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  • e-mail: dimitheo@aegean.gr

    2010

  • 2

    , , -

    . M 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

    .

    .

    .

    .

    , .

    .

    .

    .

    , , ,

    :

    [1]. . , , : , , 1994.

    [2]. S. M. Ross, A First Course in Probability, Second Edition, Macmillan Publishing Company, 1984.

    [3]. P. Hoel, S. Port, C. Stone, , : A ,

    , 2002.

    [4]. . , , , , .

    , , 2004, , 2005.

    [5]. G. Roussas, A Course in Mathematical Statistics, Second Edition, Academic Press, 1997.

    [6]. . & . , : , ,

    , , 1999.

    [7]. R. M. Spiegel, , : . , McGraw-Hill, New York,

    , , 1977.

    [8]. . & . , , 3 , ,

    1993.

    [9]. . . , , , , 1990.

    [10]. . , , , , 2000.

    [11]. S. M. Ross, , :

    , 1 , , , 2012.

    [12]. . . , , , , 2012.

  • 4

    1:

    1.0

    . .

    ( )

    ,

    . .

    .

    1.1

    :

    () k

    .

    ()

    .

    .

    . 1a 1v

    1a 2a 2v

    , ..., ,1a ...,,2a ,1ka ka kv

    , ,1a ...,,2a ka

    , kvvv 21 .

    1 ( ). T , 21 ,TT 3T

    3,2,1, =iTi i . 1 .

    2 . 3 .

    2 3222 =

  • 5

    . n

    .2n

    2 ( ). T 21 ,TT 2,1, =iTi

    i . 1

    2 .

    26 . n

    .6n

    3 , .

    ;

    . . 30235 = .

    4 }.,...,,{ 21 NsssJ = J

    ;

    . .1s

    J . .,...,2 Nss

    .22222 N=

    1 .

    -;

    . .1501015 =

    2

    ;

    .

    . .1382400001024 43 = .6120576078910222324 =

  • 6

    1.2

    n . k n (

    ).k n . k n

    (sampling). :

    () . k

    .

    () .

    . :

    () . k .

    () -. .

    . 1.1

    1.2.

    1.1 ( -). k

    (i) ( )

    ,),1()1()(, nkknnnn kkn +== L

    ,nk =

    .321!)(, nnn nnn L===

    (ii) ( ) .kn

    . (i) 1 n , 2 )1( n ,

    ..., k ))1(( kn .

    k ( )

    .)())1(()2()1( ,knknknnnn = L

    (ii) 1 n , 2 n , ..., k

    n .

    k ( ) .knnnn = L

    1.2 (- -). -

    k

  • 7

    (i) ( ) knk

    n,=

    .)!(!

    !, knk

    n

    k

    nkn ==

    (ii) ( ) ),( knN

    .1

    ),(

    +=

    k

    knN kn

    . (i) M . k

    .,kn k !k

    .k - .!,

    kkn

    - .

    k .n k

    .

    =

    k

    nv1

    !2 kv =

    .

    !21 kk

    nvv

    = . , .

    !! ,,,21 kk

    nk

    k

    nvv knknkn

    =

    =

    = ,

    !)(21

    ])1()1[()](21[

    !

    )1()2()1(

    !,

    kkn

    nnknkn

    k

    knnnn

    kM kn

    +

    =+

    =

    =L

    LLL

    .)!(!

    !

    =

    =

    k

    n

    knk

    n

    (ii) [1], . 48-49.

    ,10

    =

    =

    n

    nn ,

    =

    kn

    n

    k

    n .11 nk !n n

    Stirling : ,2! ne

    nn

    n

    e Euler,

    ...).72.2=e

  • 8

    3 .

    . ;

    ;

    . . 1T .

    . 2T .

    3

    5

    . .3

    51

    , ,

    .3

    1361

    +

    4 .

    (i) ; (ii) ; (iii) ;

    . (i)

    10

    52 .

    10

    48 . ,

    10

    48

    10

    52

    .

    (ii) .

    1

    4

    ( ).1T 2T .

    9

    48 .

    .9

    48

    1

    4

    (iii)

    ,

    . .10

    48

    9

    48

    1

    4

    10

    52

  • 9

    1.3

    k n .

    n k .

    1.3 ( ). n

    k

    (i) nk .

    (ii) ,!!!

    !,...,,;

    2121 knnnn

    knnn

    n

    L j jn ,

    kjn j ,...,2,1,0 = .1

    nnk

    jj =

    =

    . (i) ,21 knnn n ==== K

    n k .

    (ii)

    1n

    n 1n ,

    2

    1

    n

    nn

    2n , ...,

    k

    k

    n

    nnnn 121 ... kn .

    )!(!

    )!(

    )!(!

    )!(

    )!(!

    !...

    11

    11

    212

    1

    11

    121

    3

    21

    2

    1

    1 kkk

    k

    k

    k

    nnnnn

    nnn

    nnnn

    nn

    nnn

    n

    n

    nnnn

    n

    nnn

    n

    nn

    n

    n

    =

    K

    KLL

    .!!

    !

    1 knn

    n

    L=

    5 .

    ;

    . 1.3(ii) 8=n 4=k

    24321 ==== nnnn . .)!2(

    !8

    !2!2!2!2

    !84

    =

  • 10

    6 . ()

    ; ()

    ;

    . () 1.3(ii) 52=n 4=k

    134321 ==== nnnn . .)!13(!52

    4

    () . 1T

    . !1!1!1!1

    !4

    . 2T

    ( ).

    .!12!12!12!12

    !48

    !1!1!1!1

    !4.

    )!12(

    !48!4

    !12!12!12!12

    !484

    =

    1.4

    1. .

    () ; () ; ()

    ; () ;

    . () .

    5

    13 .

    () .

    1

    4 .

    5

    13 . ,

    5

    13

    1

    4 .

    () .

    2

    4 .

    ..

    . ,

  • 11

    .5

    13134

    1=

    k kk ,

    =

    4

    1 5

    1313

    2

    4

    k kk .

    () . 1T

    .

    1

    4 . 2T

    . 2T

    2

    13 .

    43 ,TT 5T .

    43 ,TT 5T

    1

    13 .

    1

    13

    1

    13

    1

    13

    2

    13

    1

    4 .

    2. ,)(0=

    =+

    n

    k

    knkn yxk

    nyx yx, n ()

    () .

    . () . 1=n .1

    1

    0

    1 0110 yxyxyxyx +=

    +

    =+ ,rn =

    =

    =+

    r

    k

    krkr yxk

    ryx

    0

    .)( ,1+= rn

    +

    =

    ++

    +=+

    1

    0

    11 .1

    )(r

    k

    krkr yxk

    ryx

    ,rn =

    krkr

    k

    rr yxk

    ryxyxyxyx

    =

    +

    +=++=+

    0

    1 )())(()( 10

    1

    0

    +

    =

    +

    =

    +

    = krk

    r

    k

    krkr

    k

    yxk

    ryx

    k

    r

    .210110

    121112 yxr

    ryx

    rxy

    ry

    rx

    r

    ryx

    r

    ryx

    rxy

    r rrrrrrrr

    ++

    +

    +

    +

    +

    ++

    +

    = ++ KK

  • 12

    1,,2,1,1

    1=

    +=

    +

    rj

    j

    r

    j

    r

    j

    rK

    )!1(!

    )!1(

    )!1(!

    )1(!

    )!1(!

    )1(!

    )!1(!

    !

    )!(!

    !

    )!1()!1(

    !

    1 jrj

    r

    jrj

    rr

    jrj

    jrr

    jrj

    jr

    jrj

    r

    jrj

    r

    j

    r

    j

    r

    ++

    =++

    =++

    ++

    =

    ++

    =

    +

    .1

    +=

    j

    r ,1

    0=

    =

    r

    rr

    .1

    2

    1

    1

    1)( 11211 +++ +

    +++

    ++

    ++=+ rrrrrr xyx

    r

    ryx

    rxy

    ryyx K

    : +

    =

    ++

    +=+

    1

    0

    11 .1

    )(r

    k

    krkr yxk

    ryx

    ,1+= rn ,

    - .n

    () . ),()()()( yxyxyxyx n +++=+ L n ,

    nyx )( + yx, yx +

    x y . x k ,

    nk ,,1,0 K= y kn

    .,,1,0, nkyx knk K= x