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Conf. univ. dr. RODICA IOAN Lector univ. dr. MANUELA GHICA Lector univ. dr. ILEANA RODICA NICOLA Lector univ. dr. VLAD COPIL MATEMATICI FINANCIARE ŞI ACTUARIALE Curs în tehnologie ID-IFR

Curs Matematici Financiare Si Actuariale

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Curs Matematici Financiare Si Actuarialematematica economica semestrul II an I

Text of Curs Matematici Financiare Si Actuariale

  • Conf. univ. dr. RODICA IOAN Lector univ. dr. MANUELA GHICA

    Lector univ. dr. ILEANA RODICA NICOLA Lector univ. dr. VLAD COPIL

    MATEMATICI FINANCIARE I ACTUARIALE

    Curs n tehnologie ID-IFR

  • Editura Fundaiei Romnia de Mine, 2012 http://www.edituraromaniademaine.ro/

    Editur recunoscut de Ministerul Educaiei, Cercetrii, Tineretului i Sportului prin Consiliul Naional al Cercetrii tiinifice

    din nvmntul Superior (COD 171)

    Descrierea CIP a Bibliotecii Naionale a Romniei Matematici financiare i actuariale/ Rodica Ioan, Manuela Ghica, Ileana Rodica Nicola, Vlad Copil

    Bucureti, Editura Fundaiei Romnia de Mine, 2011 ISBN 978-973-163-658-0 I. Ioan, Rodica II. Ghica, Manuela III. Nicola, Ileana Rodica IV. Copil, Vlad 51(075.8)

    Reproducerea integral sau fragmentar, prin orice form i prin orice mijloace tehnice,

    este strict interzis i se pedepsete conform legii.

    Rspunderea pentru coninutul i originalitatea textului revine exclusiv autorului/autorilor.

  • UNIVERSITATEA SPIRU HARET FACULTATEA DE MANAGEMENT FINANCIAR CONTABIL BUCURETI

    PROGRAMUL DE STUDII UNIVERSITARE DE LICEN CONTABILITATE I INFORMATIC DE GESTIUNE

    MATEMATICI FINANCIARE I ACTUARIALE

    Curs n tehnologie ID-IFR

    Realizatori curs n tehnologie ID-IFR

    Conf. univ. dr. RODICA IOAN Lector univ. dr. MANUELA GHICA Lector univ. dr. ILEANA RODICA NICOLA Lector univ. dr. VLAD COPIL

    EDITURA FUNDAIEI ROMNIA DE MINE Bucureti, 2012

  • 5

    CUPRINS

    INTRODUCERE .. 9

    Unitatea de nvare 1

    NOIUNI INTRODUCTIVE DE TEORIA PROBABILITILOR 1.1. Introducere ..................................................................................................................................... 13 1.2. Obiectivele i competenele unitii de nvare ............................................................................. 13 1.3. Coninutul unitii de nvare ........................................................................................................ 14

    1.3.1. Cmp de evenimente. Probabilitate ...................................................................................... 14 1.3.2. Probabilitate pe un cmp finit de evenimente ...................................................................... 15

    1.4. ndrumar pentru verificare/autoverificare ...................................................................................... 17

    Unitatea de nvare 2

    VARIABILE ALEATOARE

    2.1. Introducere ..................................................................................................................................... 21 2.2. Obiectivele i competenele unitii de nvare ............................................................................. 21 2.3. Coninutul unitii de nvare ........................................................................................................ 22

    2.3.1. Variabile aleatoare unidimensionale. Caracteristici numerice. Funcie de repartiie ........... 22 2.3.1.1. Variabile aleatoare discrete ............................................................................................... 22 2.3.1.2. Variabile aleatoare continue................................................................................................ 26 2.3.2. Variabile aleatoare bidimensionale ...................................................................................... 29

    2.4. ndrumar pentru verificare/autoverificare ...................................................................................... 33

    Unitatea de nvare 3

    SCHEME CLASICE DE PROBABILITATE 3.1. Introducere ..................................................................................................................................... 39 3.2. Obiectivele i competenele unitii de nvare ............................................................................. 39 3.3. Coninutul unitii de nvare ........................................................................................................ 40

    3.3.1. Repartiii probabilistice clasice ............................................................................................ 40 3.3.1.1 Repartiii de tip discret ....................................................................................................... 40 3.3.1.2. Repartiii de tip continuu ................................................................................................... 42

    3.4. ndrumar pentru verificare/autoverificare ....................................................................................... 44

    Unitatea de nvare 4

    ELEMENTE DE STATISTIC MATEMATIC 4.1. Introducere ............................................................................. ........................................................ 49 4.2. Obiectivele i competenele unitii de nvare.............................................................................. 49 4.3. Coninutul unitii de nvare ........................................................................................................ 50

    4.3.1. Teoria seleciei ..................................................................................................................... 50 4.3.2. Teoria estimaiei .................................................................................................................. 51

    4.4. ndrumar pentru verificare/autoverificare ...................................................................................... 53

  • 6

    Unitatea de nvare 5

    GRAFURI I 5.1. Introducere ..................................................................................................................................... 61 5.2. Obiectivele i competenele unitii de nvare ............................................................................. 61 5.3. Coninutul unitii de nvare ........................................................................................................ 62

    5.3.1. Introducere. Definiii ............................................................................................................ 62 5.3.2. Matrice asociate unui graf. Proprieti ale grafurilor ........................................................... 66 5.3.3. Determinarea drumurilor hamiltoniene n grafuri fr circuite ............................................ 68 5.3.4. Determinarea drumurilor hamiltoniene n grafuri cu circuite .............................................. 69

    5.4. ndrumar pentru verificare/autoverificare ....................................................................................... 70

    Unitatea de nvare 6

    GRAFURI II 6.1. Introducere ............................................................................. ....................................................... 74 6.2. Obiectivele i competenele unitii de nvare ............................................................................. 74 6.3. Coninutul unitii de nvare ........................................................................................................ 75

    6.3.1. Drumuri de valoare ntr-un graf. Algoritmul Bellman-Kalaba ............................................ 75 6.3.2. Flux maxim ntr-o reea de transport ..................................... .............................................. 77 6.3.2.1. Algoritmul Ford-Fulkerson ..................................... ............................................... 78

    6.4. ndrumar pentru verificare/autoverificare ...................................................................................... 79

    Unitatea de nvare 7

    DOBNZI 7.1 Introducere ...................................................................................................................................... 89 7.2 Obiectivele i competenele unitii de nvare .............................................................................. 89 7.3 Coninutul unitii de nvare ......................................................................................................... 90

    7.3.1. Dobnda simpl .................................................................................................................... 90 7.3.2. Dobnda compus ................................................................................................................ 91

    7.4. ndrumar pentru verificare/autoverificare ...................................................................................... 93

    Unitatea de nvare 8

    OPERAIUNI DE SCONT 8.1. Introducere ..................................................................................................................................... 96 8.2. Obiectivele i competenele unitii de nvare ............................................................................. 96 8.3. Coninutul unitii de nvare ........................................................................................................ 97

    8.3.1. Operaiuni de scont ............................................................................................................... 97 8.4. ndrumtor pentru verificare/autoverificare .................................................................................... 100

    Unitatea de nvare 9

    PLI EALONATE 9.1. Introducere ..................................................................................................................................... 104 9.2. Obiectivele i competenele unitii de nvare ............................................................................. 104 9.3. Coninutul unitii de nvare ........................................................................................................ 105

    9.3.1. Anuiti posticipate, temporare, imediate ........................................................................... 105 9.4. ndrumar pentru verificare/autoverificare ...................................................................................... 106

  • 7

    Unitatea de nvare 10

    PLI EALONATE GENERALIZATE. MPRUMUTURI 10.1 Introducere .................................................................................................................................... 109 10.2 Obiectivele i competenele unitii de nvare ............................................................................ 109 10.3 Coninutul unitii de nvare ........................................................................................................ 110

    10.3.1. mprumuturi ...................................................................................................................... 110 10.3.1.1. Amortizarea unui mprumut prin anuiti constante posticipate ......................... 110 10.3.1.2. mprumuturi cu anuiti constante, pltibile la sfritul anului 111 10.3.1.3. mprumuturi cu anuiti constante cu dobnd pltit la nceputul anului (anticipat) .............................................................................................................

    111

    10.3.1.4. mprumuturi cu amortismente egale ................................................................... 111 10.4. ndrumar pentru verificare/autoverificare .................................................................................... 112

    Unitatea de nvare 11

    BAZELE MATEMATICII ACTUARIALE 11.1. Introducere .................................................................................................................................... 117 11.2. Obiectivele i competenele unitii de nvare ........................................................................... 117 11.3. Coninutul unitii de nvare ....................................................................................................... 118

    11.3.1. Funcii biometrice ............................................................................................................. 118 11.3.2. Asigurarea unei sume n caz de supravieuire la mplinirea termenului de asigurare ....... 120

    11.4. ndrumar pentru verificare/autoverificare...................................................................................... 121

    Unitatea de nvare 12

    CONTRACTE DE ASIGURARE VIAGER 12.1. Introducere..................................................................................................................................... 123 12.2. Obiectivele i competenele unitii de nvare............................................................................ 123 12.3. Coninutul unitii de nvare ....................................................................................................... 124

    12.3.1. Anuiti viagere ................................................................................................................. 124 12.3.1.1. Anuiti viagere posticipate imediate ................................................................. 124 12.1.1.2. Anuiti viagere anticipate imediate ................................................................... 124 12.1.1.3. Anuiti viagere limitate la n ani i anuiti viagere amnate ............................. 124

    12.4. ndrumar pentru verificare/autoverificare...................................................................................... 125

    Unitatea de nvare 13

    REZERVA MATEMATIC 13.1. Introducere..................................................................................................................................... 128 13.2. Obiectivele i competenele unitii de nvare ........................................................................... 128 13.3. Coninutul unitii de nvare........................................................................................................ 129

    13.3.1. Rezerva matematic .............................................................................. 129 13.3.1.1. Ecuaia diferenial a rezervelor matematice ................................................... 130

    13.4. ndrumar pentru verificare/autoverificare...................................................................................... 131

    Unitatea de nvare 14

    PLI VIAGERE FRACIONATE 14.1. Introducere..................................................................................................................................... 133 14.2. Obiectivele i competenele unitii de nvare............................................................................ 133 14.3. Coninutul unitii de nvare ....................................................................................................... 134

    14.3.1. Asigurarea de pensie ............................................................................ 134 14.3.2. Asigurarea de deces ........................................................................... 134

  • 8

    14.3.3. Asigurri mixte .............................................................................................................. 136 14.4. ndrumar pentru verificare/autoverificare...................................................................................... 136 Rspunsuri la testele de evaluare/autoevaluare 139 Bibliografie ............................................................................ 155

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    : "

    8 0 1

    + 0 10 0 11 0 1

    *&

    &4 Pentru construirea liniei i din D + 1,i n= 5

    ?1 de pe linia i din matricea C

    1

    1

    1

    =

    ==

    i

    i

    i

    c

    c

    c

    1

    1

    1

    =

    ==

    i

    i

    i

    d

    d

    d

    0 (& "

    &

    C la linia i ; noile valori [email protected] & ? i a D ' k l m $

    & 8

    9 * +"

    5

    k l m C la linia i trecnd noile valori de [email protected] ? i a matricei D .&. &

    5 ijd + 1,j n= ) devin egale cu 1;"5nu mai apare niciun element egal cu 1, caz n care locurile

    & " & $ $ & ? 1+i , &

  • 6
  • 6

    .

    %#& ( )

    njiijdD

    ,1, == & indice i pentru care 1iid = A&

    ;$ & ( )1

    2

    n n elemente de 1

    & / 9

    1 este mai mic dect ( )

    2

    1nn

    &78. &

    &&

    93 onian n graf cu circuite

    * & && $ 8 ?1 n matricea

    & 8. & &$ ' " & & drumuri ntre vrfurile care corespund acelor valori de 1.

    " & & &$ $ *+4695?

    @

    , ( )1M , care, n locul valorilor de [email protected] $ " $$ & &8 $ in vrfurile care l compun. *& ( ) ( )( )

    njiijmM

    ,1,

    11

    == & ix jx ( )

    =0

    1 jiij

    xxm

    n caz contrar

    Prin suprimarea primei litere n matricea ( )1M & " ( )1~M #

    ( )1M ( )1~M ( ) ( )1 1M L M ; &

    & 8 . & &

    & ' & &

    au vrf co' $

    & &

    vrfurilor componente ale simbolurilor participante.

    : & 8 ( ) ( ) ( )2 1 1M M L M= ( ) ( ) ( )3 2 1M M L M= B * . "

    ( )1nM , deoarece ntr n vrfuri un drum 1n

  • 21

    n matricea ( )1nM & &&

    ( )1nM & $ + ( ) 0=1nM 5

    01 : & "

    + & 5 & .&

    &+&5

    ;$ & $

    & ? "@ nduce la ideea de drumuri optime ntr

    3ndrumar pentru verificare/autoverificare4( D

    ( )1 2 3 41

    2

    3

    4

    0 1 0 1 2

    0 0 0 1 1

    1 1 0 1 3

    0 0 0 0 0

    ix x x x p x

    x

    xD

    x

    x

    =

    # &$$#&"%:$ D &

    ( ) ( ) ( ) ( )3 1 2 4p x p x p x p x> > > vom scrie vrfurile n ordinea { }4213 ,,, xxxx n loc de ordinea { }4321 ,,, xxxx *8

    3 1 2 4

    3

    11

    2

    4

    0 1 1 1

    0 0 1 1

    0 0 0 1

    0 0 0 0

    x x x xx

    xD

    x

    x

    =

    &$

    0(

    ( )1 2 3 41

    2

    3

    4

    0 1 0 1 2

    0 0 0 1 1

    1 1 0 1 3

    0 0 0 0 0

    ix x x x p x

    x

    xD

    x

    x

    =

    C5 { }3 1 4 2: , , ,Hd x x x x "5 { }1 3 2 4: , , ,Hd x x x x 5 { }4213 ,,,: xxxxd H 5 &&% &&5

  • 24

    7

    81 &&

    *8 ( )1 2p x = ' ( )2 1p x = ' ( ) 33 =xp ' ( ) 04 =xp & ( ) 641

    ==i

    ixp 4=n

    $ ( )

    62

    1 =nn Deci, se poate aplica teorema lui Chen, n G & & &{ }4213 ,,,: xxxxd H

    8

    "

    (

  • 20

    8

    5

    0 1 0 0 1

    0 0 1 1 0

    0 0 0 1 0

    1 0 0 0 1

    0 0 1 0 0

    C

    =

    "5

    0 1 0 0 0

    0 0 1 1 0

    0 0 0 1 0

    1 0 0 0 1

    0 0 0 0 0

    C

    =

    5

    0 1 0 0 1

    0 0 1 0 0

    0 0 0 0 0

    1 0 0 0 1

    0 0 1 0 0

    C

    =

    5 &&%&$ 48

    5

    1 1 1 1 1

    1 1 1 1 1

    1 1 1 1 1

    1 1 1 1 1

    1 1 1 1 1

    D

    =

    "5

    1 1 1 1 1

    1 1 1 1 1

    1 1 1 0 1

    1 1 1 1 1

    1 1 1 0 1

    D

    =

    5

    0 1 1 1 1

    1 1 1 1 1

    1 1 1 1 1

    1 1 1 1 1

    1 1 1 0 0

    D

    =

    5&&

    &D

    '/&8 8.& ( ) 5=ixp 5,1=i D

  • 29

    # & 5"535954(E iid = )E iid = ( )i

    91.!1.

    4 , - % ! * , % !.$ # /( Romnia de Mine!0116

    0 , - % ! * , % ' /( Romnia de Mine!011

    9!* /( Romnia de Mine!0113

    3 , ' / ( Romnia de Mine!4

    7&/( Romnia de Mine!46

  • 6.3.1. Drumuri de valoare ntr !"# 6.3.2. Flux maxim ntr $!%%&$

    6.4. ndrumar pentru verificare/autoverificare

    ' $$( ) *+ )"

    # , $) ++ -%

    %&$! (

    * $ . $

    $ $$ $+ graf utiliznd algoritmul Bellman # * $. $ ( %%&$

  • /

    Drumuri de valoare ntr !"#

    % ( )= ,XG . : ,,,$(

    0 ( )jiij xxvv ,= ( )vXGv ,,= graful valuat. n (.()$ 1 2-$$ $

    3 { }kiii

    xxxd ,...,,21

    = n graful G .$. )

    ( ) = +

    =1

    11

    k

    hhihi

    vdv

    4$ 5 d de la un vrf oarecare ix

    vrful nx . ( )dv $ Pentru aceasta, introducem

    , ( )njiij

    vV,1, == . )

    ( )( )

    =

    =

    ji

    jiijij

    xxji

    xx

    ji

    vv

    ,,pentru

    ,pentru

    pentru0

    ( )kim d ix nx n .$ k .

    im ix nx .$ 1 2

    ! $ ( ) miim ,1= $ (( ( )

    $% & k 000'( ) ( ){ }kjijnj

    ki mvm += =

    +,1

    1 min

    $% ( k 000' ( ) ( )1+= kiki mm ni ,1=

    ( ) ( )siki mm = ( ) 1,i n = ( ) 1s k + 2 i

    ki mm = ( ) 1,i n =

  • $)6 $ ( )vXGv ,,= . { }1 2, ,..., nX x x x=

    $$+$ ( )njiij

    vV,1, ==

    6 V . $ ( )( )1im . ( )( )2im , ,

    $ )2 ( )( )1im $$ n V .( )t

    njjnv

    ,1= -

    2 $7 ( )( )1,

    ki

    i nm

    = $ (

    ( )( ) nikim ,11 =+ ( -2 $ (127

    ( )( )kim ( )( )1+kim 6 $ ix nx $ )

    $ 5 i din V ( )( )1+kim $( $ 6 $

    ( ) ( )1k ki ij jm v m

    + = + . ix nx $ ( )ji xx , -

    $ 5 j din V ( )( )1kim + 7 . +5 h , atunci al doilea arc va fi ( ),j hx x 8$$ nx

    $6$V $ )

    ( )( )

    =

    =

    ji

    jiijij

    xxji

    xx

    ji

    vv

    ,,pentru

    ,pentru

    pentru0

    6.$2

    ( )( )11,

    ki

    i nm +

    =$( ( ) ( ){ }1

    1,maxk ki ij jj n

    m v m+=

    = + *+$

    . + 7 $ $ +$ $

  • Flux maxim ntr

    ) & ( )pXG ,,=

    { }nxxxX ,...,, 21= . ( ) 1,i n = ( ) ii xx , Xx 1

    nx

    G 1x nx

    / ! :c ,,,astfel nct ( ) 0uc u - ( )uc

    ) ( " # XY .

    Y ( ) ( ){ } = YxYxxxY jiji ,, ( ) ( ){ }, ,i j i jY x x x Y x Y+ =

    $ ( )( ) ( ) ( ) = Yjxix ijpYc ,

    ( )Y

    ) *% : ,,,+ ( )pXG ,,= )

    )

    ( )( ),i jx x . ( )0 , iji jx x p !

    ( ) ix X . ( )( )

    ( )( )

    1 1

    , ,

    , ,n n

    k j j hk h

    x x x xjk j h

    x x x x= =

    = ,# & 2 7 x

    1xx nxx . $ + x $ $ +$ x

    ,# (% $ ) - ( ) ji xx , $

    n raport cu ( ),i j ijx x p = 6 astfel nct suma

    + nx $ + .

    ! . + : ,,,+ $ ( )pXG ,,= $ $ $ + )

  • 9

    ( )( )

    ( )( )

    1 1

    , ,

    , ,n n

    h n h nh h

    x x x xn nh h

    x x x x= =

    $% * % ( )pXG ,,=

    1x . nx : ,,,+ G .)

    ( )( )

    ( )( )

    11 1

    , ,1

    , ,n n

    i j ni j

    x x x xi j n

    x x x x= =

    = 12

    4 ( )( ) ( ) ( )( ) ( ) 1,, 1, ,j n ix x xx x xn n ij ix x x x+ = = $ + : ,,,+

    $% - " ( )pXG ,,= XY # 1x G 1x Y - nx G Yxn & : ,,,+

    ( )( ) ( ) ( )( ) ( ) ( )( ), ,, ,i j i jx x Y x x Yi j i jx x x x c Y

    + = 12

    &""'

    '$ $ $. 7 $ . :

    3 ( $ $ %Fulkerson pentru determinarea fluxului maxim ntr $

    $ 6 $ + 0 . $ 7 .de exemplu chiar fluxul avnd componente nule pe fiecare arc al

    . ( ) ( )( )0 , 0, ,i j i jx x x x = $ %$ (

    ;$.$( + 0 $+-

    $ )

    2 $:( $$ 1x cu semnul + ;2 vrfurile ( )1xx j + vor fi marcate cu 1x+

  • =

    ( )jk xx ( )jk xx +. 7 kx jx .;

    ' . 7 $ )

    * $ nx $. +$+$

    * $ nx $ . + $+ $$ )

    2$ 1x nx -2pe arcele drumului marcat cu + + $ (

    -

    2 +$$: -2$3$! . + + $

    7 $ nx ,# ? +$ 7

    unitate, evitndu$ $ . $ ) $$ V format din drumuri marcate cu + sau ce 1x nx . $ 7 $$

    nx 1x 0 +V ( )yx, y $

    [email protected]

  • 9A

    %

    2 { }1 2 3 7: , , ,d x x x x .2 { }1 4 2 7: , , ,d x x x x 2 { }1 2 4 7: , , ,d x x x x , $$2())$V )

    1x 2x 3x 4x 5x 6x 7x

    1x 0 2 6 11 2x 0 4 4 9 3x 0 1 11 4x 0 9 5x 6 0 14 19 6x 4 0 13 7x 0 ( )( )1im 9 19 13 0 ( )( )2im 20 13 10 9 15 13 0 ( )( )3im 15 13 10 9 15 13 0 ( )( )4im 15 13 10 9 15 13 0

    2 ( )1im V .$$$ ( )7 1,7j jv = -2 V ( )2im . ( )3im . ( )4im

    ( ) ( ){ }kjijj

    ki mvm += =

    +7,1

    1 min

    !. ( )2im . ( )21m $ 7 1 V ( )1im .

  • 9

    ( ) ( ){ }{ } 200,13,19,911,6,2,0min

    min 17,1

    21

    =+++++++=+=

    =k

    jjj

    mvm

    ( ) ( ){ }{ } 130,13,199,49,4,0,min

    min 27,1

    22

    =+++++++=+=

    =k

    jjj

    mvm

    ( ) ( ){ }{ } 100,1113,19,19,0,,min

    min 37,1

    23

    =+++++++=+=

    =k

    jjj

    mvm

    ( ) ( ){ }{ } 90,1311,19,09,,,min

    min 47,1

    24

    =+++++++=+=

    =k

    jjj

    mvm

    ( ) ( ){ }{ } 15019,1314,019,69,,,min

    min 57,1

    25

    =+++++++=+=

    =k

    jjj

    mvm

    ( ) ( ){ }{ } 13013,013,19,49,,,min

    min 67,1

    26

    =+++++++=+=

    =k

    jjj

    mvm

    ( ) ( ){ }{ } 000,13,19,9,,,min

    min 77,1

    27

    =+++++++=+=

    =k

    jjj

    mvm

    3 ( )3im ( ) ( ){ }27,1

    3 min jijj

    i mvm += = )( ) { } 150,13,15,119,610,213,020min31 =+++++++=m ( ) { } 130,13,159,49,410,012,20min32 =+++++++=m ( ) { } 100,1113,15,19,010,12,20min33 =+++++++=m ( ) { } 990,13,15,09,10,12,20min34 =+++++++=m

    ( ) { } 15190,1413,015,69,10,12,20min35 =+++++++=m ( ) { } 13130,013,15,49,10,12,20min36 =+++++++=m ( ) { } 000,13,15,9,10,12,20min37 =+++++++=m

    3 ( )4im ( ) ( ){ }37,1

    4 min jijj

    i mvm += = )( ) { } 150,13,15,119,610,213,015min41 =+++++++=m ( ) { } 130,13,915,49,410,213,15min42 =+++++++=m

    ( ) { } 100,1113,15,19,010,13,15min43 =+++++++=m ( ) { } 990,13,15,09,10,13,15min44 =+++++++=m

    ( ) { } 15190,1413,015,69,10,13,15min45 =+++++++=m ( ) { } 13130,013,15,49,10,13,15min46 =+++++++=m ( ) { } 000,13,15,9,10,13,15min47 =+++++++=m

    $ ( )3im ( )4im . $$B ( )4im ( ; 7x

  • 9

    6 1V ( )4im $(.$15 . ( )21 , xx 6 2 V ( )4im .( 13 . ( )42 , xx 6 4 V ( )4im .( 9 .$( ( )74 , xx *. 1x 7x { }1 2 4 7: , , ,d x x x x ( ) 17=dv (6$ 6 $+ 1x 6x 2 { }1 2 3 4 5 6: , , , , ,d x x x x x x .2 { }1 2 3 5 4 6: , , , , ,d x x x x x x 2 { }1 2 4 3 5 6: , , , , ,d x x x x x x , $$)2

    %

    ()! "# $$(

    V 1x 2x 3x 4x 5x 6x

    1x 0 5 8 18 2x 0 6 10 12 21 3x 0 9 11 23 4x 0 8 16 5x 0 9 6x 0 ( )1im 21 23 16 9 0 ( )2im 34 29 25 17 9 0 ( )3im 35 31 26 17 9 0 ( )4im 36 32 26 17 9 0 ( )5im 37 32 26 17 9 0 ( )6im 37 32 26 17 9 0

  • 9

    ( ) { } 340,9,1618,238,215,0max21 =++++++=m ( ) ( ){ } 29021,912,1610,236,210,max22 =++++++=m ( ) ( ) ( ){ } 25023,911,169,230,21,max23 =++++++=m ( ) ( ){ } 17016,98,160,230,21,max24 =++++++=m ( ) ( ){ } 990,90,16,23,21,max25 =++++++=m ( ) ( ){ } 000,9,16,23,21,max26 =++++++=m ( ) { } 350,9,1718,258,529,034max31 =++++++=m ( ) { } 31021,912,1710,256,290,34max32 =++++++=m ( ) { } 26023,911,179,250,29,34max33 =++++++=m ( ) { } 17016,98,170,25,29,34max34 =++++++=m ( ) { } 909,90,17,25,29,34max35 =++++++=m ( ) { } 000,9,17,25,29,34max36 =++++++=m ( ) { } 360,9,1718,268,531,035max41 =++++++=m ( ) { } 32021,912,1710,266,310,35max42 =++++++=m ( ) { } 26023,911,179,260,31,35max43 =++++++=m ( ) { } 17016,98,170,26,31,35max44 =++++++=m ( ) { } 909,90,17,26,31,35max45 =++++++=m ( ) { } 000,90,17,26,31,35max45 =++++++=m ( ) { } 370,9,1718,268,532,036max51 =++++++=m ( ) { } 32021,912,1710,266,320,36max52 =++++++=m ( ) { } 26023,911,179,260,32,36max53 =++++++=m ( ) { } 17016,98,170,26,32,36max54 =++++++=m ( ) { } 909,90,17,26,32,36max55 =++++++=m ( ) { } 000,9,17,26,32,36max56 =++++++=m ( ) { } 370,9,1718,268,532,037max61 =++++++=m ( ) { } 32021,912,1710,266,320,37max62 =++++++=m ( ) { } 26023,911,179,260,32,37max63 =++++++=m ( ) { } 17016,98,170,26,32,37max64 =++++++=m ( ) { } 909,90,17,26,32,37max65 =++++++=m ( ) { } 000,9,17,26,32,37max66 =++++++=m

    $$. ( ) ( )65 ii mm = 4+

    1x 6x $37

  • 9

    * $$+$ 2 ! ( )6im 1V .+ $37 $

    ( )21 , xx 2 ! ( )6im 2 V . + $ 32 . ( )32 , xx 2 ! ( )6im 3 V . + 26 . ( )43 , xx 2 ! ( )6im 4 V .+ 17 .$( ( )54 , xx /2! ( )6im 5 V .+ 9 . ( )65 , xx *$( . { }654321 ,,,,,: xxxxxxd ( ) 37=dv

    .

    & 6 $ 6 $

    1x 6x 2 { }1 3 6: , ,d x x x 2 { }1 2 6: , ,d x x x 2 { }1 4 6: , ,d x x x 2. Pentru a transporta n 6 $ $.

    ;$. ( ( . . + $( -$ $ ++$

  • 9/

    %

    !##

    1. Duda I., Trandafir R., Baciu A., Ioan R., Br( 6.* . B% Romnia de Mine.".AA

    * . C ,. " !. ,. . B% Romnia de Mine.".AA/

    "!.Matematici aplicate .B% Romnia de Mine.".AA

    * . # . B % Romnia de Mine.".===

    /$D:.*.B% Romnia de Mine.".==

  • 9

  • !

    "

    #$%&'(%$ (

    )

    * Romnia de Mine+,,+-+.+/ dobnda,

    procent mediu nlocuitor,

  • 0

    DOBNZI

    1$+

    2 34

    2 31 3+

    7.4. ndrumar pentru verificare/autoverificare

    1"2

    5

    +

    2 dobnzilor simple sau compuse.

    +

  • 0,

    3

    31

    6 dobnda.Dobnda

    2

    1,, 71,,

    #/ dobnda:

    100

    SpD Si= = #1/

    .

    100

    S p tD S i t

    = = #+/! 5" 38,

    3, 0S

    " " 5( )0 0 0 0 1tS S D S S it S it= + = + = + #3/

    #

    * nSS ,...,1

    ntt ,...,1 . Suma dobnzilor aduse de cele

    2 95

    1 1 2 2 ... n nS t S t S ttS

    + + += #:/ $ 1 2 ... nS S S S= + + + 5

    1 1

    1

    ...

    ...n n

    n

    S t S tt

    S S

    + += + + #-/ * nSS ,...,1 ntt ,...,1

    1 2, ,... np p p %tul mediu nlocuitor 5

    1 1 1 2 2 2

    1 1 2 2

    ......

    n n n

    n n

    S i t S i t S i tp

    S t S t S t+ += + + + #8/

  • 01

    3+

    2 5

    * 0S

    0S # 2/ tS 100p

    i =

    5

    '

    nceputul anului

    2

    1 0S iS0 ( )iSS += 101 + ( )iSS += 101 ( )iiSiS += 101 ( )202 1 iSS += t ( ) 101 1 += tt tSS ( ) iiSiS tt 101 1 += ( )tt iSS += 10

    ui =+1 2 ,...3,2,1=t

    5

    ( ) ttt uSiSS 00 1 =+= #/ . ntreg:

    ( )[ ] ( )111 00 =+= tt uSiSD #/ 5

    ( )01

    1

    tt tt

    S S S vi

    = =+ #0/

    vi

    =+11

    "&

    &

    #0/

    '5 0S nu este, n genera

    2 k

    hnt += '

    5# #/2

    0S

    5 ( )nn iSS += 10 ' nS 2

    k

    h

    k

    hiSn ' 5

    ( )

    ++==+ k

    hiiSSS n

    k

    hn

    t 110 #1,/

    reprezentnd " cnd se

    0S k

    hnt += 2

  • 0+

    # 0S

    k

    hnt += ( ) ( ) khntt iSiSS ++=+= 11 00

    51 4

    +

    1 i u+ = este n tabele financiare att pentru puteri ntregi,

    3 ! 0S 2 2

    2

    % " ($

    1p 2p

    1t 2t 1 1

    2 2

    t pt p

    = '* ai si

    ' ai si 1 2s ai i=

    ! 1 2

    ai ( )1 ai+ 1 2

    si ( ) ( )1 2 u.m. 1 u.m.s ai i+ = + 1 2

    si

    ( ) 2 221 1 1 12 4a a

    s a ai i

    i i i + = + = + + > +

    %" )$

    1p 2p

    1t 2t

    ( ) ( )1 21 21 1t ti i+ = + * 1 21 2;100 100p pi i= =

    #11/

    2 2

    an se ia dobnjk

    jk

    5

    1 1k

    ji

    k + = + #1+/

    ! ( reprezentnd suma dobnzilor percepute n cele /

  • 03

    !

    1 1k

    ji

    k = + #13/

    2 k 1 ji e+ = ( )ln 1j i= +

    ! 1 2 1 ji e+ = este dat n fiecare interval de timp

    ( ),t t dt+ ( )ln 1 i = + ;2 # /

    + ( )ln 1 1i i e = + = e 2 i >

    :ndrumar pentru verificare/autoverificar

    ($ +,,,, :->4 ?#2 1738,/

    /+,,9/++,,,9/+,+,,9/+++,,% 5/+&

    0D=S it # /

    ,7+,,,, 8% 0,08p i= = 9":-7 45360 8 45D 20.000 D 200

    100 360= =

    f 0S =S +D

    f 0 0S =S +S it

    ( )f 0S =S 1+it fS =20.200 % 5/)$Ce devine suma de 20.000 u.m. n regim de :

    8>>>0>?/+,,,9/+,,,9/+8,:9/+,3% 5/

  • 0:

    #

    ( )( )( )( )f 0 1 2 3 4S =S 1 1 1 1i i i i+ + + + 1 1=6% =0,06p i $2 2=7% =0,07p i $3 38% 0,08p i= = $4 49% 0,09p i= = $fS =20.000 1,06 1,07 1,08 1,09 fS =20.000 1,06 1,07 1,08 1,09

    20.000 1,3352

    ==

    fS =26.704 % 5/

    ,

    ($ 1 + 5 +,,, -,,,1,,,,0>1,>1+> #/383'

    2#2

    2 /5

    /:9/9/++,9/1:+3)$ 4 2

    :

    1,>

    --8:?

    0S = $ 1,,,,

    --># // 5

    812

    15082,35S + = 9/ 5812

    23082,35S + = 9/ 5812

    65083,75S + = 9

    / 58

    12

    13508,85S + =

  • 0-

    -$ 1,,,, --># /

    / 58

    12

    12077,77S + = 9/ 5812

    15077,97S + = 9/ 5812

    18077,97S + = 9/ 5812

    25077,37S + =

    ./"/

    1 $ & % (' $% ( )

    * Romnia de Mine(+,,8

    + $ & % ( ' $ % % )

    * Romnia de Mine(+,,-

    3(' )

    * Romnia de Mine(+,,:

    : $ % )

    * Romnia de Mine(1000

    - @ )

    * Romnia de Mine(1008

  • 8.4. ndrumar pentru verificare/autoverificare

    !

    " # $

    # % & ' ' '%

  • (

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    x n x x nn x n x

    x x

    D M MP E A

    D D+ += + = +

    ' S ( "

    x x n x n

    x

    M M DP S

    D+ + += ,2-

    ndrumar pentru verificare/autoverificare

    * /3 (

    33333334, 50 2.416M = 50 7070,2D = -#5

    6 #5723 0 ,-

    50

    5010.000.000 3.417.160

    MP

    D= =

    * /3 (

    3333333 ( * * 23 4 , 50 2.416M = (

    60 1.863M = ( 50 7070,2D = -#56 #57823 0 ,-

    50 60

    5010.000.000 782.160

    M MP

    D= =

    * /3 (

    3333333( e n *23 *234, 50 2.416M = 9 60 1.863M = 9 50 7070,2D = 9 60 3.885,7D = -

  • 7

    #56 #527833 0 ,2-

    50 60 60

    5010.000.000 6.278.040

    M M DP

    D += =

    &!

    . * , $-"

    - ( )1/ 2 1x xA v ia= 9 xa ( a x

    - ( )1/ 2 1x xA u ia= + 9- ( )1/ 2 1 ax xA v i= . ,x xC D , ( ) 1/ 21 xx x xC l l v ++= ( xx xD l v= -- ( ) ( )1/ 2 11x x xC i vD D+= + 9- ( ) ( )1/ 2 11x x xC i vD D += + 9- ( ) ( )1/ 2 11x x xC i D D+= + . ,x xM N

    - ( ) ( )1/ 2 11x x xM i vN N += + 9- ( ) ( )1/ 2 11x x xM i N N += + 9- ( ) ( )1/ 2 11x x xM i vN N += + +

  • 8

    / n xA * ( (

    !

    / /n xA * ( (

    ! *

    20 ( * (

    ,-( ! * (# "

    - /x n x x n

    n x n xx x

    D M MP E A

    D D+ += = 9

    - /x x x n

    n x n xx n x n

    D M MP E A

    D D+

    + += + = + 9

    - /x n x x n

    n x n xx x

    D M MP E A

    D D+ += + = + 9

    - 7:

    xyD introdus n cazul

    (

    "- x yxy x yD l l v

    += - ( )1 2x yxy x yD l l v + += - ( ) 2x yxy x yD l l v += -

    '( !$)!( $!

    ' ( %$ 6( ;( 6( ;*! .( (

  • ( ) ( ) ( | )P A B P B P A B =

    ( ) 0,2 1( ) 0,5

    ( | ) 0,4 2

    P A BP B

    P A B

    = = = = ( )( ) ( ) ( )( )

    ( ) ( )( ) ( )

    ( ) ( ) ( ) 0,2 0,25 0 0,45.

    C C

    C

    P A P A P A B B P A B A B

    P A B P A B P A B A B

    = = = == + = + =

    !"

    # $% &&% #$ %& &&&& %& &$& $%&&'#() & $&

    3

    4p = & 1

    4q =

    ( ) 150M = ( )2 37,5D = *%& +&&,-..& ( ) ( ) 37,5100 120 100 10 1 0,625

    100P P< < = < =

  • "

    *%& /$&0$

    (