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tinU gnilpmaS: pop lacitsitatS.: gnilpmas & elpmaS: elbairav modnaR: naem citemhtirA: noitaiveD dradnatS & ecnairaV : noitalerroC & ecnairavoC pop .retemarap: citsitatS elpmasA: ytilibaborP: noitatcepxE: noitubirtsid laimoniB: noitubirtsiD lamroN : tsiD tnedutS: pop fo noitamitsE .sretemaraP: rotamitsE & etamitsE: rotamitsE tseB: etamitse lavretnI & tnioP:

tinU gnilpmaS:

] ] ] " . ] ] ] ] ] ] ] " ] ] ]

. ] ( . ) ( ] ] ] . ]

)] . ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] )] ] ] ] ( ] ] ] ] ] ] ] ]

( .

.pop lacitsitatS:

" " ] ] ] ] ( )] ] ] ] ] ] ] ] . ] ] ] ] ] ( )] ] ] ]

] ] .pop etinifnI ] ] ] ] ] .pop etiniF] ] ] ]

.

gnilpmas & elpmaS:

. ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ...( ] )] ] ] ] ] ]

. .

" ] " ] ] ] ] ] ] ] " ] ] ] . "

:

] ] ] ] ]

. . ] ] ] " . "

N] ] ] ] ] ] ] ] ] ] ] ]

n < 03 03 . n .

] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ]

n f ] f] ] ] ] N

... 2 1+ + + =L n n n n ( ) ] ] ] =

i ( ] ] ] ] ] ] ) ] ] L] i

iN i

n = f

( : L) i in i i N

fn

Nf

n

Nf

n

LNL

L1

1

12

2

2

..., ,= = =

elbairav modnaR:

. ] ] ] ] ] ] ] ] ] ]

( ...,z,y,x) " " ] ] 3 2 1,..., , , :N X X X X X X N ] ]

( ) ( ) snoitavresbo] ] ] ] ] ] ]

.

: elbairav modnaR etercsiD ( )

]

01,...,4,3,2,1,0 :X elbairaV modnaR suounitnoC ( )

. ]] ]] ]] ]] ]] ]] ]] ]] ]] ]] ]] ]] ]] ]] ]] ]] y . ] ] ] ] ] ]

. 2.83,83,1.93,5.73,73:Y 14 63

] ] ] ] ] ] ] ] ] ] ] ] ] ] ".: "

] ] ]

2,1,1,2,1=X

. 2 1

naem citemhtirA:

] ] ] ] ] ] ] . N X .

= = )1( X N

ii

N

1

i X i

N

=

1

= i xi i . N

x x n,...,2,1

:

)2( n

xx

n

ii

=

= 1

] ] ] ] ] ] . naeM .

: noitaiveD dradnatS & ecnairaV

] noisrepsiD fo erusaeM ] ] ] ] ] ] ] ] ] . ] ]

. ] ] ] ] .s2 2

:

x) ( )3( N

i

i

N

=

2

1

. N

)4( =

=

n

i

i

nSxx

1

22

1()

. 1-n n 03 n ] ] n ] ] ] ] x] :

:

= )5(

=

u X) (N

ii

N2

1

: s

S )6( x x

n

ii

n

=

=

2) ( 1

1

S )7( n

ixn xi

n

=

=

2 1 12

1

) (

: rav( x) v(x) )X(raV V(X)

) ( =x sav s

noitalerroC & ecnairavoC

]] ]] n ]] ]] ]] n ]] ]] y x]] ]] ]] ]] ]] ]] ]] ]] ]] ] 2 2 1 1n ny x y x y x) , ( ,...,) , ( ,) , (

: voC( y,x)

)8( =

=

n

n ivoCxyxxyy

1 1 (,)(())

. n03 1-n n ] ] ] ] ] ] . y x] ] ] ] ] ] ] ] ]

.

] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ]

:

y x, (voc )9( y xn yn

= ) i ix

1

:

)01( SS xy =(voc,) rxy

1 1-]]] ]]] ]]] ]]] ]]] : ]]] ]]] y X ]]] ]]] ]]] ]]] ,x yS S]]]

] ] ] ] + 1 ] ] ] ] x,y ] ] ] ] ] -1 ] ] r 11 . . ] ] ] ] ] ] ] ]

:

y y x x r )11( S S ni i

iy x

n

=

=

11) () () (

y xn y x r )21( S S ni i

iy x

n

=

=

11) (

xn x S

n

yn y Sn

xi

i

n

yi

i

n

=

=

=

=

2 2

1

2

1

1

1

.

retemarap .pop:

. ( ) ] ] ] ] ] ] X] ] ] ]

] .

. ] ]] ]] ]] 2 ]] ]] ]] ]] ]] . ]] ]] ]] ]] ]] ]] ]] ]] ]] ]]

:

==

==

N

ii

N

iNX i

X1 N

22

1 1, 1()

citsitatS elpmasA:

. ] ] ] ] ] ] ] ] ] ] ]

: 2S x .

x

nx

Sn

x x

ii

n

ii

n

=

=

=

=

1

11

1

2 2

1

) (

ytilibaborP:

07 % 05

] ] ] . ] ] ] ] ] % . ] ] ] ] ] ] ] ] ] ] . ] ] ] ]

)E(P-1=)E(q: )E(q ] P(E) ] E] ] nA]] ]] . ]] ]] ]] ]] ]] ]] ]] ]] ]] ]] . ]] ]] ]] ]]

elbissoP]] ]] ]] " ]] ]] ]] ]] ]] ]] ]] ]] ]] ]] " ]] tnemirepxE . tneve nA emoctuo

)E(n ( E )] ( ] ) ] ] N) ( ] ] ] ] ]

)E(P ] E] ] ] ] :

E n E P N

n

N) ( ) (

) (= =

] . ] ] ] ] ] ]

01 ] ] ] ] ] 002] ] ] 1 E ..., ,2 1E E] ] ] ] ] ] ]

01=n 002=N ] ] ] ] 2 E] ] : 1) (E P

E n E P N

n

N) ( ) (

) (1.

01 1002

= = = =50

% . 5 qEPE === 11 ()1()1.50.59 % 59

noitatcepxE:

] 2 1,..., , :nx x x X ] ] ] ] ] ] ] X ] ] ] ] : )X(E ( ) f(ix) ix

)41( =

=

n

iEXfxx ii

1 ()()

:

=) ( ) (i ixd x f x X E )51(

. X] ] ] ] ] )x(f]

] ] f(ix)] ] fn

. n / i in x f =if n i ] E(X) ] ] ] n ] ] f(ix) ] ] ] /in f ] ] .

:

: 5 4 3 2 1 X

.50 .03 .53 .2 .50 .50 fxi) ( ( )

:

i ix f x X E i

n

=) ( ) (=

1

50.x5+3.x4+53.x3+2.x2+50.x1+50.x0 = 3 = 59.2 =

.

: . .

t) ] ]

] ] ] ] ] ] ] ] ] ] ] ] ] ] ( .

noitubirtsid laimoniB:

] ( ] ] ] ] ] ] )] ] ] ] ] ] ] ] ] ]

X n] ] ] . ] ] ] ] . . n ( )

] ] ] f(fxi) () ] ] ] ] X] ] ] ] ] ] ] ] .

: )x(f

pq xnx )61( xnx

fxn

=

!(!) ()!

: . P 1=q+p q . n x 4321 !4 n 1 ] ] ] ] ] ] ] n ] ] !n

. !O

: pn = )71(

: qpn = 2 )81(

: noitubirtsiD lamroN

] ]] ]] ]] ] ]] ]] ] ]] ]] ] ]] ]] ( ]])] ]] ] : . . + x . . ]] ]] ]] ]] ]] 2 ]] ]] ]] ]] ]] ]] ]] ]] ]] ]]

. lamroN N 2 ) , (N .

:

()/ 22 )91( 2

2 ()11

= fxex

X 82817.2 e 95141.3

)x(f

0 = noitubirtsid lamroN dradnatS] ] ] Z ] ] )1,0(N ] ] ] 1

:

)02( 2

21

2= fxez ()1

=x Z

2 = Z % 72.86 1 = Z] ] ] ] ] % 37.99] 3 = Z] % 54.59

] Z ] ] ] ] 0=Z] ] ] . 0=Z

(1)

tsiD tnedutS:

2 ] ] ] ] ] ] ] . 2S ] ] ] ] ] ] ] ] ] ]

t t ( n

t n] ] : t

tY x f )22( n

n) (

) (=

+

0

121

t t . 1-n] ] ] ] 0Y]

. ] ] ] ] ] ] ] . ] ] ] ] ] ] ] ] 1-n] ] ] ] . t ] ] ] t]

. n 51 5

(2) t

sretemaraP .pop fo noitamitsE:

,2 ] ] ] ] ] . ] ] ] ] ] ] ] ] ] ] ] ] ]

.

rotamitsE & etamitsE:

] ] ,..., ,2 1nx x x] ] ] ] ] ]] ]] ]] ]] ]] ]] ]] ]] ]] ]] ]] ]] ]] ( ]])]] ]] . ]] ]] ]] ]]

.

. )=x () ( )] ] ] ] x ] ] ] : rotamitsE

) == xfxxxn 12 (,,...,)

=

==

n

ixx ni

1

)1

rotamitsE tseB:

] ] ] ] ] ] ] ] ] ] ] ] ] ] ]

. ] ( ] )] ] ] ] ] . ] ] ] ] ] ] ] ] ] ] ] ] ] ]

. tseB .

: ssendesaibnU ycnetsisnoC ycneiciffE ycneiciffuS

) :

()= )32( )

E .

: : u x

=

==

n

ixx ni

1

)1

:

: 2 x

s) (n

ix xi

n2 2

1

11

=

=

: 2S

= =12 2 2S NN s E) (

] ) n] ] ]

)

:

>= mil0 ) ( )42(

np

>0 :

) (mil0 ) (mil

nV

nE

:

. x 2 n] ] :

milmil/0 ) ( ) (mil

2

=

nVxnn

nx

. x

,1 2

) (.636 ) (

2==

VEMVx

. EM x

] )

)

)

) .

f(,x) ] ] ] ,..., ,2 1nx x x ] n ] ] ] ]

: = , ... , , , ; ,..., ,2 1 2 1n nx f x f x f x x x g) ( ) ( ) ( ) (

== 1212 ,,...,;,,,..., gxxxhkxxx nn) ( ) ( ) ( . ,..., ,2 1nx x x k) (:

: . a x ( )

ne x x x g) ( ) (n

ia x2 1

12

212

2

; ,..., ,

=

: x

) ( ) (kxxxhxa) ( ) (

exxxan

i

n

,,...,,2

-n2

12

1

12

2

21

2

=

=

. x

etamitse lavretnI & tnioP:

] ] ] ] ] ] ] ] :

etamitse tnioP -

.] ] ] ] ] ] ] ] ] p . x ( )] ]

. P

etamitse lavretnI ecnedifnoC - . ] ] ] ] ] ] ] ] ] ] ] ] ]

. stimil ecnedifnoC] ] ] . ] ] ]

. h h ] . ] ] ] . ] ] ] ] ]

] h h h ] h h ] ] ] ] ] ] ] ] : x .

x Z )52( n

=

/

x . ]

. Z n 2 ) n (3)

:

=

121/2 /

zn

pzx

2 1 2 1 Z ] ] 2 ] ] ] ] ] 2 Z]

1 ] ] ] ] ] ] ] ] :

z x )62( n

z xn

+ +

2 1 2

% 59 ] 2 1 Z ] ] ] ] ] ] ] ] 2 Z] )

( . -69.1 2 Z 69.1 2 1 Z

]] ]] ]] ]] ]] ( ]] )t ]] ]] ]] ]] ]] ]] ]] ]] ]] ]] ]]

ns

x=

: )

x) (tx

=

: 1-n t

n ) ( ) ( )72( xts

n ++ /2,11/2,1 nn xts

:

s) (n

ix xi

n2 2

1

11

=

=

1-n ( 1%) ( t) t

n N N

1 :

t x) ( ) (sn

s t x u fn

1 11 2 1 1 2, / , / + +n nf

s] ) n

xn 1f = 22

:

n xn NN

sn

n NN

22 2

1=

=

]]] s ]]] ]]] ]]] ]]] ]]] ]]] ]]] ]]] ]]] ]]] 2S ]]] ]]] ]]] ]]] ]]] 2s ]] ]] ]] 2s ]] ]] )x]] ]] ]] ]] ]] ]] ]] ]] ]] ]] ]] . ]] ]] ]]

. 2s 2

]]] ]]] ]]] ]]] n]]] ]]] ]]] ]]] ]]] ]]] ]]] ]]] ]]] ]]] ]]] ]]] ]]] . ( 50.= % )59