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  • -

    E-mail: [email protected]

    2007

    1

  • 2

    , , -

    . Laplace

    .

    ( ) ,

    .

    . 1000

    .

    . ,

    , , , , ,

    . .

    tests.

    , .

    3000 .. , ,

    .

    1600 .. 7 10 ..

    ().

    .

    Pascal (1623-1662) Fermat (1601-1665)

    1650 .. .

    Pascal Fermat Chevalier

    de Mere. .

    . . On calculations in

    Games of Chance Christian Huyghens (1629-1695).

    , , James Bernoulli (1654-1705), Abraham de

    Moivre (1667-1754), Pierre Simon Laplace (1749-1827), Simeon Denis Poisson (1781-1840) Karl Friedrich

    Gauss (1777-1855). Pafnuty Chebyshev (1821-1894), Andrei Markov (1856-1922),

    Richard Von Mises (1883-1953) Andrei Kolmogorov (1903-1987).

  • 3

    .

    I.

    3 , .

    4 . 5

    .

    6.

    .

    , , :

    () . , , : , , 1994.

    () S. M. Ross, A First Course in Probability, Second Edition, Macmillan Publishing Company, 1984.

    () P. Hoel, S. Port, C. Stone, , : A ,

    , 2002.

    () . , , , , .

    , , 2004, , 2005.

    () G. Roussas, A Course in Mathematical Statistics, Second Edition, Academic Press, 1997.

  • 1

    1.1 , ,

    .

    (random experiment). , ,

    .

    , ,

    .

    (set) .

    (sample space) . ,

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

    .

    .

    . ,

    .

    .

    (..

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

    , ),( yx x y

    .

    ,

    }.2,1,0{= .

    (sample points).

    .

    . (subset)

    '

    'S S

    SS ' Ss .Ss 'S S

    SS ' SS ' Ss '.Ss

    ( ).

    .

    .

    4

  • (complement) .,, CBA A (

    ) cA

    }.:{ AAc = I ., IjAj

    (union)

    (intersection)

    IjAj , UIj

    jA

    jIj

    j AA =

    :{U

    }.Ij IjAj ,

    (difference)

    IIj

    jA

    jIj

    j AA =

    :{I }.Ij BA, BA

    ABA = :{ .} cBAB =

    : ABBA = ABBA = ( ).

    )()( CBACBA = )()( CBACBA = ( ).

    )()()( CABACBA = )()()( CABACBA = ( ).

    IUIj

    cj

    c

    Ijj AA

    =

    ( De Morgan). UI

    Ij

    cj

    c

    Ijj AA

    =

    . ()

    ( ) (events) ( ) (

    ) ( field) .

    1.1 ( -). . -

    ( ) (i) (ii) A cA (iii) ,

    (

    jA Ij

    =U

    1jjA

    ).

    1.1 .

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

    (power set)

    ,

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

    1.1, .

    5

  • A A (occurs). ,

    A .

    .

    }{=A }6,4,2{= (outcome) 2

    4 6, , , 2

    A2 . 4 6.

    }{

    .

    (null event), , .

    .

    .,, CBA

    () A , .cc CBA

    () , CBA ,, .CBA

    () , ).()()( ACCBBA

    () , .CBA

    () , ).()()()( ccccccccc CBAABCCABCBA

    1.2 ,

    },,{ 1 n K=

    nii ,,1},{ K= () .D

    .

    , .}{1U

    n

    ii

    =

    = .1})({1)(1 n

    DnDPPn

    ii ====

    =

    B ,s }.,,,{ 21 sbbbB K=

    =

    =s

    iibPBP

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

    nDbP i ==

    , n . .1)(ns

    nssDBP ===

    .

    6

  • 1.2 ( ).

    },,{ 1 n K= nii ,,1},{ K=

    . )(AP A : =)(AP ,##A

    A# # A , .

    .

    (sampling).

    .

    .

    -

    .

    n k n

    k

    k

    -

    k

    n

    kn

    .)!(!

    !knk

    nkn

    =

    1.2 .

    () ; ()

    ;

    . .

    () . , ,

    . , ,

    . ,

    1P

    211

    26

    .

    21126

    1

    =P

    () , , ,

    2P

    .

    211

    15

    16

    2

    =P

    7

  • )(AP A

    (limiting relative frequency)

    .A n

    An .A nnA

    Af A

    .

    ,

    n

    .A

    A

    .

    Von Mises .

    1.3 ( ). A

    . An A ,

    n

    ,0 nnA A .limlim)( AnA

    nf

    nnAP

    ==

    .

    n

    Von Mises

    . ,

    , , .

    .nnf AA =

    Kolmogorov .

    . Kolmogorov A

    )(AP

    .A

    ,

    Kolmogorov .

    .

    8

  • 1.4 ( Kolmogorov).

    :

    P

    :P

    (P1) -, 0)( AP A . (- ).

    (P2) ( .1)( =P ).

    (P3) , ()

    (mutually exclusive events) ( ,jA Ij ,, Iji ,ji = ji AA )

    ( .)(11

    =

    =

    =

    jj

    jj APAP U ).

    Kolmogorov

    ( )

    .

    1.5 ( ). ,

    . P ,( ,

    (probability space).

    )P

    :

    () . (P ) 0= ,

    () cA A ).(1)( APAP c =

    () P - , 21 AA ).()( 21 APAP

    () A : .1)(0 AP

    () P ,

    Boole (Booles inequality).

    =

    =

    11

    ).(j

    jj

    j APAP U

    () ., BA )()()()( BAPBPAPBAP += ( ,

    Additive law).

    CBA ,,

    ).()()()()()()()( CBAPACPCBPBAPCPBPAPCBAP ++=

    9

  • 1.3 .

    BA, ,21)( =AP

    31)( =BP .

    61)( = BAP :

    () T .

    () M .

    () O A B .

    () T .

    . () .32

    61

    31

    21)()()()( =+=+= BAPBPAPBAP

    () .31

    61

    21)()()( === BAPAPBAP c

    () .31

    321)(1])[( === BAPBAP c

    () .65

    611)(1])[( === BAPBAP c

    1.4 A

    .B

    ).(2)()( BAPBPAP +

    . , ).()( BABA cc

    = )()( BABA cc (P ())()( PBABA cc = .0) = ,

    (P ).()())()( BAPBAPBABA cccc +=

    , )()()()()()( ccc BAPBAPAPBABABBAAA +====

    ).()()( BAPAPBAP c = , ),()()( BAPBPBAP c =

    == )()()()( BABABABA cc .

    , ).(2)()())()(( BAPBPAPBABAP cc +=

    1.5

    ( ). ()

    . () :

    :1A 3 5 .

    :2A 10 .

    10

  • :3A 6 8 15 20 .

    :4A 7 10 25 .

    :5A 18 .

    () ,51 AA ,52 AA

    11

    ,43 AA ( ,) 425 AAA .)( 453 AAA

    . ()

    ( ).

    k x

    )}.,0[,,2,1,0:),{( == xkxk K

    () : 51 ,, AA K

    )},,5[:),3{(1 = xxA )},,0(,101:),{()}0,0{(2 = xkxkA ]},20,15[,86:),{(3 = xkxkA

    ]},25,10[,7:),{(4 = xkxkA ]}.18,0(,1:),{()}0,0{(5 = xkxkA

    () () ]},18,5[:),3{(51 = xxAA ]},18,0(,101:),{()}0,0{(52 = xkxkAA

    ]},20,15[:),6{(43 = xxAA ]},18,10[,11:),{()( 425 = xkxkAAA

    ]}.18,15[:),6{()( 453 = xxAAA

    1.6 70%

    , 40% ,

    30% .

    .

    :A :B

    ,BA ,cc BA ,cBA ,cBA

    .BAc

    . ,

    BA, BA

    ,7.0)( =AP ,4.0)( =BP .3.0)( = BAP , ,8.0)()()()( =+= BAPBPAPBAP

    ,2.0)(1])[()( === BAPBAPBAP ccc ,

    .

    4.0)()()( == BAPAPBAP c

    ,9.0)()()()( =+= ccc BAPBPAPBAP 1.0)(1])[()( === cccc BAPBAPBAP

    1.3

    . ( ) .61}2{ =P

    , ( ) .31}2{ =P .

  • 1.4 ( ). A

    .0)( >AP B , (conditional probability)

    B A (given : )A .)(

    )()|(AP

    ABPABP =

    2. }6,4,2{=A }.2{=B

    .31

    21

    61

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

    )()()|( ====

    PP

    APABPABP

    1.7 . A K

    .

    .

    }.,,,{ KKKAAKAA=

    =B },{ KKAA= =C }.,,{ KAAKAA=

    ;

    . ).|( CBP ,31

    43

    41

    )()()|( ===

    CPCBPCBP .

    21

    42)( ==BP

    1.8 , .

    .

    ;

    . K M , .

    ).|( cMKP .52

    305130

    10

    )(130

    10

    )()(

    )()()|( =

    =

    ==

    =

    MPMPKP

    MPMKPMKP cc

    cc

    1.4 Bayes

    A B

    ,

    (Total Probability Theorem).

    )|( BAP ).|( cBAP

    ;A

    12

  • 1.1 ( ). ,

    nAA ,,1 K

    B

    (partition)

    .)()|()(1=

    =n

    ii