABPAPABPAPABPAPBP +++= L .

7.3. ( () Bayes). A }...,,,{ 21 AAA

0)( >AP , ...,,2,1=

0)( >BP ,

=

=

rrr

ABPAP

ABPAPBAP

1)|()(

)|()()|( , ...,,2,1=r . (7.5)

.

=

==

rrrr

ABPAP

ABPAPBP

BAPBAP

1)|()(

)|()()(

)()|( , ...,,2,1=r .

7.1. ) )( AP , ...,,2,1= ,

, (a priori) , )|( BAP r ,

, (a posteriori) .

) (...) Bayes 2=v , AA =1 AA =2 ,

1)(0 BP

)|())(1()|()()|()()|(

ABPAPABPAPABPAPBAP

+= , )|(1)|( BAPBAP = .

) 2

vAAA ...,,, 21 , (7.4) (7.5)

35

, vAAB L1 (.

)...,,1 vAA .

) (7.4) (7.5) ...,...,,, 21 vAAA ( . =v ).

7.3. 25 . 3 2 . .

() ,

253)( =AP ,

253)( =BP .

() ,

253)( =AP

... :

253

243

2522

242

253)|()()|()()( =+=+= ABPAPABPAPBP .

7.4. 5% . 30% 1% , . () , () () . , .

05.0)( =AP , 95.0)(1)( == APAP , 30.0)|( =ABP , 01.0)|( =ABP .

() :

0150.030.005.0)|()()( === ABPAPABP .

36

() ... :

0245.001.095.030.005.0)|()()|()()( =+=+= ABPAPABPAPBP .

() Bayes :

6122.001.095.030.005.0

30.005.0)|()()|()(

)|()()|( =+

=

+=

ABPAPABPAPABPAPBAP .

7.5. , AIDS, : , : : , 25%, 25% 50%, . 5% , 1% 1. () AIDS; () AIDS, ; . () 05.0)|( =A , 01.0)|( = 001.0)|( = ,

...

)()|()()|()()|()( ++=

50.0001.025.001.025.005.0 ++=

%55.10155.00005.00025.00125.0 ==++= .

1.55% . ) , )|( ,

Bayes:

3125

0155.00125.0

0155.025.005.0

)()()|()|( ====

.

,

316)|(1)|( == .

(prevalence) (??) :

tt

""#""#

=

37

# = .

, 100 10 .. 4 ()

%4100/4 = %10100/10 = . . . , . , (sensitivity) (specificity) . :

+ : : + : : .

dcba +++= :

+

+ a b ba + c d dc +

ca + db + dcba +++

() :

ca

aATP+

== ++ ]|[ , db

dATP+

== ]|[ .

38

, . , (predictive value) :

ba

aTAP+

== ++ ]|[

dc

dTAP+

== ]|[ .

, Bayes :

)()|()()|(

)()|()|(++++

+++++

+=

APATPAPATPAPATPTAP

)-)(1-(1)()(

)()(+

=

dcba

caAP+++

+== + )( .

, . :

db

bATP+

== + ]|[

ca

cATP+

== + ]|[ .

7.6. () 5%. 80% 10% .

39

;

+A , A

, +T ( ) T ( ), :

05.0)( =+AP , 95.005.01)(1)( === + APAP ,

20.0)|(1)|(,80.0)|( === +++++ ATPATPATP ,

90.0)|(1)|(,10.0)|( === ++ ATPATPATP .

Bayes,

)()|()()|()()|()|(

++++

+++++

+=

APATPAPATPAPATPTAP

%3030.090.010.005.080.0

05.080.0=

+

= .

, ( , a posteriori) 30% , a-priori ( ) 5%. 8. BA , .

() , =AB , 0)|( =ABP ,

, () ,

BA , 1)|( =ABP ,

. . ,

)()|( BPABP = .

. , ,

40

)|()()|()()( BAPBPABPAPABP == ,

)()(

)|()()()()|( AP

BPABPAP

BPABPBAP === ,

)()()( BPAPABP = .

. .

8.1. () ( ) BA , .

)()()( BPAPABP = . (8.1)

8.1. , B . : () () B .

)()()( ABPAPBAP = , )(1)( BPBP =

,

)()()( BPAPABP = ,

:

)()()](1)[()()()()()()( BPAPBPAPBPAPAPABPAPBAP ==== .

A , , A , A B , ( 10).

8.1. 3 . . . AB . :

41

83)( = BAP ,

43

821)(1)( === APAP ,

21)( =BP .

E (8.1) .

.

AAA 321 ,,

)()()( 2121 APAPAAP = ,

)()()( 3131 APAPAAP = , (8.2)

)()()( 3232 APAPAAP = .

1A 2A 3A -

1A 32 AA (. 8.2).

, (8.2),

)()()]([ 321321 AAPAPAAAP = , (8.3)

)()()()( 321321 APAPAPAAAP = . (8.4)

, (8.2), (8.4), (8.3),

)()()]([ 312312 AAPAPAAAP = , (8.5)

)()()]([ 213213 AAPAPAAAP = . (8.6)

.

8.2. () ( ) AAA ...,,, 21 . AAA ...,,, 21

( )

)()()()(2121 iiiiii APAPAPAAAP LL = (8.7)

}...,,,{ 21 iii }...,,2,1{

...,,3,2= .

, 3= (8.2) (8.4).

42

8.2. . 1A

, 2A

3A

42

.

21 , AA 3A .

366)( 2 ==N ,

6 () }6...,,2,1{ 2 .

),3,4(),2,4(),1,4(),6,2(),5,2(),4,2(),3,2(),2,2(),1,2{(1 =A

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

)6,3(),4,3(),2,3(),6,2(),4,2(),2,2(),6,1(),4,1(),2,1{(2 =A ,

)}6,6(),4,6(),2,6(),6,5(),4,5(),2,5(),6,4(),4,4(),2,4( ,

)5,3(),3,3(),1,3(),6,2(),4,2(),2,2(),5,1(),3,1(),1,1{(3 =A ,

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

321323121 AAAAAAAAA ===

)}6,6(),4,6(),2,6(),6,4(),4,4(),2,4(),6,2(),4,2(),2,2{(= .

,

21

3618)()()( 321 ==== APAPAP ,

41

369)()()( 323121 ==== AAPAAPAAP ,

41

369)( 321 ==AAAP

),()()( 2121 APAPAAP = )()()( 3131 APAPAAP = , )()()( 3232 APAPAAP = ,

)()()()( 321321 APAPAPAAAP .

21, AA 3A

.

8.3. . jA j

43

(), 3,2,1=j . 21, AA 3A

.

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

)},,( ),,,( ),,,( ),,,{(1 =A ,

)},,( ),,,( ),,,( ),,,{(2 =A ,

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

)},,( ),,,{(21 =AA , )},,( ),,,{(31 =AA ,

)},,( ),,,{(32 =AA , )},,{(321 =AAA .

,

21

84)()()( 321 ==== APAPAP ,

41

82)()()( 323121 ==== AAPAAPAAP ,

81)( 321 =AAAP

),()()( 2121 APAPAAP = )()()( 3131 APAPAAP = , )()()( 3232 APAPAAP = ,

)()()()( 321321 APAPAPAAAP = .

21, AA 3A .

9. () . 1 2 . (

) () . ( )

44

} , :),{( 22112121 = .

. ,

=1 =2 ,

,

}2 ,1 , :),{( 212 == ii .

iiA ( i ),

2 ,1=i . 21 ,

, 21 Bi , 2 ,1=i , 211 AB =

212 A =B . 1B 2B

, . , =1 =2 1B 2B

, . iB

i- ( i- ), 2 ,1=i .

. :

1 2

)()()( 2121 BPBPBBP = (9.1)

211 AB = 212 A =B (

21 ) , .

, ().

)(BP 21 B

(9.1) ( ). , )( iAP iiA 2 ,1=i

)(BP

21 B (9.1). ,

45

, .

1 2 .

)(BP 21 B

)( iAP iiA 2 ,1=i ,

, : , (9.1), )},{( 21 21 :

})({})({)}),({( 2121 PPP = .

)(BP 21 B , ,

,

=B

PBP),(

2121

)}),({()(

.

211 AB = 212 A =B ,

)()( 11 APBP = , )()( 22 APBP = .

)()()( 2121 APAPAAP =

)()()( 2121 BPBPBBP = .

, , ( ).

9.1. 5 . 2 . , ,

}6,...,2 ,1 ,6,...,2 ,1 :),{( === jiji ,

46

366)( 2 ==N .

,

}5,...,2 ,1 ,6,...,2 ,1 :),{( =++== iiijjiA ,

15)( =AN .

A , },1 {= .

125

3615})({ === Pp ,

127

3621})({ === Pq .

, 5

},{:),,,,{( 543215 = i , }5,4,3,2,1=i .

To 5 ():

== iB :),,,,{( 54321 }}5,4,3,2,1{i

k5

,

5 . , 5 ,

})({})({})({})({})({)}),,,,({( 5432154321 PPPPPP =

=

5

127

125 .

)(BPp =

=

5

127

1255p , 5...,,1,0= .

2 , 2Q ,

2 ,

6913.02412.00675.01127

1255

12711

45

102 ==

== ppQ .

9.2. N Mendel. .

47

, , . , ( ). , , . . , p, 2q r 12 =++ rqp

.

, : 21, AA 3A

, , 21 , BB 3B

, , . ( ) , . , ,

qpqpAAPAPAAPAPAP +=+=+=2121)|()()|()()( 2211

rqqpAPAP +=+== )(1)(1)(

12 =++ rqp .

qpBP +=)( , rqBP +=)( .

21 , 3

( ) , , . AB =1 , BABA =2

BA =3 . ,

B A A B (. 8.1).

21 )()()()()( qpBPAPABPP +===

48

)()()()( 2 BAPBAPBABAPP +==

))((2)()()()( rqqpBPAPBPAP ++=+= ,

23 )()()()()( rqBPAPBAPP +=== .

. 1 1. , ,

. . . BABA ,, BA .

2. 4 : A , ( ) . .

3. ,

)(4)(3)(2 ABPBPAP == , 2/1)( = ABP .

)(AP , )(BP )(ABP

)( BAP , )( BAP , )( BAP , )( BABAP .

4. 4/3)( =AP , 3/2)( =BP 5/3)( =ABP

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

5. A 5/1)(2)( += APAP )(AP .

6. 3

)()(2)( BPAPABP += , )()( BPAP = .

49

7. 2/1)( =AP , 3/1)( =BP 3/2)( = BAP

. 2/1)( =AP , 5/1)( =BP

5/3)( =BAP .

8. 4/3)( = 8/3)( = , :

() 4/3)( ,

() 83)(

81

.

3/1)( = , 4/1)( = .

9. De Morgan: () vv AAAA = LL 11 )( ,

() vv AAAA = LL 11 )( .

10. , () B , () A , () A B .

11. (i) (ii) 6.2 6.1.

12. . ( ) , 6,5,4,3,2,1= .

13. 10 . , 10...,,1,0= ;

14. ( de Mr). , 4 , 24 ;

15. . .

16. A . - . () , () rrr ...,,, 21

...,,2,1 , , rrr =+++ L21 .

17. . r

50

18. }...,,,{ 21 =

. })({2})({ 1+= ii PP , 1...,,2,1 = i ,

})({ iP , ...,,2,1=i .

}...,,,{ 21 = , .

19. 5 . )6,5( , )5,6(

)6,6( .

20. 0.001. 0.97. ;

21. , , 80% , . , , 60% , 99% . . 1% , 30% . ;

22. : , : : 50 . : 30.0)( =AP , 25.0)( =BP , 40.0)( =P , 15.0)( =ABP ,

20.0)( =P , 10.0)( =P 05.0)( =P .

)( BAP , )( BAP , )( AP )( BAP .

23. },...,,,{ 121

.

51

1

}...,,,{ 121 .

24. 10 . 100/ , 10...,,2,1= .

55/ 10...,,2,1= .

, 10...,,2,1= ;

25. 4/5. 3/5 2/5 .

26. : , : : . ,

)()|( BPABP < , )()|( PAP > , )()|( PP = .

27. . . . 28. 0.01. 0.95. () () .

29. 0 1 2 1/4, 1/2 1/4, . , . A

, 2,1,0=

rB r,

6...,,1,0=r , )( 0BP , )( 1BP )|( 12 BAP .

52

30. , , 50%, 40% 40%, . , 35% . 25% 20%. , 15% . ;

31. 1A 1 1 , 2 2

2 . ( ) 1

( ) 2 . 2

1A . , 1A .

() ;

() , ;

32. ( ) . () ; () ,

; ( !)

33. )521( v ,

4/52, v ...,,2,1= .

34. 1 . , - ( ). 1/ v ...,,1= .

35. , s ( ), 11 + s ;

36. () . () . , 95%; 95%;

53

37. , . , .

38. ( Bonferroni). vAA ...,,1 ,

1)()()...( 11 +++ vAPAPAAP vv L .

39. 75% . 20% , 10%. , ; ;

40. . 2 1 , 1 2 . , ( !), ( ) ( ). () ; () ,

;

41. , p, 2q r 12 =++ rqp

. ( ) , Mendel, . .

42. 7% 2% . 48% 52% .

43. 50% . 1/2. 1/3.

44. 95% , () 95%

54

, () 5% . )

10%, ;

) , () . , . ;

45. , ( ). , 10%, 40% 50%, . , . , , . , 1/2. . () , ; () ,

;

46. () ( ). () . ()

0)( =AP 1)( =AP .

47. ( v2 ) . ; r )2( rvr ;

48. v...,,2,1 x . ,

y x...,,1 . y

1; y 1, x 1;

49. , , :

55

() B . () B . () B .

50. {=A

}, =B { }, {=

}. BA ,, (, ,

, A, B, ), , B, .

51. 5%. 80% , 10% . , ;

52. 21 , A 3A .

() 1A 32 AA , () 2A 31 AA () 3A 21 AA

.

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

1.1. () . (..). )(Xx = .

WZYX ,,, XXX ...,,, 21

wzyx ,,, xxx ...,,, 21 .

RRX

() ( ). ],( x .

XRB ( )

. .

1.2. F

}))(:({)()( xXPxXPxF == ,

58

(..) (...) .. .

.. XF x )(xFX .

, , ]1,0[ :

1)(0 xF ,

59

==

=),(,2

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

)(

X

}2,1,0{=XR

. F .. :

60

X =)( , , .

F :

61

1.3. x ]1,0[ 1)10( = XP

]1,0[ (

0 1). ],( 21 xx 10 21 xx

12 xx ,

)()( 1221 xxcxXxP =< ,

c .

0)0( =

62

2.1. ( ) , 1, ( )

...},...,,,{ 10 xxxRX = . f x , ...,2,1,0= ,

}))(:({)()( xXPxXPxf ==== , ...,2,1,0= , (2.1)

.

.. Xf x )( xf X .

, = L}{}{ 10 xxRX

L}{ x , )( XRXP

1)(0

==

=xXP .

, ,

0)( xf , ...,2,1,0= 0)( =xf , XRx (2.2)

==

01)(

xf . (2.3)

, }...,,,{ 10 xxxRX = , (2.3)

=

=

01)(xf .

)()( xXPxf == , ...,2,1,0=

)()( xXPxF = ,

63

)()( 00 xFxf = .

=xx

xfxF

)()( ,

64

0)( 0 == xXP )( 0xf

. , (2.9), . 0>x

xxfxxXxP )()( +< .

2.1. , 1.1, . }2,1,0{=XR (). ..

:

41)}],[{()0()0( ==== PXPf ,

21)}],(),,[{()1()1( ==== PXPf ,

41)}],[{()2()2( ==== PXPf .

=

=++=2

0

141

21

41)(

x

xf ,

.

2.2. x ]1,0[

(. 1.3). .

1.3

65

2.3.

2

)(2)(

xxf = , x 0 ,

0> . )(xF ,

66

})(:{ yxgx .

)(xgy = XR

YR .

)(1 ygx = .

yxg )( )(1 ygx ,

)(xgy = )(1 ygx , )(xgy =

))(()]([)( 11 ygFygXPyF XY == ,

)(xgy =

))((1)]([1)]([1)]([)( 1111 ygFygXPygXPygXPyF XY ==

67

xxgy +== )( ,

0 , yygx /)()(1 ==

dyydg /1/)(1 = . .

67

3.1. )(xf X ,

XRx . Y += , 0 ,

||1)(

yfyf XY

= , YRy . (3.2)

3.1.

= 2

2

2)(exp

21)(

xxf X ,

68

})(:{ yxgx ,

, . .

3.2. () )(xf X , XRx )(xFX , Rx .

2XY = .

0

69

. . . .

4.1. () )()( xXPxf == , ...,2,1,0= . ,

)( ,

=

=0

)()(

xfxXE . (4.1)

() )(xf ,

70

=

==+++++

===6

1 27

621

6654321

61)(

xxXE .

, , .

4.2. ],[ . )(XE .

..

xf

21)( = , x .

, 4.1 (),

=

====

xxdx

dxxxfXE 042

1)()(2

.

0)( = , .

. .

, )()( xXPxf X == , ...,2,1,0= )(xf X ,

71

. (. 4.2) . , ,

2)()( = XXg ,

)(XE= (4.3)

(4.4) .

4.2. )(XE= .

, )(XVar 2X 2

,

])[()( 22 XEXVar = . (4.5)

)(XVar ,

)(XVar X = (4.6)

.

(4.3) (4.4), .. )()( xXPxf == , ...,2,1,0= ,

==

0

2 )()()(

xfxXVar ,

.. )(xf ,

= dxxfxXVar )()()( 2 .

.

.

4.1. XE =)( , 2)( XVar = ,

.

XE +=+ )( , (4.7)

)]([)]([)]()([ XhEXgEXhXgE +=+ , (4.8)

72

22)( XVar =+ , (4.9)

22 )()( XEXVar = . (4.10)

. )()( xXPxf == , ...,2,1,0= . (4.3)

=

=

=

+=+=+=+0 0 0

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

XExfxfxxfxE

=

=

=

+=+=+0 0 0

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

xfxhxfxgxfxhxgXhXgE

)]([)]([ XhEXgE += .

, (4.4), (4.7) (4.8). 4.2 (4.7), (4.8),

])]()[[()( 2XEXVar ++=+

222222 ])[(])([ XEE === .

2)( X , ..

)(XE= , (4.7) (4.8)

)2()()2(])[()( 22222 XEXEXXEXEXVar =+==

2222 )()(2)( XEXEXE =+= .

4.1. () .

=)(XE 0)( 2 >= XVar ,

=XZ (4.11)

, (4.7),

0/])([]/)[()( === XEXEZE

, (4.9),

1/)(]/)[()( 2 === XVarXVarZVar .

73

(4.11) .

() 0)( =XVar ,

c 1)( == cXP .

4.2. . (4.10) , . ,

])[( 2)2( XE= ,

)1()( 2 = XXX .

22 ])[()( XEXVar += . (4.12)

(4.10)

XEXXEXXEXE ==== )()()]1([])[( 222)2( .

4.3. . )(XVar .

..

61)()( === xXPxf , 6...,,2,1=x .

, (4.3)

=

=+++++

==6

1

22

691

6362516941

61)(

x

xXE .

(4.10) (. 4.1) 2/7)( == XE ,

1235

449

691

27

691)()(

222 ==

== XEXVar .

4.4.

xf

21)( = , x .

(4.4)

74

=

===

xdxx

dxxfxXE362

1)()(23

222

(4.10) (. 4.2) 0)( == XE

3/)()( 22 XEXVar == .

4.5.

2

)(2)(

xxf = , x 0 ,

0> (. 2.3). )(XE

)(XVar .

)(XE , 4.1,

=

===

332)(2)()(

00

2

32

2

x

xdxxx

dxxxfXE

.

(4.4),

=

===

6232)(2)()(

2

00

2

432

222

x

xdxxx

dxxfxXE

1896)]([)()(

22222 XEXEXVar === .

4.1. XR g

XR )]([ XgE .

(i) xg )( XRx

XgE )]([ . (4.13)

(ii) xg )( XRx

XgE )]([ . (4.14)

(iii) xg )( XRx

XgE )]([ . (4.15)

75

(iv) h, )]([ XhE , )()( xgxh

XRx

)]([)]([ XgEXhE . (4.16)

. (iv) )()( xXPxf == , ...,2,1,0= . (4.3)

)]([)()()()()]([0 0

gExfxgxfxhhE

==

=

=,

)()( xgxh , K,2,1,0= , (4.16).

, (4.4).

(i) (4.16) xh )( .

(ii) xg )( XRx , (4.13),

XgEgE = )]([)]([ ,

(4.14).

(iii) (4.13) (4.14).

A . 2

1.

77

() c )3/10(1

3

1. , 2. . () () . , . . . . 2. BERNOULLI 2.1. Bernoulli . A , ),( AA ,

= AA AA = . A . },{ = .

Bernoulli.

pP =})({ , qpPP === 1})({1})({ , (2.1)

.

2.1. Bernoulli p ( pq = 1 ). (-)

(-) Bernoulli p. ( ))(~ pbX .

80

, Bernoulli .

2.1. Bernoulli p

xxqpxXPxf === 1)()( , 1,0=x . (2.2)

81

() . p. ( )),(~ pvbX .

, .

2.2. , . yx, ,

=

=+

v

k

kvkv yxkv

yx0

)( , ...,2,1=v .

.

),(),(),()())(()( 21 yxpyxpyxpyxyxyxyx vv LL =+++=+ ,

yxyxpp ii +== ),( , vi ...,,2,1= .

kvkvv yxyxx ...,,, 1 , vv yxy ,..., 1 , kvk yx vk ...,,1,0= . ,

, vyx )( +

=

=+v

k

kvkkv

v yxCyx0

,)( ,

=kvC , kvk yx .

kvk yx k

vpp ...,,1 , x ( ,

kv y). ,

=kvC , k

=

kv

pp v...,,1 ,

4.1 () . 1.

.

2.3. p

xxqpx

xXPxf

=== )()( , x ...,,2,1,0= (2.5)

83

(2.5) 20...,,2,1= 50.0...,,10.0 ,05.0=p .

5.0>p 5.01

84

222 )1()()1( pqpp =+= .

,

pqpppXEXVar =+=+== 222222 )1(])[()( .

2.1. ),( 11 zy ,

),(...,),,( 22 zyzy . zy >

, zy , ...,,2,1= .

p pq =1 2/1=p .

. , ,

===

21)()(

x

xXPxf , x ...,,2,1,0= .

( ( ), ( )). () 2 () 7

8= .

=

=++=

=

2

0

8 1445.01094.00312.00039.0)5.0(8

)2(x x

XP ,

=

=+=

=

8

7

8 0351.00039.00312.0)5.0(8

)7(x x

XP .

2.2. AAA, qp 2 , r ( 12 =++ rqp ),

. ( ) , Mendel, . ( ) AA . 1

AA , Bernoulli (. 9.2 . 1)

85

211 )()(})({ qpPPp +=== ,

211 )(1)(})({ qpPPq +=== .

Bernoulli AA

xxqpx

xf

= 11)( , x ...,,1,0= .

4/1=== rqp , 4/11 =p , 4/31 =q ,

AA , 4= ,

xx

xxf

=

4

43

414)( , 4,3,2,1,0=x .

4 AA

6836.0256175

431)0(1)1(

4

==

=== XPXP .

AA

1414)( === XE .

3. PASCAL 3.1. 3.1. Bernoulli p ( q),

pP =})({ , pqP == 1})({ ,

() . . p. ( ))(~ pGX .

.

3.1. p

86

1)()( === xpqxXPxf , ...,2,1=x (3.1)

87

.

=

=

===1 1

11)(x x

xx xqpxpqXE

=

=

===2 2

2122)2( )1()(])[(

x x

xx qxxpqpqxXE .

, q

=

=0

1)1(x

x qq ,

=

=1

21 )1(x

x qxq ,

=

=2

32 )1(2)1(x

x qqxx .

pqpxqpXE

x

x 1)1(

)(1

21

=

=

=== ,

22

32

2)2(2

)1(2)1(])[(

pq

qpqqxxpqXE

x

x =

===

=

,

222

22 112])[()(

pq

pppqXEXVar =+=+== .

. .

3.3. (3.1).

)()|( rXPXrXP >=>+> , ...,2,1 ,0, =r . (3.4)

. })(:{ rX +>

})(:{ X > , })(:{ rX +>

})(:{ X > (3.2),

88

)()(

)(),()|(

XPrXP

XPXrXPXrXP

>+>

=>

>+>=>+>

rr

qq

qF

rF==

+

=+

)(1)(1

rqrFrXP ==> )(1)(

(3.4).

: r ( ) r . () .

3.1. 1= XY (3.1) :

yY pqyXPyYPyf =+==== )1()()( , ...,2,1,0=y . (3.5)

.. p. H (3.3):

pqXEXEYE === 1)()1()( , 2)()1()( p

qXVarXVarYVar === . (3.6)

3.1. 100 . , 20 .

5/4=p . ()

4 () .

()

1

51

54)()(

===

x

xXPxf , ...,2,1=x

91

prXE == )( , 2

2 )(prqXVar == . (3.10)

. ..

=

==rx

rxr qprx

xXE11

)( ,

,

=

=

=

rxx

rrxr

xrrxr

xxrx

x)!(!

!)!()!1(

)!1(11

(3.9),

prqrpq

yyr

rpqrx

xrp rr

rx

y

y

rrxr ==

+=

=

=

=

10

)1( .

..

=

+=+=rx

rxr qprx

xxXXE11

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

,

+

+=+

++=

+=

+rx

xrr

rxrxrr

rxrxxx

rx

xx1

)1()!()!1(

)!1()1()!()!1(

)!1()1(11

)1(

(3.9),

=

=

+++=

+

+=+=rx y

yrrxr qyyr

prrqrx

xprrXXE

0]2[

1)1(

1)1()]1([

22 )1()1()1( +=+= prrqprr rr .

..

22

2

222 )1()]1([)(

prq

pr

pr

prrXXEXVar =+=+== .

3.2. r- Bernoulli p. rXY = (3.7) .

92

yrY qpy

yryrXPyYPyf

+=+====

1)()()( , ...,2,1,0=y . (3.11)

.. Pascal r p. (3.10) :

prqr

prrXEYE ==== )()( , 2

2 )()(prqXVarYVar === . (3.12)

3.3. Pascal .

prX , r-

Bernoulli p ( ),(~, prNBX pr ), pY ,

Bernoulli p ( ),(~, prbY p ).

)()( ,, rYPXP ppr = , ,,...,2,1 r = (3.13)

r- r.

)1()1( ,, ==+= rYpPXP ppr , ,1,...,2,1 += r (3.14)

r- 1+ 1r 1+ . (3.14) Pascal.

3.3. . 49.0=p .

() 4 () .

() . .. Pascal ,2=r 49.0=p

=

=++==4

2

2222 67.0})51.0(3)51.0(21{)49.0()51.0()49.0)(1()4(

XP .

93

() , (3.10),

08.449.02)( === XE .

3.4. Banach. Banach, o Steinhaus Pascal. . Bernoulli

2/1== qp .

. . z ...,,2,1,0= .

)()( zZPzf Z == , z ...,,2,1,0= .

. z

)1( + 12)()1( +=++= zzx .

. , (3.7),

z

Z z

zXPzZPzf

=+====

2

212)12(2)()( , z ...,,2,1,0= .

4. , , . , . . .

4.1. , , . () o

94

. .. ,

. ( )),,(~ .

.

4.1. ,

+

===

x

x

xXPxf )()( , x ...,,2,1 ,0= . (4.1)

.

+=

N )(

, - . . }{ xX =

x

x

- x x .

, ,

+

===

x

x

xXPxf )()( , x ...,,2,1 ,0= .

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

Cauchy,

=

+=

x

x

x

0, (4.2)

= =

=

+

=

x

x

x

x

xf0 0

1)( ,

.

x 0 , x 0 , x 0

x

},min{},0max{ x .

95

.

4.2. (4.1).

XE+

== )( , 1

)(2++

+

+

==

XVar . (4.3)

. .. , ,

=

+

==

x

x

x

xXE1

)( .

=

=

=

11

)!()!1()!1(

)!(!!

x

xx

xx

xx

x

(4.2) Cauchy,

+

=

+

=

==

y

y

x

x

y

x

1

01 11

11

+

=

+

+

=1

1.

H ..

=

+

==

x

x

x

xxXXE2

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

=

=

=

22

)1()!()!2(

)!2()1()!(!

!)1()1(x

xx

xx

xxx

xx

(4.2) Cauchy

=

=

+

=

+

=

x

y

y

y

x

x

2

2

0)2( 2

2)2(

22

)1(

)1)((

)1()1(2

2)1(

++

=

+

+

=

.

96

222

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

+

++

++

=+==

XXEXVar

1+

+

+

+=

.

, N += .

4.3.

(4.1) N += . N ,, pN

N=

lim ,

xx

pp

x

x

x

=

+

)1(lim , x ...,,2,1,0= . (4.4)

. pN

N=

lim

N

N = 1

pN

N

NN==

1lim1lim .

0lim = N

cN

( N) c.

x

Nxx

Np

Nx

N

NN

N

N =

=

11lim)(lim L ,

x

Nxx

Np

Nx

N

NN

N

N

=

= )1(11lim)(lim L ,

111111lim)(lim =

=

N

NNN

N

NL .

xxv

xx

xvx

NN

NM

N

x

M

x

x

x

)()()(

)()()(

=

+

=

+

(4.4).

4.2. (Feller, 1968). A N . . .

97

N Np ,

.

() (4.3) Np ,

=

N

N

p N, .

N

)/()(1)/(1

)())((

,1

,

NN

NNNN

pp

N

N

=

+=

1 )/()()/( NN < 1

)/()()/( NN > . Np , N

/N < , /N > ][ /N = , ][x x.

Np ,

.

5. POISSON 5.1.

!)(

xexf

x= , ...,2,1,0=x , (5.1)

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

ze ,

=

=0 !x

xz

xze , (5.2)

98

=

=

===0 0

1!

)(x x

x

eexexf ,

.

..

99

( 0p ) Poisson

. .

Poisson .

5.2. Poisson (5.1).

XE == )( , XVar == )(2 . (5.5)

. .. , ,

=

=

===

1 1

1

)!1(!)(

x x

xx

xe

xxeXE ,

, (5.2) (5.5). ..

=

=

===

2

22

2)2( )!2(!

)1()]1([

xe

xexxXXE

x

x

x

, (5.2)

2)2( )]1([ XXE == .

XXEXVar =+=+== 2222 )]1([)( .

5.1. () Bernoulli. 01.0=p .

100 .

100 .

xx

xxXP

== 100)99.0()01.0(

100)( , 100...,,2,1,0=x .

100

100= 01.0=p 1== p

10, Poisson

!/)( 1 xexXP == , ...,2,1,0=x

.

7358.03679.022)1()0()1( 1 ===+== eXPXPXP .

,

7357.03697.03660.0)1()0()1( =+==+== XPXPXP .

5.2. () Poisson. () (, ). Petri ( ) ( ). . , , . tX

t. t, tX

...,2,1,0 , t , tX ,

0t , ( ). tX

],0( t t/t = .

() t/tp = , 0> ,

() pq =1 .

tX (

)

101

xxt qpx

xXP

= )( , ...,2,1,0=x ,

tp .

0t , tp =lim ,

!)()(

xtexXP

xt

t== ...,2,1,0=x , )0,0( >> t . (5.6)

)(~ tPX t .

Poisson.

() . t Poisson. Rutherford, Chadwick Ellis (1920) 2608= 5.7 . Poisson 87.3= .

() , . Poisson. Poisson.

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

() . .

() . t Poisson.

102

.

() Petri , , . t Poisson. , , Poisson.

Poisson.

5.2. 80 4 . () () ;

Poisson

!4)( 4x

exXPx

== , ...,2,1,0=x .

, ()

9084.00733.00183.01)1()0(1)2( ===== XPXPXP ,

. ()

=

====4

03711.01954.01954.01465.00733.00183.01)(1)5(

xxXPXP

.

5.3. 3 . : () 2 , () 4 2 , () 2 2 .

tX t

Poisson

!)3()( 3

xtexXP

xt

t== , ...,2,1,0=x .

103

, ()

=

=++==2

0

31 4232.02240.01494.00498.0!

3)2(x

x

xeXP

()

=

=++++==4

0

62 2851.01339.00892.00446.00149.00025.0!

6)4(x

x

xeXP .

2 ),( pb 3= 4232.0=p ( ()),

yy

yyYP

== 3)5768.0()4232.0(

3)( , 3,2,1,0=y

()

3857.0)4232.0(33

)5768.0()4232.0(23

)2( 32 =

+

=YP .

. 3 1. 12 . .

2. 10 5 4 . 5 .

3. 3.0=p .

0.9.

4. r

. p . . r .

r .

r , , r

104

. , r . . r . . () r . )(XE

)(XVar . ()

r . )(YE

)(XVar . ()

5= , 3=r 1.0=p

)(YE

15. )(YVar .

5. () a a

105

() .

7. Bernoulli p. () () r .

8. 125 50 . 5 . 5 2 , () 5 () .

9. 1 , . . ()

)()( xXPxf == () )(XE )(XVar .

10. 350 42 . () x () 10 3 .

11. 0.1% . 5000 () 3 () 2 () 4 .

12. Poisson. () () .

4 1. H () () ( ) . , ] ,[ , < .

)()( 1221 xxcxXxP =< , xx 21 , (1.1)

c . x =1 , x =2

1)()( ==< XPXP

c

=

1 . (1.2)

, , 0)( == xXP Rx ,

. . , (1.1) (1.2),

108

> FFXPXP

31

321 == .

() (1.5)

0)( == XE , 12/10)( 82 == XVar .

1.2. 10 , 5 .. 7:20 7:40 () 4 () 7 . , 7:20. ..

]20,0[

110

103)}10()13({)}0()3({)1310()30()( =+=

111

===0

220

222 1)()( dyey

dxexdxxfxXE yx

=++=+==0 0

20 0 0

222 2]22[2][ yyyyyyy eyeeydyyeeydeydyey

22 2)(

XE = .

222 1)()(

XEXVar == .

. .

2.2. (2.1).

)()|( yXPxXyxXP >=>+> , 0x , 0y . (2.4)

. }{ yxX +>

}{ xX > , }{}{ xXyxX >+>

(2.2),

)()(

)(),()|(

xXPyxXP

xXPxXyxXPxXyxXP

>+>

=>

>+>=>+>

yxyx

ee

exF

yxF

+

==

+=

)(

)(1)(1

yeyFyXP ==> )(1)(

(2.4).

2.1. Poisson tX , 0t ,

tXE t =)( (. 5.2 . 3)

( ). }{ tT > ,

t,

112

}0{ =tX , t ,

(5.6) . 3,

tt eXPtTP

===> )0()( , 0t

.. ,

113

,...2 ,1 ,)!1(0

1 === dxexI x , (2.7)

=

=

=0 0

11 1)!1(

1)!1(

)( dyey

dxexdxxf yx

,

. ,...2 ,1 , =I ,

:

+ +=== 0 0 01

01 ][ dxexexdexdxexIxxxx

,...2 ,1,1 ==+ II . (2.8)

==01

1dxeI x

(2.7).

2.3. Erlang (2.6).

XE == )( , 2

2 )(XVar == . (2.9)

. ..

=

===

0 0)!1(1

)!1()()( dyey

dxex

dxxxfXE yx

(2.7)

=

=)!1(

! .

+ +

=

==

01

20122

)!1(1

)!1()()( dyey

dxex

dxxfxXE yx

222 )1(

)!1()!1()(

XE +=

+

= .

114

..

22

2

2222 )1()()(

XEXVar =+=== .

2.2. Poisson tX , 0t ,

tXE t =)( (. 5.2 . 3) T

- ( ). }{ tT > , -

t }{ X t < , t

, (5.6) . 3,

=

=

===1

0

1

0 !)()()()(

ttt

teXPXPtTP , 0t .

.. T

=

=1

0 !)(1)(

tetF t , 0t , (2.10)

0 ,0)(

115

() , , 12 () .

() tX t

Poisson tXE t =)( , 3/124/8 == .

3T Erlang

=

=2

0

3/

!)3/(1)(

tetF t .

=

==>2

0

43 !

41)12(1)12(

eFTP

Poisson

7619.0)1465.00733.00183.0(1)12( 3 =++=>TP .

() 3T , (2.9),

93)( 3 == TE .

2.3. , ),1()( E , Erlang ,

),( ,

0> 0> , ),( . ,

X 0> 0> ( ),(~ X ), (.

(2.1) (2.6))

116

Erlang (, , ). )( , 0> ,

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

, }...,3,2,1{2/1 a (.

). : 0> ,

)()1( =+ , (2.14)

( , . (2.8)). ,

=)2/1( , (2.15)

( (2.15) Euler), ...),2/5(),2/3(),2/1(

(2.14) (2.15). ,

2/)2/1()2/1()12/1()2/3( ==+= ,

4/3)2/3()2/3()12/3()2/5( ==+= ,

8/15)2/5()2/5()12/5()2/7( ==+= ,

... ),( ,

Erlang ( 2.3). , (. (2.9))

XE == )( , 2

2 )(XVar == . (2.16)

, 2/ = ( ) 2/1= , )2/1,2/( - (chi-square)

(degrees of freedom),

2 . , -

( 2~ X ),

117

, -, , . 3. ANONIKH KATANOMH

, . :

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

. , .

.

, , .

De Moivre Laplace ),( pb ( )

Gauss . "" (Normal) Karl Pearson.

3.1. X

2 )0,( 2 >

118

),;( 2xf

.

3.1. () f ( ) x =

21),;(max 2

xf

x=

119

(, 2).

(1.5, 2) =0.5, 1 2.

1,0 == ,

, ),(~ 2NX

)1,0(N .

)1,0(N .

)1,0(N

Z. )(z )(z ,

,2

1)( 2/2zez =

120

2/2

)2/1()( xex = (0, 1).

)(z . ,

121

==

z z dyydyyz )()()(

yt =

===

z

z

zdyydttdttz )()())(()( .

1)()()()()( ==+=+

z

zdyydyydyyzz

.

0=z 5.0)0( = .

1)1(2))1(1()1()1()1()11( === ZP ,

1)2(2))2(1()2()2()2()22( === ZP ,

1)3(2))3(1()3()3()3()33( === ZP ,

1

%686826.01)8413.0(2)11( == ZP ,

%959546.01)9773.0(2)22( == ZP , (3.1)

%7.999974.01)9987.0(2)33( == ZP .

),(~ 2NX

.

122

3.4. X ),( 2N

() XZ /)( =

)1,0(N .

() ,)(

=

XP

=

)(XP ,

=

=

XP 1)( .

. () )(zFZ

XZ /)( =

),;()()( 2zFzXPz

XPzFZ +=+=

=

)2/(])[(2 22

21),;()()( zZZ e

zfzFzf +=+== )(2

1 2/2 ze z ==

)1,0(~ NZ .

()

=

=

Z

P

X

PXP )(

XZ /)( = )1,0(N )(z .

=

=

Z

PXP )( .

:

=

=

=

ZP

XPXP )( ,

.111)(1)(

=

=

==

ZP

XPXPXP

3.5. ,

X ),( 2N 2,

,

123

XVarXVarXE === )(,)(,)( 2 .

. XZ /)( =

)()()( ZEZEXE +=+=

)()()( 2 ZVarZVarXVar =+= .

)()( zzzg = )()( zgzg = , g

,

=== 0)()()( dzzgdzzzZE .

=== dzezdzezZEZVar zz )(

21

210)()( 2/2/222

22

=+=+= 110)(][2

1)( 2/2

dzzzeZVar z .

22 1)(,0)( XVarXE ===+= .

3.1. X 270= 30=

. 7

)2()2(30

27021030

270)210( =

124

(3.1)

%,68)11()||()( ==+ ZPXPXP

%95)22()2||()22( ==+ ZPXPXP ,

%7.99)33()3||()33( ==+ ZPXPXP .

68% , 95% 99.7% .

.

3.3. )1,0(~ NZ z )( z=

zZP => )( , 10

z = )(1 z =1)( .

01.0=

99.001.01)( ==z

1

33.2z .

125

05.0=

95.005.01)( ==z 645.1=z ,

10.0=

90.010.01)( ==z 28.1=z .

z

zZP => )( , 10 XP

)1,(~ 2NX .

999.0001.01)75(1)75( ==>= XPXP

999.01

751

=

XP

999.01

75 =

.

126

1

09.31

75 =

09.375 = . 91.7109.375 == .

3.5. X min30=

min2.1= .

33min

=

==>

2.13033

2.1301)33(1)33( XPXPXP

%6.00062.09938.01)5.2(1)5.2(1 ==== ZP .

28min

%50475.0)67.1(1)67.1(2.13028

2.130)28( ===

= ZPXPXP .

10 2 28min

Y = ( 10) 28min.

),10(~ pbY 05.0)28( == XPp

)1()0(1)2(1)2( ===

127

, 100= .

, p ),( 2N

, p = , )1(2 pppq == . 100= .

128

De Moivre 1733 5.0=p

p )10(

129

4.2. ),(~ pbX

pq

ppq

pXP )( .

. . , )( kXP = , ...,1,0=k

),( pqpN k,

21

k 21

+k

+

=pq

pk

pq

pkkXP

2

121

)( .

4.3. ( ). ),(~ pbX ( p )

0 ,

+

pq

p

pq

pXP 2

121

)( .

p 2/1 . , , Poisson. , Poisson ( =)(XE , XVar =)( )

),( 2 = , = .

k

e

kXP 2)( 2

21)(

=

130

XP )( ,

+

=

k

kkXP 2

121

)( ,

+

XP 2

121

)( .

4.1. 20%. 100 26 ; 100 ,

2.0=p 100= .

kk

kkXP

== 100)8.0()2.0(

100)( , 100...,,1,0=k

=

=

100

26

100)8.0()2.0(100

)26(k

kk

kXP . (4.1)

. ,

=8.02.01002.010026

8.02.01002.0100)26( XPXP

0668.09332.01)5.1(1)5.1(16

2026====

ZPZP .

131

)375.1(8.02.0100

2.01002126

)26( =

ZPZPXP

0845.09154.01)375.1(1 === .

)26( XP

(4.1) 0875.0 .

4.2. p . p 1% 95%; 03.0p

( ) ; , p. /X ,

95.001.0

pXP .

=

=

01.001.001.0 p

XPp

XP =+ ]01.001.0[ pXpP

+=pq

pppq

pXpq

ppP )01.0()01.0(

101.0201.001.0

=

pq

pq

pq ,

95.0101.02

pq

132

975.001.0

pq .

1

96.101.0 pq

pq 38416 . (4.2)

4/1)1( = pppq ( 2)1()( pppppg ==

5.0p 5.0p 25.0)5.0()(max == gpgp

)

(4.2)

96044138416 .

03.0p

0021.0)03.01(03.0)1( == pppq ,

(4.2)

810021.038416 .

4.3. Poisson 200 .

() 170 ; () 11

170;

A 1 , Poisson 200=

),( 2N 200= , 200= .

()

=

= )12.2(

200200170

200200)170( ZPXPXP

)12.2()12.2(1)12.2(1 === ZP 983.0= .

,

133

)16.2(200

2005.0170200

200)170(

= ZPXPXP

%.5.989846.0)16.2( ==

() ( ) 170. ),(~ pbY ,

12= , 9846.0=p .

1211

1212

1112

)12()11()11( pqpYPYPYP

+

==+==

1211 )9846.0(0154.0)9846.0(12 +=

9859.08301.01558.0 =+= . 5. , . Xlog .

. Xlog

( ) . Xlog

SGPT (serum glutamic pyruvic

transaminase) , Xlog .

, .

,

5.1.

(lognormal) 2 )0,( >

134

XY log=

),( 2N .

, )(xF ( 0>x ) :

===

x

XPxXPxXPxF loglog)log(log)()(

=

x log (5.1)

=

==

x

xx

xxFxf log1loglog)()( , 0>x .

5.1. 2

2

2

2)(log

21)(

x

ex

xf

= , 0>x .

r ( )

22

21

)(rrr eXE

+= ...,2,1=r

2

21

)(

eXE+

= , 2222)( eXE +=

)1())(()1())(()()(222 2222 === + eXEeeXEXEXVar .

(5.1).

135

SGPT 25.

2 .. ,

2

21

)(

eXE+

= , )1())(()(22 = eXEXVar

54.182

21

=+

e , 03.14)1()54.18(22 =e .

04.0)54.18(

03.141 22

==e

04.004.1log2 == .

92.254.18log21 2 ==+

9.204.02192.2 == .

==

2.09.225log

2.09.2log)25log(log)25( XPXPXP

9452.0)6.1(2.0

9.222.3==

= .

%5.94 25 SGPT . . 4

1. ],[ . 1)( =XE 3)( =XVar ,

) ,

136

) || XY =

) )(YE )(YVar .

2. ]1,0[ . )(XgY = ,

yxg =)( 111 +

137

i) 140 150, ii) 130 160.

8. 295 240 . 9 10 , .

9. 250 50. )

200 260. ) c 10%

c.

10. 0.04. 02.0

( 02.0 02.0+ ).

) ; ) 20 .

i) ; ii) 2 ; iii) 3 ; iv) 6 ; ) i) 20 ; ii) 10 ;

11. K cm175= cm5= .

) i) 175 cm; ii) 180 cm;

138

iii) 170 cm 180 cm; ) 6 i) 180 cm; ii) 4 ;

12. X 2 c

)(2)( cXPcXP =>

c =+ 43.0 .

: K 110 mg/dl

2)/5( dlmg , c

.

13. )1,0(~ NZ z zZzP = 1)( ,

10

139

p 0.05 0.99. 80% ;

18. .

73.2)( =XE , 075.0)( =XVar

. )

2.71 2.74; ) 10

2.71 2.74; ) 10 2

2.71 2.74;

19. 5.1 SGPT 64.34)( =YE

113)( =YVar .

25 , : 25X , 25>X . ;

20. (). , c, %100

. c ; %1= , %5 , %10 . ;

5 , 1. 1 () BA, ,

. .

1.1. () YX , () ,

)()(),( yYPxXPyYxXP = , (1.1)

x y. () , vXXX ...,,, 21 ()

)()()()....,,,( 22112211 vvvv xXPxXPxXPxXxXxXP = L (1.2)

vxxx ...,,, 21 .

1.1. () (1.1) : .. , .. 2X , , (1.1),

)()2|( yYPXyYP = ,

y. .. .. ( ). () (1.1) ,

)(),( BAPyYxXP = ,

})(:{ xA = , })(:{ yB = .

142

, (1.2)

})(:{ ii x , vi ...,,2,1= .

() (1.2) : vBB ...,,1 ,

)()()...,,( 1111 vvvv BXPBXPBXBXP = L .

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

.. . , , .

1.1. () .. vXXX ...,,, 21

vfff ...,,, 21 , ,

)()()()...,,,( 22112211 vvvv xfxfxfxXxXxXP L==== ,

vXvXX RxRxRx ...,,, 21 21 , iXR iX ,

vi ...,,2,1= .

() .. vXXX ...,,, 21 vfff ...,,, 21 , ,

)()()()...,,,( 221121...,,, 21 vvv xfxfxfxxxf v L= ,

vxxx ...,,, 21 ,

)...,,,(...

)...,,,( 221121

21...,,, 21 vvv

vXXX xXxXxXPxxxxxxf

v

=

vXXX ...,,, 21 .

: vXXX ...,,, 21 ,

vggg ...,,, 21 , ..

)(...,),(),( 222111 vvv XgYXgXg ===

143

. , .. )( iii XgY =

.. iX ,

. , , .. ..

)...,,( 111 kXXgY = , )...,,( 112 vk XXgY += , 11 vk ,

vXX ...,,1 , .. 1Y 2Y

.. ( .. ),( 2111 XXgY = ),( 3122 XXgY = , .. 1X

.. ), 21 YY .

..

1.2. .. vXX ...,,1 ,

(i) ][][][ 11 vv XEXEXXE LL = ,

,

(ii) )]([)]([)]()([ 1111 vvvv XgEXgEXgXgE LL = ,

( ).

H .

1.1. vXX ...,,1 ,

(i) )()()( 11 vv XVarXVarXXVar ++=++ LL ,

(ii) )]([)]([)]()([ 1111 vvvv XgVarXgVarXgXgVar ++=++ LL

( ).

. )()( 11 vv XgXgY ++= L .

22 )]([)()( YEYEYVar = .

)]()([)( 11 vv XgXgEYE ++= L

)]([)]([ 11 v XgEXgE ++= L

++= L1 ,

)]([ iii gE = , (. 4.1, (4.8) . 2),

144

= =

=++=v

i

v

jji YE

1 1

21

2 )()]([ L .

211

2 )]()([ vv XgXgY ++= L = =

=v

i

v

jjjii XgXg

1 1)()( ,

=

= =

v

i

v

jjjii XgXgEYE

1 1

2 )()()( = =

=v

i

v

jjjii XgXgE

1 1)]()([ .

,

22 )]([)()( YEYEYVar = ])]()([[1 1

j

v

i

v

jijjii XgXgE

= =

= .

1.2 (ii), ji

jijjiijjii XgEXgEXgXgE == )]([)]([)]()([ ,

ji XX , . ,

{ }=

=v

iiii XgEYVar

1

22]))([()( =

=v

iii XgVar

1)]([ ,

(ii). (i) (ii)

iii XXg =)( , vi ...,,2,1= .

1.1. Bernoulli vXXX ...,,, 21 ,

p ( ), . pXP i == )1( ,

qpXP i === 1)0( , vi ...,,2,1= . ..

vXXX ++= L1 (1.3)

, , p, ),(~ pvbX . }...,,1,0{ vRX = . vp =

vpq =2 . 2.

(1.3)

vp =++=++== )()()()( 11 LL

( pXE i =)( , )...,,2,1 vi = . 1.1

, vXX ...,,1 pqXVar i =)( . ,

145

vpqXVarXVarXXVarXVar vv =++=++== )()()()( 112 LL ,

. 2. ,

vv XaXaY ++= L11 ,

vaa ...,,1 , :

= = ==

===

=

v

i

v

i

v

iiiiii

v

iii apXEaXaEXa

1 1 11)()()( ,

==

=

=

v

iii

v

iii XaVarXaVarYVar

11)()(

= ===

v

i

v

iiii apqXVara

1 1

22 )(

( .. ii Xa , vi ...,,2,1= , ). ,

0)( 21 = XXE , pqXXVar 2)( 21 = .

1.2. iX i

2i , vi ...,,2,1= ( )),(~2iii NX ,

==

=

v

iii

v

iii aXaE

11

===

v

iii

v

iii aXaVar

1

22

1.

,

2121 )( XXE = , 22

2121 )( XXVar += .

1.3. i .. Poisson, )(~ ii P ,

vi ...,,2,1= , 0>i ,

==

=

v

iii

v

iii aXaE

11

===

v

iii

v

iii aXaVar

1

2

1

( iii XVarXE == )()( )(~ ii PX ).

2.

.. , .

2.1. vXXX ...,,, 21 ..

146

(i) ( Bernoulli ( )). ),(~ pvbX ii , vi ...,,2,1=

= =

=

v

i

v

iii pvbXX

1 1,~ ,

, iX Bernoulli, ),1()(~ pbpbX i ,

),(~1 pvbXX v++L .

(ii) ( (Pascal) ). ),(~ prNBX ii , vi ...,,2,1= ,

= =

=

v

i

v

iii prNBXX

1 1,~ ,

, iX , ),1()(~ pNBpGX i ,

),(~1 pvNBXX v++L .

(iii) ( Poisson). )(~ ii PX , vi ...,,2,1= ,

= =

=

v

i

v

iii PXX

1 1~ .

(iv) ( ). ),(~ aX ii , vi ...,,2,1= ,

xa

i

a

X exaxf i

i

i/1

)()( = , 0x ,

dueua uai i = 0

1)( , 0>ia ,

Euler (. 2.3 . 4),

= =

v

i

v

iii aX

1 1,~ .

, iX 0> , .

),1(),1()(~ EEX i , ),(),(~1 v ++L .

(v) ( ). ),(~ 2iii

= = =

v

i

v

i

v

iiii NX

1 1 1

2,~ ,

147

,

= = =

++

v

i

v

ii

v

iiiiii X

1 1

2

1

2,~ .

, ),(~ 21 NX ),(~2

2 ( ),

)2,3(~3 221 .

, . 3.

, , ,

vXXS ++= L1 ,

v . , v , ..

vXX ...,,1 .

3.1. vXXX ...,,, 21 ..

F (, )~...,,, 21 FXXX v . vXXX ...,,, 21

. vXXX ...,,, 21

(= , . ), i.i.d.= independent, identi-cally distributed.

3.1. vXXX ...,,, 21

F. XE i =)( 2)( XVar i = ,

148

vS

)(1 =++= L , v XXS ++= L1)( .

.

)()( 1)( v XXESE ++= L vXEXE v =++= )()( 1 L

,

SEv

S

==

= )(11)( )()( .

, , 2

11)( )()()()( vXVarXVarXVarSVar vv =++=++= LL

SVar

vS

vVarXVar

2

)(

2

)( )(11)( =

=

= .

XXVar )(

)()(

2

=

=

,

vS

SSVarSES

=

= )(

2)(

)(

)()(

)()(

,

S

v

S

v

S

Xv

=

=

= )()()(

)( .

(3.1). ,

)()(

XvE =

0)())(( ===

,

=

)()( vVar

XvVar )()( 2

2

XVarvXVar

v

=

=

1)(2

22 ===

vXVar

.

1.1. ( ...,,1 ) )(S

...,,1 . 3.1

149

Xv

XVar )(

)()( =

vS

SVarSES

= )(

)(

)()(

)()(

.

, , ( , ) .

3.2. ( , ...). vXXX ...,,, 21 .. F

( ) XE i =)( , 2)( XVar i = ,

150

(ii) ),(~ pvbX , ( p),

=+

== + rv

uurvu

rrr

u 22

12)1()(

)(

=++

=+

=+== ++ rv

xxzzrvxz

rrrr

rr

22

2][2

12)1()(1

)(

=+

=+= + rv

xxrvx

rrr

22

12)1()(

)(

+= .

0 (ii) 5.2.

) 5.1, 5.2 (;) (;)

5.1 . (coding).

1 . 0y .

2K .

cyy

u ii0= , ki ...,,2,1= .

3K . u 2us

=

=k

iiiuvv

u1

1 ,

=

=

k

iiiu uvuvv

s1

222

11 .

4K . x 2xs

0yucx += , 222ux scs = .

199

2K ,...,2,1,0 =iu

.

5.1. 2.2 ( 2.3) ( ) .

iy iv iN ii yv 2iy 2ii yv

5.5 8.5

11.5 14.5 17.5 20.5

8.5 11.5 14.5 17.5 20.5 23.5

7 10 13 16 19 22

4 16

3 4 0 1

4 20 23 27 27 28

28 160

39 64 0 22

49 100 169 256 361 484

196 1600

507 1024

0 484

28 313 3811

178.1128

313==x , 56.11

283131811

271 22 =

=s .

310

= iiy

u , 6...,,2,1=i

iy iv iu iiuv 2iu 2iiuv

7 10 13 16 19 22

4 16

3 4

Documents

Ι∆ΑΚΤΙΚΕΣ ΗΜΕΙΩΣΕΙΣ - users.uoa.grusers.uoa.gr/~npapadat/BOOKS/PithStatOpen.pdf · ΕΙΣΑΓΩΓΗ ΣΤΙΣ ΠΙΘΑΝΟΤΗΤΕΣ ΚΑΙ ΤΗ ΣΤΑΤΙΣΤΙΚΗ