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I J ~)-·/-~ ~ ~) Ot::: x --

'f):; X \)

I

~ rr<··fi·l) ~J 0<::;. X --

• /';/O 0

' ((X")hr>/ 7 v:A)d -

0 <: )( -

(x~-d)d -(x.:; ··-x() 1-) d b/ •t: 7 v~llx -

(h) O?J o2o z-~ ..7 )d == ( Doo-z < J'.?) d

I:S

(\:~I=OOy::._' ) lo· .I-1 =

( £10o·J

I o · < ~000"+iTDoo ·t)

(I o 'I --I ) -2-S d =

~hOO. =

( ~ _o ?__ozr \ 1 0 · ) I o • +-£i-S:)c1 = ( B > S ) cl :::: S b. 9 z . rt,

\A --"'k:> -z-

?; 1?' (~\A?+ u q)-+ 2! _.D ~c ?-tg)...r-w (lAo) .::

(.--" U4-• w. u) "') -j. ( .. W+ ~..D) u 9 + ),'!A l> = ( .. n ? + M q + n b) 3

("XX} 7) 3 ~ ( "x-'i ~ IP := (-zh)3

( zw-+-~J:>) \.1 = c. xR 1..1 =( ""l:J wu _:-\ n):3 b~·Cb

( _., ). -;h ~ J j'>~ S i:> '9 '-''l S,l ). ".5 ...,_. -~ ......1::. '-

( n-~ .;) _r -== (-'A) d = ( .6) 3

;w O.,;w aw•s "''1 ~ s10~ )

( ~ ~;) ):£ ::;

( ).,.~ ::> 7 ;: 1,() d .:: (-:> <.() d .:-( ~) ::r :>~. g

~-thh'O =

( Ool < (oc)z:Z) J (hz'5J~) E --

00/ OQ! )

( (hZ'SJ?) i ~ !172 cl _... -

( ( hZ'59£)_s '/ -

l2)d (

00\

(oz) .. ,-x /"'J ?.l JG q~·g

·c'¥'e>wtt9 ,.V<O (t9Jottt~ JOt .::i5)1W d1+ d(;ls :(o' 'ool)tvtt9 b S'o

( Dot 0 D/ )

_( /-.2 ·sy2) h < ?1_ ~ 2. J _:-

((hZ:S.JZ)Z < 7_1_ ~ )d 7 Sl/ =>s·g

Note that with n=13, we have 12 spares. Since the question specifically asked for the number of spares needed, 12 is the correct answer.

~s~ No fhe d;ff'er-e~ce ot ,·VlJepeo~~dev?t Xz. r.v.s ca~ be neJCJf,ve. A- chi-s'l~qr~d r.v, 1s

~.Cf

So,

hon-n~yq'/,ve. Th;s he1s nofhiny io do ~,vifh vY1 qnd V1.

Mx-~-lt)= /1jt)J4y(-tl af\d

f1x+Yct:) = (.J_ \~ l-2t)

fVly(t J:: ;t1X+Y (t)

JVlxt:t)

T~ lltS y "V ~~(v,).

hy J'hdepf',..,cfenc ~

:= /_ 1 )~ ~1-2t

__.. -( \ :f-

.. 1- 2t ) 'Z.

/. I )~ • :=. U-2t: ~

( or;S6~z 0~ = ~ (_=-

( ~ > ( 0?_)/;i )d =

( o?z > ?)!; -z)d ~

( e~r / x_,i)J ~ ( i; 7f )d :: (e > 1/)d = Sb · 01·2-

( .. (s:,,) ~ (z),~ )cJ =

(Js:.zl) 7 J·~ +.[~}d ~

(s:ZI 7 J~~J~ ~ ::

IOSb'O :::-(s-z/(5/hZ)d-== (.sz> ,(?-7i-):( )d Cl

OOhb'O ::: (z~?(Jih'Z)d = ( U? ,1~j)J

u:1v 6·' 11 ::: -:; v

/9 z ~:ix s T ( ~-iXJl) w/-3 \A z -

f e 1. ":'x JV ~ _ .'x -z e z Jl _:;Ce) l -5

I a_ 5 '"f Fos e -\(x) e) ::- 11-1 ~ X-(lf ~ f

lf b

I'\

Theri LCG) = 7T 1 i -Jz_.!x;-e ~-H ~ = \

- i_j --± = X:.," - eJ 1.1 Xn:o - G = -} J - 1~ V -' ~ eLx '] - ~ "\fl: r'\ z - - I: Y\ 4- 2.

r-vhic.~ 15 VVI q Y1' VV11' te cJ ~or- anJ fJ E: (x"'" -i ~ X,,. 11 -rf_)

s"'f pose

Then

f ~; e) ::: ~ 11 G ~ Y ~ z e J,

L(e) = iJ-n .11~~ G !: X;:•}.

Thvts- Xfl:Y\ -2.. m ax,·~,- t es l ( e ) q f'\ d ; .s q JV\ L e s --I,· n1(r-.J e

of El. It is c.lec.rlj ncd +he o"'IJ ML. es/,·,,.,,crte,

u

/a:-.1)/ 7 = e ("-

0.::· [zt-::r/7 + eu-~

f>e ---- le 5 I

/2'-·"k(? LA d1/V1 p;r~ 'vw~Xbtu St ("ZL'I) 1 ~ ~~J_

. I = e asoJJ,,5

[U~x1 -~ + (e)bol IA-(-zfao1lA-= (u'G)"/

(i7£~;j_) dxe> JJ "~ = ( 7l 'e)l

90/

(1a-~x1-)dxd ~ ::: \y'g~)()j-vOI

)3 X X "" i.cd I 1 • • • / VI

N(M/o-;')

N ( ,t{, er; ')

) ,'""depe"'de,,,-t Sq,-,f /es

Y. I •• 1 / YM A/ ~~d

-(>ti_-~:.. M

L(µ_jo-;' oz_?_) = rr" I e 2.a-;'- IT~ " ~ NZrro-i.z.. -- Zrr....- 2 k=r rr z..

J:,-' VI

-('j1< -M lz e ~l

.RJ µ,a-;i o-~L) = ~El/oJ2n-)-l foa(o-;i) -J_ (x~ -M. )z.j i L z. 'J 2- J Zcr i.

I i_,:= I .

1- f \-z.I 1°3(2.rr) -l lo3(o;_')-l z (~.._-,,,.) 2] I<,: I l 2.rr,

::: C -_!1 {onfo-,.,_) - ~ /oq(crz2\ -j__ .l_(X;-M)"-_L 2.&,.,-4

z J" z _) . } Z.Oj z z <:r:" z 2

To .\',,,,d /t1LF.s -\or ~, cr,"'-1

q 11 d ~ 2 ) vve t.vc..., Id ne<:>d -fo

solv-~ o_ s~si~vVI cf S e_i~ctilons. :C-t t;volA/d be 1.Anpl~C<SqYJi

1'j ~cud/ So T°M j'.,(st Jo)~

o...re kY'\owri avid es--+1.v01°'1~

·fu q 5· Sl4. ~t'

µ_,

i "1C1 t 0-:- '?.

'

---- 1: .2_ 0-- 2.

l(x:-,<.{) +- ;,_ l(:Jk -~) ~ O ~

'dJ -a M

:;;.)

__:_)

-;

I

nx -tJ,t{ - -- m j ..... """' A.{_.

-t- - ::::: 0 o-; t. er, z. a-2.~ ~ 2.

4(~ -t-.0- J:: er.' ~'2. I '-'?...

I'\ A -

nx (V\~ ..-- 1-- _.;->­<Tl "2. a-;.. i.

h + vY)

-nx a:- z.

I

crz.. I

a-: 2. L

~

rvi::; T -

<:l"i z

is a.. JV\ L- E

+ + ~

f o, g I

C(nJ 0-: z z_

"' ~

15 q

e;{flt-P).:: F (f-X'-) = p -Elf.?.)

- P - -b E(f f ~ = P - ~"L (0~-~J P2

-r- np) - p - {"L(n"'pz -'1p"'~X;~z~d BER(p)

- p[l-~(np-P+I)]

- p[\-p+~-~1

- p(l-p)+-~(p-1)

= (1-p)(e-~J == p(l-r)(I-~)::: p(l-p)(~)

lh"'s (i:-1) P(1-P) is uvibiqsed -'oc f(t-p)

l S-6 VC\<' (X):: n pcl-p)

I ~c

Ol V1 J nz. f ( I - p) is L1 "' b i a s e cf to \ V cu-- (X ) . ( n- IJ

L X'-- i's 'v(n11'0.5ed ~o\ p w ~~('e Y~ A-~~d f3I!V(n1p) nN

Y -/ fS _LM

I re l...IS' '"' j / .. Jius1'"'j fl-,e n oiqi 1'01-1 X 1 V\ I S- C\ T u s ~ d t l, is a .s i. i"1 13 E R( p ) ,

Let

lh1.1_s

P= LX· ~

(llN) 2 p(f-p) --(ntJ - ')

is u n biC(.sed ~ t V°'r (X).

s /ri c e

z 9 ~ --z-v:~)3 -ez + (v:S():r :=

\"A/3 ~V:'x)3 c= ( v:.-X~";'X)3 "j0J4j"H

I ( ~ 11

'A-)3 -0 z :: ( ~ 11

~ X) 3 p ~ 0

(v:'K-8)3:: (e-v:~X)~ ~n~1L 'G t-n°1P ') .tJj-a ~t..t-ihS s I ~x Jf"lJ vtO,lf-h~ ~ Jt-S!f J~-/-+c11+ d-Jltou

/ ~-,:"'/-a f' d '1 + i1b,,,p.11 i-,(., ~ "-_,o,-,,, _r° F"df-&"' {

\-8

11 If=: \+eJ

h ------., __ ____,...---::::::--. ---G z +--"Z e -e-z + I + '2 e -

I,

. ( 1 -eJ' -( 1 +e) z G.

() t I

[,1:~ J 8(= l1~4 -1] e-c=

ll 0 -~l -~l= [ :-1 (62/~~.J ~)] -e]z~

[xr~(::~)) -~z~

[xr }x~ :r -J }xfi j ?::

Xt IA I -1< [(XJj](x)} X s z :: (":'"Xfl z ~ ( ":'.K-":i-X)-=1 (=

(v1f,{) / _ =("'Y/~ (=:

( "' ; '.,( _ 0 /0 -~ ( 0-~: .. ;\") --::4 G + .-w ~ h s h ~

( () 10-)=JI(\Jn f!:/V )' bf

. (d~'.2±_ (d-1)d

t ··-J -\-d

d-l -l -dT-_ ~(~=~ -+ ~ _ = [<d :X)j-Co/~~ 3

r ~(J-1) d . de ( X-1-17 -+ Y-= ( J '.y)j h

0

I ;f

, d-1 d . de X+t-+ f =-( J :X)J bo\ e

[(J:Y)j-bo; ~?~]] ~ _ ---- 2 l_(dLt] 'O) z

2 7_ ~ x A- .- ~d N (µI 3 ;a_)

!he c RL B -t'ti\ (..<vi boase~{ es.- --1,.vtqfor-S tif µ

- C( -YI E[~L IQ) t(X;-«)J -

vi

51\·1ce

-(X-k)<) l co

e /oCf +()"'4) ~ /o.oc/-_l _ J ) Jl .[2 rt 9

= -{ loj(l?rrr) -0'-A)<

2 /oq +(X_;~) ::o +z (!-A) dA J

;>~

Jµ" I °J %-;) = - ' 4

I~

2 2 b x ,· ~ °' u M v Ll E -\ 0 ' A

l 8"

l..S

Si V1C t' F{X) :'.:"' ~ Of///J VO!. r(X) = 1 = C:. RL 13

V\

Z 2c The ;VJ LE for ttie ?s ifi pe re ev1 i ,·le '·s X- t- 3 t _ ·b

T~e, Cf<..L i3 {or 1;r11b>cised E>s-/,v>tq-/ors o-t' "1 +3~.<t~

i-; !._ . E( r~ 3 ~.rt;)= µ 't 3~.q, a"" d ~

VGJ.r(f-t-3-c.~s) ~~.Thus -ft1efa1L.E is

4 U/vl VL{ E fo~ A+ 3-C.ci.r I

/\. z..

2-3~ e = Z Y~ Y\

A) . A

E(e = G 1 so e r.5 a V\ \A 0 b,· qse_J e s'·h,•wct-lor· for 6J ,

231 CRL IS -rc;r qv] (..,(;/ asr::J es-:l1""q'for c:if e IS

--=----c "" ... - _,J2 ' ·-- "' L -.. - -fl't.· 6"\ l d;/ ''') '\

-....J

- _L = [zGz] -V\(-1 ) Y\

2e~

s ,·,('! (- (' ·e~ - / (: '. ._(X)') 'J a; <2-Ze tf2ne

= - ~ /ay{Lrr) --!_ foy{EI) - X <-z.e I

d ()G l c.~rt(X; e) -

z..

=-1- + L ze; 2e 21

"2...

d 2 /aq -f(XjG) = _L ~ J6<- J wz 63 ) qvtd

E (~L {) f&Y;G)) ~ f6, - ~< = _l (-J) 6z. "z .

Nc:,w, vor(8)-:- VVlr(Xz)::: VDl<~i@cr] = 8'-vq_\(lL) h n n

~ ~ z( E(~ "i ) - \ ,_ ) =: 2<( 3 - I) ; ZG 2

t'\ V)

Th"!s 0 t.s Cl, Lffa1v'\."1E to< e.

I (;) = }y-

s r , os-

x-

w --

;X . ' lx

Fl -d] ~ + 'z. LI ~}'}/ .,..z- , _

·x )

)7 x w

j-

-w ~~ 2_eJ +N-

d d xz-

~ ) d ( d--/ ~­ \ / -

d -

( e) J,..., I\ --~ '" #"""V

( z+4 i .,.._:y -( 7. .;..ith:(i H1}

z 92 -( i h-i) Z -z. 6 - \.

(z +IA) -i \ I +1,1) ) C +-lA f ~--~;f~. /k'~/ .. ,-.-lA ·t-.. ~ Z j-j/ 'z <3 =: ?' c /~ l/ .... ~/ ,,. ,,/ ~' ,

I (z-+VtJ2(1+vi) ) ~ --\ (?+LA)-+ (z+1A)-z1.A -1(1 t\J) u '"z 7

r .. ?.c~~) + -i(I~ -l,tv) 6 --z. \A \) -z

( Ji_ i+j + -zt12+~-z~)ze ~ \

2[$-e\~]+ /3_7-(1~-~e ~ ~ (,"~"Y)]SW

-u

J z+ vi ~6 ~·'

- x ---·-z ·rvt \..A

0

x )~ ~ ( v ~ .... x-)~ /+U l) Z e

Z bet f x; 0 ) -=- 5 ~ ex 'T ~- e 1

2 0 ol<v 0

L-(G)= TT i 1( x":n ~e5

- l 5 t <d:/' .1 zX"':.1 ~- G5

8 : t:11 1 s t he I-A. L E +c.1 ( 18 .

Uh s ::: i -) CF 2X 2-

/'-·, - . MM- r So G :::: 2X \ s o.. 1 \ \ l- i··a c 8

Z6c i e i==(X,,:"") < e 1

I\

So G IS biased

•

f, (~ ·) = n f (X) I-(X j (\ - l

= \ Y\ ~ ~rl x.,..,: {\

()

) ~ Y\ Se n I - V\ F(' y I -- -Y\ x d,x - -;,, ~ 1! 11 ,., e G

. 0 ~lb> P\ -t I

0

~ (-£;-l) e.

{J(XLGJ

o/~,

r- ""+~] = :~L ~t-,

1..J)•?'V'.,1 ~ s I e 05 v

l-l --ze -\ • h

( X-z) j o I\

(\ ~~) J'kJf\

L S~ S IC>, 1_[·. J·,, :' 6~ .J b(\ -;, 8 :::J:; w _, ~I \~J

-z. -'-

j + IA '2. ( ( +1vt)

c.8 -/0 ( i +u) -

.f:e::;) )J ], l1 )' I 'J , I ly' l /Cl '13 f'' v // !\

\A -

I+\) )u .:· gr;)

·.- 'v

2

~e

:( eg) JS hi V""

G7 ><

e2x

<97 >< 0

. e .____( -d

~l ~

d

(

"' : jt' c_jjOJ.C)jd~ J.

? (\x7 ~;~)cJ /1-i (J~ e I. x \ )

e < x z-'X -..e-I .: CxJ~ d.)1A1<;

f 0< x ~ 1 I~ [ L -x G 8 J -!) :: '-

f (X;j -11 -l ~ \A \,-. _J

J ( )( ( Y)cU -l -(x /_ v:'X)d ~ l :: ( x 7 vJ)d l~

----

Vf-) \A -

l

~ r ~e f -t I -~ ( w :NJ bq I f

(i"X) Col -(wfofx+Y-~ \ rt~X~ Col

iX w d

x r(-

<;;

x-

,,r -

0= Wf: + ~----

Y\

- --~- -~----- - -

33e E <~-i) ::: E ( rr ex~ I :: TT E(e-k/n) ~~ (e-x~T

j>3f

E(e-V.,):: 4 _X/Y\ -K X

L-- e e 4

x.: o I

-Ne_Y"u'\ ~<i"~A) ( ~~ )x. ::.e LC e /{

x. -J-

V\

-M- + e A -.M-/n - e ~ e

5o -X e IS bio5e-J ~c\

--X P -µ

_µ

e "

- p l~A ) C: --?e

So ~% e 1'5 $ ;,..,py C 0 VI 5 I•) -If' Y/'i -re\

x!

-~

e

Of vt d i he r@ ~ ' ~ Cts 'j M f JoJ/c- q I!; '-M j,/c,s;.Pd,

,4/ff.r111<:,J-/;ve~1 OVl'f> C'c,vi 5hovv ihcd:.

/1·rv1 r -j_ ] "' Le-µ +e ..,1<

V) -:> Q)

-- -~ e

•

~-d -

( \A )-vr+ ~ -1-U

~ /1\)] ~

7X:/ ~] ~ ff?_

•

(~) 3 -x u '

s

---------

}·--'i' j~rc( ?x n __ Yt d

(~) ~ 7/y~ "

------- -(~)1

( d-l J d ( 5) S-IA jl > lA

(JD d(S' ~-v -/ 5 IA)

(x)~ ( d -1) d s --1-t . s

H(d-/)sd (~)

(d\x~

--

l..

10, 3 f(x ~G) ~ x~

TT_\__ -w J- e ---------

tp1 ~ ~'

l ( _!. ) n /z - \ - S/ze

G z "lz rf~\ (~I c: 1J s/05

I - l'l/; - \ s c2 TJB) z e ze e 2 \'1

12 rf !2 ~ ,_ ~

(.S~Yt/z_ -) -S/ze -- e 1-ls ?oJ g

(2 if r" /z z 'l/z r{~) -~-----·----~--~-

S~;z. -I i[ S" ?OJ

Vv~ic..~ does vto1 c:lepe,d.. of 6,

~ oi ~ -/hot 'Z.

L x~ "'-- ~i-(n) e

a v'1 d l-f VJ /\/ -F()< )

i i,E'"" e w ;'--- ~ -t(~)

Vvh le.. !,., 1· <, h ocv ]::' 9 o-I t (s)' s

IQ.~ Fr,;.,(x) = P (Yi:,. ~ v} = l - P(X,:., / x)

N~vJ

~ l - ~?(X > x )T ~ ! - L l - F(x)r

(

-h (x-)2_) \ - I - e ) i~x 7 :lJ/

~ ..... (t-?l)

s i V\C E-' F(~) ~ if OJZJ j ~ dt

T- ~~r- e fo'{p

+(~) n}

+tf>l - 2-x~

e -ns

he

7l

I t 7l1i-=K - .ilx 71l] (--e- t

I 1::= r 7l

=: 1}olZj (l -e-x-r12) ~

- vi ( )(-72.)

tX,:,, Q<).: n e. 1 i o ?lJ

- (x~ 7l) :--L)(,_' r n7l L ~ { n- e :1- l~x,: /n_ J _ e i is )72.j

_ Vl ( $ - 7l) - vi (s-7t) V\. e · 1~s/J'2} ne j_£r 7 /lj

\,--./ ~ i C h J CF·) in.:1 dfl'pp,,1J c~-i 7l to\·

Vq{U~-.5 o{' ~ """" h e r e + C~ l s) e :x \ s -1 s. ~\S

'1 t ----------- ~ d ( 7X J..1) ( (}J e =(Tu .•x~ -\ 1..1-1Az---~

l-Vt7' $ 4(2.)J

{ 0 t_ u~;;--~ -i -0 X Ji} Cc• 2)J

O> ~\A Z)J vzG

s ---\ 6Ys-\ -lA 2

c;\x JO '--:' X$1 ;> :y (-z)J ?~ Jl ~ ()1 X)j--

eAx -\-~

(Vlz1e)wV<:J -v 'X~ -<;

\ Z I fJ)Jtv V 9 f1

:' ~ :x 5 • 01

I • I

ign . .i

1

'''

1

1'0J7 ?x ?r . (d-D, d (:~)Ji =CX)J q·o1 ( ·X-?k,.1 ·X 11

/

I:]

{u:(. !_(~li d J1 ~ (?t ~~j ·i ( f.t-1~-[,,(

I --, -..

'ci j "'j-j-VI d '. :J !J:t h S <; I A" ? 6 \A

d "Z.J'JZ

-z ( ),y--:x) -r-

SI

. Q\

'f) l ~I \

d t: -

SI

'·(})97f ,,b j01 / sI ·vt}

SI

( Ll)bG/-2l = dx f:::

(d-~~(;QI =-dX-1{ 6

cl-1 = 2 <;-dx-~

e -l

0 -=

u( d -

<:lx u+><-

)t/

------------- -------

lo. A. fQ<) 6):

-\ lte ce~-o'~ X z.. is 5 IA f .\!' c}e Yl i. f u r e ~

It { C>' ci 11 G 7 d J

z..

o ~ E. 4(X 2)) .=. ft<Cx')~ e-J;- d.l( N2 rt6

Thev> P ~(X7-) ~o) -= I

-) xz. - ' f /.of·&' - l s co vi.·1 . \._ ' \.... .

l<6b t r+ ~(k) = x)

\Vle~ F( L{ (X)) ~ s u(x) ~G IK

T t, e re -f o, e N( q e) 1 s "'

-%

rrz

i CJ)(

r ov'·? le-/ e.

=:: 0