Ho28-hw7solutions

18

1.00

16

0.90

0.80

14

0.70 12 0.60 10 0.50 8 0.40 6 0.30 4

0.20

2

0.10

0

0.00 -INF - 0

0.5 - 1

1.5 - 2

2.5 - 3

3.5 - 4

4.5 - 5

5.5 - 6

Number of Grades within Range

6.5 - 7

7.5 - 8

8.5 - 9

Cumulative

Question 1

10%

50%90%

Question 2

10%

50% 90%

Question 3

10%

50%

90%

Question 4

10%

50%

90%

Question 5

10%

50% 90%

Question 6

10%

50%

90%

Question 7

10%

50% 90%

Question 8

10%

50%

90%

Question 9

10%

50%

90%

Question 10

10%

-2.00

-1.50

-1.00

-0.50

0.00

50% 90%

0.50

1.00

Score on HW Question (./1)

!" # $ %

& %

9.5 - 10

'

( " $

) )

"

$

( , ) ( ( )

$ , ,

(

($ ,

$

$$

,

" $ (

( "

(

* & ( . ) / $ (

* "

)

+ (

(

(* *'

(

"

(

$$

*

* " $

( 0 (

$

$

(

0

*

*'

(

'

" $ $

$ " (

) , (

1 ,

)

"

*

" $ "

"

)

*

$

(

!"



%$2 *

*

& %

"

0.5 Smarty Jones 14 0.3 Lion Heart

Bet Smarty Jones 2

-10 0.2 Imperialism -10 0.5 Smarty Jones -10 0.3 Lion Heart

Bet Lion Heart 2

1

2

30 0.2 Imperialism -10 0.5 Smarty Jones -10 0.3 Lion Heart

Bet Imperialism 2

-10 0.2 Imperialism 50

4)

$ 5

"

Lion Impe

'

)

$

*

Prior Smar

)

0.5 "Smar" "Lion" "Impe" 0.3 "Smar" "Lion" "Impe" 0.2 "Smar" "Lion" "Impe"

1 ,

Cond'l Prob Joint Prob 0.8 0.4 0.1 0.05 0.1 0.05 0.1 0.03 0.8 0.24 0.1 0.03 0.1 0.02 0.1 0.02 0.8 0.16

$

Pre-Posterior 0.45 Smar Lion Impe "Lion" 0.31 Smar Lion Impe "Impe" 0.24 Smar Lion Impe "Smar"

(

!" 3 $ %

2 *

Posterior Joint Prob 0.889 0.4 0.067 0.03 0.044 0.02 0.161 0.05 0.774 0.24 0.065 0.02 0.208 0.05 0.125 0.03 0.667 0.16

*

& %

0.889 Smarty Jones 4 0.067 Lion Heart

Bet Smarty Jones 1.333333

-20 0.044 Imperialism -20 0.889 Smarty Jones -20

0.45 "Smarty Jones" 1.333333

1

0.067 Lion Heart

Bet Lion Heart -17.3333

20 0.044 Imperialism -20 0.889 Smarty Jones -20 0.067 Lion Heart

Bet Imperialism -17.3333

-20 0.044 Imperialism 40 0.161 Smarty Jones 4 0.774 Lion Heart

Bet Smarty Jones -16.129

-20 0.065 Imperialism -20 0.161 Smarty Jones -20

0.31 "Lion Heart" 8.8

10.96774

2

0.774 Lion Heart

Bet Lion Heart 10.96774

20 0.065 Imperialism -20 0.161 Smarty Jones -20 0.774 Lion Heart

Bet Imperialism -16.129

-20 0.065 Imperialism 40 0.208 Smarty Jones 4 0.125 Lion Heart

Bet Smarty Jones -15

-20 0.667 Imperialism -20 0.208 Smarty Jones -20

0.24 "Imperialism" 20

3

0.125 Lion Heart

Bet Lion Heart -15

20 0.667 Imperialism -20 0.208 Smarty Jones -20 0.125 Lion Heart

Bet Imperialism 20

-20 0.667 Imperialism

!"



%$40

& %

6 '

, $ 7 8 9 # : 36 ;# 1 , $

(

62 *3 *

0.5 Smarty Jones 0.785202 Bet Smarty Jones

$

u-Value 14 0.785202

0.3 Lion Heart

-0.6074

-2

-10

-2

-2

-10

-2

-2

-10

-2

0.2 Imperialism

0.5 Smarty Jones

0.3 Lion Heart

Bet Lion Heart 1 -0.6074 -4.32015

-1.11111

0.962963

30 0.962963

0.2 Imperialism -2

-10

-2

-2

-10

-2

-2

-10

-2

0.5 Smarty Jones

0.3 Lion Heart

Bet Imperialism -1.40082

0.2 Imperialism 0.995885

' ( (, $$

1 , ) )

$

( "

> "

$

$ (

50 0.995885

62<*<=* ' $ " " ,

*

!"



%$& %

$ *

0.889 Smarty Jones $ 0.355606 Bet Smarty Jones

u-Value 4 0.355606

0.067 Lion Heart

-0.57279

-8

-20

-8

-8

-20

-8

-8

-20

-8

0.044 Imperialism

0.889 Smarty Jones

0.45 "Smarty Jones" -0.57279

1

Bet Lion Heart

0.067 Lion Heart

-7.40741

0.888889

20 0.888889

0.044 Imperialism -8

-20

-8

-8

-20

-8

-8

-20

-8

0.889 Smarty Jones

Bet Imperialism

0.067 Lion Heart

-7.60055 0.044 Imperialism

0.987654

40 0.987654

0.161 Smarty Jones 0.355606 Bet Smarty Jones

4 0.355606

0.774 Lion Heart

-6.65232

-8

-20

-8

-8

-20

-8

-8

-20

-8

0.065 Imperialism

0.161 Smarty Jones

0.31 "Lion Heart" -1.0864 -6.69426

-1.11828

2

Bet Lion Heart

0.774 Lion Heart

-1.11828

0.888889

20 0.888889

0.065 Imperialism -8

-20

-8

-8

-20

-8

-8

-20

-8

0.161 Smarty Jones

Bet Imperialism

0.774 Lion Heart

-7.42015 0.065 Imperialism

0.987654

40 0.987654

0.208 Smarty Jones 0.355606 Bet Smarty Jones

4 0.355606

0.125 Lion Heart

-6.25925

-8

-20

-8

-8

-20

-8

-8

-20

-8

0.667 Imperialism

0.208 Smarty Jones

0.24 "Imperialism" -2.00823

3

Bet Lion Heart

0.125 Lion Heart

-6.88889

0.888889

20 0.888889

0.667 Imperialism -8

-20

-8

-8

-20

-8

-8

-20

-8

0.208 Smarty Jones

Bet Imperialism

0.125 Lion Heart

-2.00823 0.667 Imperialism

!" < $ %

0.987654

40 0.987654

& %

!

" '

1 , " * @

,

( " ( , p 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

,

" )

"

(

1 .$ )) / ) 1 , $+

) " ?#

#

( (

, " ( "(

$ $ " * '

Sensitivity to p for Risk Neutral Case

CE 6.6 6.54 6.48 6.42 6.36 6.3 6.24 6.18 6.12 6.06 6

4

$

CE

$

,

6.7 6.6 6.5 6.4 6.3 6.2 6.1 6 5.9 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

p

$ $

" $

( ,

$* '

1 , , A B (

,

$ 9*

# " , ?# p 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

( $

, $ 7 8 9 # : 36 ;# ) 1 , , , $ ( , " , $ .$ )) /)

CE -7.18709 -7.82247 -8.36703 -8.79941 -9.08972 -9.19426 -9.04544 -8.53039 -7.43855 -5.30113 -0.64981

$

$ 1 "

) " *

" )

$

Sensitivity Analysis to p for Risk Averse Case 0 -2

CE

6

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

-4 -6 -8 -10 p

!" % $ %

& %

0.8

0.9

1

'

1 , (

( (

C

" , :

9# ( 1

$

( (

" 5 6,

* ,

1 , $> $ 9#*

( ( (,

)

$ *<3 7

, ( ) " 36% $ $ ( $ ", , $ $ ( " * 7& ) 6, 36#3 $ *8

!"



%$$

" (,

* ) , 1 , $62 *3 , " $$ ) "

( 8* ' $

6,

( $

"

$

& %

$

) *

( ) )"

0.79 S Outdoor

100 0

$49.80 0.67

$85.00 0.92

0.21

$85.00

R -$15.00 0 0.79

-$15.00 -0.31

S

0.43

$75.00

"S"

Porch 0

$53.70 0.70

2

90 0

$53.70 0.70

$75.00 0.86

0.209302326 R

$5.00 20

$5.00 0.09

0.79 S

$25.00 Indoor

40 0

$26.98 0.42

$25.00 0.39

0.209302326 R

$35.00

50

$35.00 0.51

0.11

$41.74 0.59

S $85.00 100 0

-$9.08 -0.18

$85.00 0.92

0.89 R

-$15.00 0 0.11

-$15.00 -0.31

S

0.57

$75.00

"R"

Decision 13

90

3 0

$33.88 0.50

0

$9.87 0.17

$75.00 0.86

0.89 R

$5.00 20

$5.00 0.09

0.11 S

$25.00 Decision 14

40 0

$33.88 0.50

$25.00 0.39

0.89 R

$35.00 50

%

! "" " )

&

,

'($)

* $+

$ $

) 6,

$

)

$35.00 0.51

(

+

" $

( *

, * = -

*<3

>

*<< !" = $ %

& %

>

*

!

-

$ 8* D

7 * "

( "

(,

$

) ) " $ 1

$ 7( (

(

,

) , " "

$ , " 8* & 8 8*

)

* F $

*

$ *> $" " 6 , A) $

2

2#

8*

(,

72#

8

72#

( *-

, $ ( 1 ,

E

, *E 06 ) ) 7 6 , $ 2 ( *

2

8?* 72 8 9 7> 8

2 + $ G ( >

$> , ( , 5 ,

"

( H B *' ) 1

(

$

( $ ! -! $

B *

:' 6, L

:' 1 I J K + 9 ? % MI J K* ' 6, , , ) $ , ) $

(

$ I J K 9 *% -

, (

", 6, $ , $ $ $ $ I J K9 ;3 9 ;3 ( * $ , $I J KN ;3 ( , $ I J K O ;3 , L

!" #



%$& %

K

:' 6, $ ( $ + 4> 9 ? M I J K* > ) ) $$ ( , +

", ,

$

, •

P

, Q * R $ $

A

'

•

P , Q* I J K* ' A I J K9 ;3L

•

P

$

(

$

4>

9

#*

;3R $

, Q ;3 #R

,

@>

$

9

9 $

( :

4>

9 M I J K* ' $ I J K9 * L

@>

9 ? MI J K # ? ;3 (

4>

"$

@>

,

$I J , $ (

06 6,

(

@>

:

>

,

A

:

@>

9# ? $

$I J K*

$ I J K9 ;3*

"

$

: 4 )9 * +

( ")

1 9 *= E 0 5

“i0” i0

0.5

0.4

“i1”

$

)

0.2 0.2

0.5 “i0” i1

0.1

0.6

“i1”

0.06 0.54

0.9

4 )) "

+ i0 “i0”

0.77

0.26

i1

0.2 0.06

0.23 i0 “i1”

0.27

0.74

i1

0.2 0.54

0.73

&

)

(, K9 *%%

$ I J K9 *

,

, I J.#/ K9 * %* ' , - ( (

$( $

! " ## $ %

$

*

" , ./

I J. /

,

L & %

$ :4 ( " ) ) , E 0 5 ) , $ , .#/L $ )

: E 0 5

$

" , - $ (

"

, , ./

)

$

( "

" E 0 5 $ * I J. / KS *%

$

+

“i0” i0

p

0.4

“i1”

0.4p 0.4(1-p)

1-p “i0” i1

1-p

0.6

“i1”

0.6(1-p) 0.6p

p

4 )) "

+ i0

0.4p

“i0”

0.4p / (0.6 - 0.2p)

0.6 - 0.2p

i1

0.6(1-p)

0.6(1-p)/ (0.6 - 0.2p) i0

0.4(1-p)

“i1”

0.4(1-p) / (0.4 + 0.2p)

0.4 + 0.2p

i1

0.6p

0.6p / (0.4 + 0.2p)

&

(

$

,

(

( " $ ( ( $ I J.#/ KT *#*

$) S * 0 , $)S *

>

•

, "I J. / K9 *% + 0.4p / (0.6 - 0.2p) = 0.75 => p 0.82;

•

, "I J.#/ K9 *#+ 0.4(1-p) / (0.4 + 0.2p) = 0.1 => p 0.86;

' $ )) A

,

$) S * , $

F* , - ,

,

(

0 , " . /(

$ E 0 0

*

.

!" #



%$& %

( *4 $

( (

(

$

$) -

($

,

",

* ( $ ($ 0 ( $

( $

> > (

,

0

* "

($ $

) *

> ( $

, $$ ( 2 *

$

$

,

/ : ( "

$ ,

"

*

( (

(

* $

: (

"

, 0 ( "

$

$ "

(

*

* : C

$ ) +

)

A ) $( " , $ ) ,

", $

) , (

"

+ High Superlight **

-65

0.3

-200

0.5 Medium

-20

0.2

25

-65 Low

-125 Basic

1 ,

1

( 62< *

)

! " #3 $ %

)

"

& %

*

"+ Superlights -200 High

-125 0.3 ** Basic Superlights **

-42.5

0.5 Medium

-125 -20

-20 -125 Basic Superlights **

0.2

25

25

Low -125 Basic

, ( , 1 7−2 * 8 − 7−2< 8 = 2 +

$ , − 2< * C* , )

−2 $

1 , $

>

$

,

*

, (

1 , *

0.0 0

0.2

0.4

0.6

0.8

1

-50.0

)

"

-100.0

-

-150.0

-200.0

-250.0

'

")

)

≈ *<#*

!" #



%$(

& %

,

+

)

$

GP>*

50 45 40 35 30 25 20 15 10 5 0 0

'

0.2

0.4

0.6

0.8

≈ *<#*

A ) ,

1

(

)

)

U

+ 4 ( ) )

0 *

)

0

",

)

Pass 0.3

0.3 High 0.7 No pass Pass 0.5 0.8 Medium 0.2 No pass Pass 0.2 0.9 Low 0.1 No pass

'

( ) "

High 0.09 0.21

0.13 0.60 0.67 Medium 0.27 Low High 0.64 0.33 0.30 No pass Medium 0.06 Low Pass

0.4 0.1 0.18 0.02

()

*

!" #



%$& %

0.09 0.4 0.18 0.21 0.1 0.02

High Superlights ** Pass

-32.09

0.13

-200

0.60 Medium

-20

0.27

25

-32.09 0.67

Low -125 Basic

-62.75

High Superlights 0.33 No pass

-131.82

0.64

-200

0.30 Medium

-20

0.06

25

-125 Low ** Basic

'

,

( GP 1

-125

$ 1 − 2 < *% 7−2< *% 8 − 7−2< 8 = 2 * "

+

( (

,

(

$

($

)

*

) " -

2

) )

) ) (

(

1

C*

(

$

"

( "

"

! " #< $ %

*

& %

− 2<

) " -

2

V)

(

) )

(

"

) A (

$

$ ,

, ,

) ) )

"

*& , (

$

(

)

L

0 *

0 *

*& (

$$

+

,

) " U

V)

U

2

) )

5

$ ( $

" ,

7$

"

, (

(

0 0

) " -

"

$

) $

"

$

2

) )

&+ '

8( ( "*

$

$

"

0

! " #% $ %

0

$

, & %

*

,

) " U

V)

U

2

'

"

5

$

) )

(

"

$

$

"

(

0 0

) " U

'

-

$ $

8* ' *

(

2

) )

$

7

"

)

( "

"

, $

$

$

*

(

$)

) )

($

"

$

*

!" #



%$)

& %

Sun

Rain

"Sun" p 0.4 1-p "Rain" "Sun" 0.6 1-p p "Rain"

Sun "Sun" 0.4p / (0.6 - 0.2p) 0.6 - 0.2p Rain Sun 0.4 + 0.2p 0.4(1-p) / (0.4 + 0.2p) "Rain" Rain

0.4p 0.4(1-p) 0.6(1-p) 0.6p

"Sun" 0.6 - 0.2p

Outdoors

Sun 0.4p / (0.6 - 0.2p)

Porch

Rain Sun 0.4p / (0.6 - 0.2p)

0.4p 0.6(1-p) 0.4(1-p) 0.6p

Rain Sun 0.4p / (0.6 - 0.2p) Indoors Outdoors

0.4 + 0.2p "Rain"

Rain Sun 0.4(1-p) / (0.4 + 0.2p) Rain Sun 0.4(1-p) / (0.4 + 0.2p)

Porch Rain Sun 0.4(1-p) / (0.4 + 0.2p) Indoors

100 0

1.00 0.00

90 20

0.95 0.32

40 50

0.57 0.67

100 0

1.00 0.00

90 20

0.95 0.32

40 50

0.57 0.67

Rain

W

0 $

(

) $ , ) $

H

0 0 ( )L$ )

* (, A ) *

! " #= $ %

(

$$ 0 (

5

& %

@

( $

) "

$$

, "

$

$ ) (*

$"

,

$

25 Value of Information

20 15 10 5 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Accuracy

( , $ ,

$$ + 3F

" #%F* @

, (

$ $

1

2# (

( $$ ( " H

" *

C

! + ' E)

(

( E

) C , 4 $ E) *

7>> $ $ " ( "

!"

+ 4

0 0



%$&& 8 , ;")

*

& %

$ ;

,

E)

E)

>> &&

2

E 4

@

, $

" $ E &

"

"

"

"

$

(

. $

$

L$ / B ) )

1

)

( 1 )

( L$ A ) $ " " $C 4 *

$

(

( 4

)

( 4* ' (

,

'

)

0

) ) ( " ( 5

4

" *

E "

(

4 ,

$

A ) ( $ L

.$

( (

$

* )

(

/ 0 $C "

" ( " E) * " $

*

"+ 4

$C 0 ,

(

5 )(

0

* 6

$

(

6) )

,

−

7 8 = #− ρ ( ρ C ) ( ) *

5

0

*4 *4 $ ρ = 2=%* 3 C &

(

!" # $ %

( ρ = 2 # *# C ρ = 2# C $

& %

0 $ $

Driver 0.1 $ 62.03 0.46 Bay Area 0.3

CCI **

62.03

0.462

0.22

$ 42.00

0.34

New Client 0.4 $ 53.00 0.6 $ 32.00

0.41 0.27

0.34 0.5 San Francisco

$

$ 25.00 San Jose 0.5

0.6

0.33

0.57

Japan

Senior 0.85 $ 95.00 0.15 $ 35.00 Associate

0.61 0.30

$ 60.00

0.45

WWS

" ( + 4 C

$$ $

, ( "

(

&& *

>> *

(

,

$$

2< * 3C

$

+ ,

$>

0

$

$70

(

, *

>>

Ce rta in Equiva le nt

$60

&&

$50 $40 $30 $20 $10 $0

200

400

600

800

1000

1200

Risk Tole ra nce

' + @ (C $ *

, 7

(

8

))

(

0 "

"

( "

!"

$



%$ρ ≈ 2< C*

( ( "*

)

$

,

& %

, E)

&

$ U

E)

>> &&

V)

2

E 4

'

,

)

(

,

$

4

( 4

,

*

Driver 1 $ 25.00 0.22 Bay Area

CCI

0

0.34 0.5 San Francisco

"Driver"

$ 25.00

0.22

$ 42.00

0.34

New Client 0.33 0.4 $ 53.00 0.6 $ 32.00

0.41 0.27

San Jose 0.5

0.451 0.1

Japan

0

0.57

** WWS

Senior 0.85 $ 95.00 0.15 $ 35.00 Associate

0.61 0.30

$ 60.00

0.45

$ 25.00

0.22

$ 42.00

0.34

New Client 0.4 $ 53.00 0.6 $ 32.00

0.41 0.27

Driver 0 $ 40.93 0.34 Bay Area

CCI

0.34

San Jose 0.5

1 0.5 San Francisco $

73.36

0.519815 "Bay Area"

0.3

0.451

0

0.33

Senior 0.85 $ 95.00 0.15 $ 35.00 Associate

0.61 0.30

$ 60.00

0.45

$ 25.00

0.22

$ 42.00

0.34

New Client 0.33 0.4 $ 53.00 0.6 $ 32.00

0.41 0.27

0.57

Japan VOC =

11.33

** WWS Driver

San Jose 0.5

$ 83.37 0.57 Bay Area

CCI **

0.6

0 0.34 0

0.5 San Francisco

0.566

"Japan"

Japan

1

0.57

Senior 0.85 $ 95.00 0.15 $ 35.00 Associate

0.61 0.30

$ 60.00

0.45

WWS

!" 3 $ %

& %

Driver 0.1 $ 62.43 0.46 Bay Area 0.3

CCI **

0.34 0 San Francisco

"San Jose"

$ 25.00

0.22

$ 42.00

0.34

New Client 0.33 0.4 $ 53.00 0.6 $ 32.00

0.41 0.27

San Jose 1

0.464 0.5

Japan

0.6

0.57

Senior 0.85 $ 95.00 0.15 $ 35.00 Associate

0.61 0.30

$ 60.00

0.45

$ 25.00

0.22

$ 42.00

0.34

New Client 0.4 $ 53.00 0.6 $ 32.00

0.41 0.27

WWS $

VOC =

62.03

0.462233

Driver

0

CCI **

0.5 "San Francisco"

0.1

$ 61.64 0.46 Bay Area 0.3

San Jose 0 0.33 1 San Francisco

0.33

0.460 Japan

0.6

Senior 0.85 $ 95.00 0.15 $ 35.00 Associate

0.61 0.30

$ 60.00

0.45

$ 25.00

0.22

$ 42.00

0.34

New Client 0.41 1 $ 53.00 0 $ 32.00

0.41 0.27

0.57

WWS

Driver 0.1 $ 64.36 0.47 Bay Area 0.3

CCI **

San Jose 0.5 0.38 0.5 San Francisco

"New client"

0.475 0.4

Japan

0.6

0.57

Senior 0.85 $ 95.00 0.15 $ 35.00 Associate

0.61 0.30

$ 60.00

0.45

$ 25.00

0.22

$ 42.00

0.34

New Client 0 $ 53.00 1 $ 32.00

0.41 0.27

WWS $

62.03

0.462233

Driver 0.1

VOC =

0

$ 60.51 0.45 Bay Area 0.3

CCI **

San Jose 0.5 0.31 0.5 San Francisco

0.6 "No new client"

0.27

0.454 Japan

0.6

0.57

Senior 0.85 $ 95.00 0.15 $ 35.00 Associate

0.61 0.30

$ 60.00

0.45

WWS

!"



%$& %

Driver 0.1 $ 67.50 0.49 Bay Area 0.3

CCI **

0.34 0.5 San Francisco

"Senior"

$ 25.00

0.22

$ 42.00

0.34

New Client 0.4 $ 53.00 0.6 $ 32.00

0.41 0.27

San Jose 0.5 0.33

0.491

Senior

0.85

Japan

0.6

0.61

1 $ 95.00 0 $ 35.00 Associate

0.61 0.30

$ 60.00

0.45

$ 25.00

0.22

$ 42.00

0.34

New Client 0.33 0.4 $ 53.00 0.6 $ 32.00

0.41 0.27

WWS $

66.34

0.4849

Driver 0.1 $ 35.68 0.30 Bay Area 0.3

CCI VOC =

San Jose 0.5 0.34 0.5 San Francisco

4.31 0.15 "Associate"

0.451

Senior Japan

0.6

0.30

0 $ 95.00 1 $ 35.00 Associate

0.61 0.30

$ 60.00

0.45

** WWS

$

,

$ 1 , $

$ , $ ( $ ( , $

2##*33C 2 2 , , ( 2%3*3
!"



%$2 *3#C* C )

)

E) * 2##*33C

& %

+ @ (

$ (

*

"

,

"

C

,

, E)

-

$

U

W " ) , )

0

$ VOI =

&

2

,

E 4

(

( * "

) 1

65.84

0 7

6,

( 4

GP>5 (

8* 0.25 0.25 0.25 0.25

0.482295

$

4

$ $ $ $

73.36 62.03 62.03 66.34

0.520 0.462 0.462 0.485

3.80

, $ $

C

E)

>> &&

V)

D , ,

)

*

) $

* & 5$ 1 23* C*

8* & $ $ $$ "* '

1

, " , $ 0 ( , ( $ , ( $ 2< * C ,

!" < $ %

7 ) , , $ $

& %

$ 0 5

!" % $ %

& %