T So•~) J hreeder/5090/5090solutions.pdf · z5"A ' P (X,:~ i.x) = l-P( X,;., "7 x) ::: l-( \ -...

81
Sl 9 7.. ·oz = o 5 \ b·1 + j > )J ( 7 }:w ) d - b · b . g I = \ 8 z I I -) + ?.. . b I ::: 0 + f. . b I 0 5 !!_. \ ( w ? .,.;_z } + 1) d = > 01·£-) d = o b 1 ( E 5 · o-z 1 .J 90 ·g 1) :: ( ( .1 h .n) } + r · h 1 1 7 ·I) !f- £ 'b I) -= (s•·z. lga <- E "b) S, I "W »J I? Y, () J, "' "S - -. ( 5b"2 h / T ) J W-tX- /G = ""Ob .f '6/ : ')(' (-zE 1 )y) N p:: -v ?f ·vr

Transcript of T So•~) J hreeder/5090/5090solutions.pdf · z5"A ' P (X,:~ i.x) = l-P( X,;., "7 x) ::: l-( \ -...

Page 1: T So•~) J hreeder/5090/5090solutions.pdf · z5"A ' P (X,:~ i.x) = l-P( X,;., "7 x) ::: l-( \ - F(X) r ;: t 't x > e J l' -(' -[1 -ee{x-e1r] - t1s

Sl

9 7.. ·oz = (gz·t)~-+ ~·bl o5

\ b·1 ~ + j > ~ )J ~ ( ~·t 7 }:w ) d -b ·

b ~ ~ . g I = \ 8 z I I -) ~ + ?.. . b I ::: 0 I'~ ~ + f. . b I 0 5

!!_. \ ( w ? .,.;_z } + 1) d = X~"" > 01·£-) d = o b • • ~ 1

( ~-, E 5 · o-z 1 .J 90 ·g 1) ::

( ( .1 h .n) } + r · h 1 1 (.5~> 7 ·I) !f-£ 'b I) -=

(s•·z. ~-;..[',!./ lga ·~ ~ <-E "b) S, I "W »J I? Y, () J, "' "S

--.

( 5b"2

h / T ) So•~) J

W-tX-/G = ""Ob

.f '6/ : ')(' ~/::tA (-zE 1)y) N p:: -v ?f ·vr

Page 2: T So•~) J hreeder/5090/5090solutions.pdf · z5"A ' P (X,:~ i.x) = l-P( X,;., "7 x) ::: l-( \ - F(X) r ;: t 't x > e J l' -(' -[1 -ee{x-e1r] - t1s

l 0 t • Ol 1 Oh. t:\) =

( (-s 1:' I) ~~:·~! r + ~. bl c \ .S!'I-) 0:·1o, r + n I)

),1 w jQ.t -r? V.Qj; 0 Qs

(sJf•·1 ¥·1 >w>(s.J"'+ f .. x)d ;;: ( \•-0'~ > 1-f~ f > lt-~"?) d = Ob •

Qt

t'

I z = r j-1

(

Page 3: T So•~) J hreeder/5090/5090solutions.pdf · z5"A ' P (X,:~ i.x) = l-P( X,;., "7 x) ::: l-( \ - F(X) r ;: t 't x > e J l' -(' -[1 -ee{x-e1r] - t1s

I 0&' zz ) ( h-t-'01} 5.t

Page 4: T So•~) J hreeder/5090/5090solutions.pdf · z5"A ' P (X,:~ i.x) = l-P( X,;., "7 x) ::: l-( \ - F(X) r ;: t 't x > e J l' -(' -[1 -ee{x-e1r] - t1s

( (bsyf;;h~-Z: / ~7.;bS 7 (bf}:1J ::: bb'

( t£-z•s I b2./ '_5) ::

{(-' I ) ~ /( , 03.Y -' ;; z -bhJO'O + I I z·c; JJ Z -) hhJO·o + 11 z:s;

: $' h' j'?j .I? ~bb v

( (_b5/""·~ r-+1 >w> (hsiop~ W-+ _r)d = ,,

( (b5)5""'f > --;!:;;; / (bt)~ d ::-,U.

1 z ,.., o o ·o = '? s bh90"0::: s

r"' '""J-5 i 'ti bwsn DJ=: u

( 2-0/w)N f?7-v 7X

Page 5: T So•~) J hreeder/5090/5090solutions.pdf · z5"A ' P (X,:~ i.x) = l-P( X,;., "7 x) ::: l-( \ - F(X) r ;: t 't x > e J l' -(' -[1 -ee{x-e1r] - t1s

f'1A» 0( + j! (-r { J1>£'!-;Y=1--"CJ) '.)I

(Cb I' JJ,C/_. h I) s o/"11\ e Jo; ~ v fJ~,1 v;;b;;; v / d. /I-' t ~ so.J

r.,,,. ! $ "10..l?"P I c; "'·'~DJ_\) I.A·' '1rJ•IA<;>r~ .... OJM '"

9h£ 'J-,( -

_x:01 )J = Sb.

f)

(ooJ)-z~ --v Xoot -

b't:.l = y 05 =u

(I 1C9) w V9 f~,1 ~ ( (j )dX-3 f17/\J J vr

·(sio·o'7so'o) s;~ .-0 ->0J-:r? f.bb 0 pv1::1

( t-t:/,E -zl:..'Ok )

( I -ewo< i:-zoo·9 :::: · (, z.i.00·9) J.i s C ~zi.oo·g) 6S \

._c; J "'.->I·· '?: v

Page 6: T So•~) J hreeder/5090/5090solutions.pdf · z5"A ' P (X,:~ i.x) = l-P( X,;., "7 x) ::: l-( \ - F(X) r ;: t 't x > e J l' -(' -[1 -ee{x-e1r] - t1s

OS ' 1i :::

l ~b'it) 0 -(}, 'tl) 0 Q/

dx--a -puo

~ ~~

T--

9 J OJ-' 17 X_<;);, J<:)ddh £h 'ti

lAb S. I ("b.i.1)90l

.Si

Page 7: T So•~) J hreeder/5090/5090solutions.pdf · z5"A ' P (X,:~ i.x) = l-P( X,;., "7 x) ::: l-( \ - F(X) r ;: t 't x > e J l' -(' -[1 -ee{x-e1r] - t1s

~A. XL ;V.: i-d exr( t, n)

p ( ;(" _ 7L ~ >< J = p ( X,:" L x t n) ~ {- p (X,:~ 7 x + 7l )

== I - ( l - R'x· :>t) )"

==(\ _ (t~ (1-e-~)j")i~x;-oj == (1 - e-.,><) i [ x ~ oj

(""' r- < f X, - (t -Jt) t: +?t/ i:-.=.J( Ji""" ru): 1tK??lJ n e dt:: ::: -e- j_fx?~}

-l::n

:: 1fx>7(J( -e

Avtd. ""'-"~ Se'P' fh.ert

-><+7l) •

y -7l II tt111 t s p/v d It:~ I,,

5 B, 0:: p ( "' ( X:" -1' < b ) t.-._..t, ere

f-'ls -1'\a.. -z:- = 1- e , a-~cl Jt::. I -e._,,~,

y -1'\0..

=- l -l + ~ -V\ b

- \ -l -I e t.:to'\d. e 2 2.. - z_ 2.

) -V\C\ = t~J(~J ctvtd -V\ b - lo3(l?) -

) ~ ~ -1 l 0o+t) V\ c~z ~VI.d b = --k lo{~)

Page 8: T So•~) J hreeder/5090/5090solutions.pdf · z5"A ' P (X,:~ i.x) = l-P( X,;., "7 x) ::: l-( \ - F(X) r ;: t 't x > e J l' -(' -[1 -ee{x-e1r] - t1s

( £.& I . I~ $: 0 ·) ::::

O..S.&

Vi = \,1 :I~ \.4~'11

OSj

f.io ~~J I os -1 X :: ~A + al

( u ( O-S~)d'X3 f?~ /V 7X

s, .

((-3-\bol \A v:•v ( z :-bo \.f ~:'X) '·+-9.) r + A ) l.t I r-ir~lr-+ 0 :::

( "'-"11X '/7t) ~ -w:rx )d =

(·:·t -9 > 7L-> ·:·x -v ) d d 'os

Page 9: T So•~) J hreeder/5090/5090solutions.pdf · z5"A ' P (X,:~ i.x) = l-P( X,;., "7 x) ::: l-( \ - F(X) r ;: t 't x > e J l' -(' -[1 -ee{x-e1r] - t1s

J-, .9 ·y OS = \OS I -hi: '19 2.) 2. z ~~ 9 ja_j-l:::> j,#'V)ol ~)b lO 'os

( L~£)sb·x) ( ~ \ @'? (~s~-_x)g£ d::: (:t~)!"z > (os.1-_J) g.~)d ::: Sh'

'U~~U)j Sf iAO!f-t"11~JfS.'fJ 2i1+ dJ>":~.; d$/i Of

J"'".: .. s.oc; :;/ hj'lvoonl, /bfO~'~.:d S.,r'JI"-aj.d d~f Jti"'ld/VIrrl

e e '9 }bt~,.,,~d

~

jJ ~I ( QSt -_X)~Z -~ ~z f.P~ ~ 05 •

-(\A-z).,~-v

e ).v.z fl/to (v'9)Wf/S> ~ Au \Ad~_j_

(l'e)wv~ __, (e)d Y3 F :: _, ~ } 0'1-1-0$ 05{-~ == ~ +d7

Page 10: T So•~) J hreeder/5090/5090solutions.pdf · z5"A ' P (X,:~ i.x) = l-P( X,;., "7 x) ::: l-( \ - F(X) r ;: t 't x > e J l' -(' -[1 -ee{x-e1r] - t1s

i.~

1-A. Xi f\/ W £ :r ( e, 2 )

fen=- ~ x. e -%)'-.1[ooJ ez

P(~f ~ ~) = e (X z ( <tr) = e( X~ ff) 1-{K;>oj

1-B.

tPv. tz 1:.:. ~) - t~ ~~ ~a - ;::;:-;:;:-. ~

~t,r{ ~, -t e cit == !l [ oo 5 ( -~ Clt_,_ a

0

::: ( \ _ e - G:~ • }1-f-)( 7 oj _:- j_ f X ?o J( 1- e' --: )

T~ is Is f~e cd f of EX P( i) "--- X- z(Z),

So ZK£ z. "'- .:~J y "- (z 1 e"l-

Y! t z. qV\ d ';- ~ /V 7(2~).

~ 6/ L::: I

0 ~ p ( ~~'_~(2") < 3._ Z)(: • ( ~~+If( L "') \ ~ -r e 2 - J 2

\ D( -= I ;z7+?r(Z") < -

~z -2 2A:'

. < 1/~ (2.~) 1'-'f.::r

'2. "2.

= A( z zr{ <- e'- ( 2Z-1:·._ \ 11!\rCz..,) ~·~~zr~))

= P( ?- ~x~ ~ e < --~i:tr(z,) N~«B-(2'"')

~ ~

)

This should be EXP(2).

rreeder
Highlight
rreeder
Highlight
Page 11: T So•~) J hreeder/5090/5090solutions.pdf · z5"A ' P (X,:~ i.x) = l-P( X,;., "7 x) ::: l-( \ - F(X) r ;: t 't x > e J l' -(' -[1 -ee{x-e1r] - t1s

r ' e

( .. zJ:Z ~ e 7 ~::Y.?Z / d -

">c J

((uz);Z:> ~;;z)d =)( ·Jt

4 .o{" 00.'1 -.· ·"". · . ., 'J'/0. -~~~on ha /tJ -'-_,., >-a-. ' t ' .

Page 12: T So•~) J hreeder/5090/5090solutions.pdf · z5"A ' P (X,:~ i.x) = l-P( X,;., "7 x) ::: l-( \ - F(X) r ;: t 't x > e J l' -(' -[1 -ee{x-e1r] - t1s

Sl •

( ~·::1fol(e ( OT-1) '0 I ,. EJ--b (:- -- - ...

( 0;/ -J)bol ze -....._ -o-(- 'l.

()g_j_ l -d (.:- - d -7/~)o-

;:> -I ..... -

~%-

0 00/ ze

j= - 01 :Xp C) X -J d---z

._ I

'at e z~x-'0 ~-z b

7:

Page 13: T So•~) J hreeder/5090/5090solutions.pdf · z5"A ' P (X,:~ i.x) = l-P( X,;., "7 x) ::: l-( \ - F(X) r ;: t 't x > e J l' -(' -[1 -ee{x-e1r] - t1s

~(fJ = )o (-

~(f)~ l = )0-1 8g

LA ( ~ )-I ;:

} ( ~) ::1 ) -I ~ ( § _:; ~. "t) cJ -I =

( v' ~x > E§-)cJ = ( v: "X") e) d = ( "'"X? >e> :"X) d

Page 14: T So•~) J hreeder/5090/5090solutions.pdf · z5"A ' P (X,:~ i.x) = l-P( X,;., "7 x) ::: l-( \ - F(X) r ;: t 't x > e J l' -(' -[1 -ee{x-e1r] - t1s

z-

rd~_v a.-""-1 05 ~~G 1-a7

0 > ~-5§ o··~ + J) J = ?J 0 I '2-) d :::; 0 b • x-w >

it (,'o)N l:-w-X f

·z1

' II

Page 15: T So•~) J hreeder/5090/5090solutions.pdf · z5"A ' P (X,:~ i.x) = l-P( X,;., "7 x) ::: l-( \ - F(X) r ;: t 't x > e J l' -(' -[1 -ee{x-e1r] - t1s

.>Y

51 '

J "J I ;J L.:l '"r;/

re Jor 1 -:J

~0/;

if > )'-;r

-

( ' t ,""1. ' (·,, I \ I · Jl \1 -fl ~ _lJ

p W-X "A()b t-cdho ""' s1 b9h'l =,Jli~·D

J i! ""' "" J % 0), . XI JJJo ~" 5 I 2 I 2 . )

.21 + 3"0 rd ~.~s dP S'P rvl

e d~+ tY\4 {

lr.'J""i"'"'' [ Oh'J-> 9 ..} OJ 5 ?/0'1 O.J.Ja

.:)~~ )-Jo 1 J-a~o(\j

ZIZ'I = ~z

9~~z t.... 8 ->oJ ~Fl()~

1 d [lA k>l/1--;::> -vt/,!S. u ,Jo~? I y

Page 16: T So•~) J hreeder/5090/5090solutions.pdf · z5"A ' P (X,:~ i.x) = l-P( X,;., "7 x) ::: l-( \ - F(X) r ;: t 't x > e J l' -(' -[1 -ee{x-e1r] - t1s

1

1 \LCu Let t=(r-) p_) ' 1he c. elf ot PO.I(M) be.:

C?vtd H (j) ") fhf c,d.f of ~:z.(v ') ..

1- ~E' II\ Cf cc ord.i 1/1 J +o e xc~rc /_"i e ~' 2 1)

F (X j ..... I = \ - \-t ( 2. ;<( • ) 2(' X -1 I) )

\o j e-l o <:l"' e - 5 ;d., d c::r: ~o .- ,«- we <l o -! ~ ; ., :

.~ L r(~s-x ~ ") whef e ct ; a.~) 1' 5' q-f lr:a5t

·1 ~( ro-t;:_ pe r c e r1 ·I ; l e of Pol-(£-/5'~).

T'-'~1 is ,. 9 £_ f( "- i '-IS" ,oL) ~ avtd

• C( ?._ f(a-1; l-15A)

The~ t-ltt e ci to<"" t<A.. I S 1- L, e 5e1 of ~

5, t. Lj ) x Z aft<-}

No+e -l~t.ot l-(~s-x.,.l ; Lf5f<-} I_·"\ iff '-t ~ ,( f._ a, (M ) •

~o, Vv-~ c Qv1 5o/ ~ F ·+ ~- ,· s 5 .

'',,17 F ( '1) ( Ll)-1; 'i > ,k) c \ - H (10 p. j '!O(I:t) ! I

I :::;;;) H ( qo~--t , ro(l /:t)- ) I . t 0

I -;::;;.,) ;t_"' (10(/.1) ) < 10A

• tO - 5 I 'iS{, '' " c o• se<•)i" 10 '(,

I 7 A I ~~~o ( qo(l.-:r) )/era

o, • lowe\" C.. L-,

Page 17: T So•~) J hreeder/5090/5090solutions.pdf · z5"A ' P (X,:~ i.x) = l-P( X,;., "7 x) ::: l-( \ - F(X) r ;: t 't x > e J l' -(' -[1 -ee{x-e1r] - t1s

0);

( z + ( '='1)06 ) O.b: ~ / )?" (_:::;

"'Oh < (Zt(t'/)01,)1'7/( (='

.b • > (z+(i.'J)o); S >?Ob)J-1 (='

( Z + ( t: 'J) 0 b t -w 0 )) , /) rl -1

1\

0 I . < ( ¥ Sh ~ (t·J)s!J_j

Page 18: T So•~) J hreeder/5090/5090solutions.pdf · z5"A ' P (X,:~ i.x) = l-P( X,;., "7 x) ::: l-( \ - F(X) r ;: t 't x > e J l' -(' -[1 -ee{x-e1r] - t1s

1-1

(

s, ;

51 •

p/1 -v :v I . !\

Page 19: T So•~) J hreeder/5090/5090solutions.pdf · z5"A ' P (X,:~ i.x) = l-P( X,;., "7 x) ::: l-( \ - F(X) r ;: t 't x > e J l' -(' -[1 -ee{x-e1r] - t1s

zo .( "-- , ~ d E x e (e , ) .:: -" I, ·I Jll,

'(;. A- Lid F x f'( G..) ~.---= I, .. · 1 n ~

n, i ~ GAM(e,, n,) -

Zn, X ,___ ;;t.'-(2.".) 8,

-S ;, ; /«r lj ~ "•! "- ;t:} (Z "L)

e z..

5o - ·~ ZV1, X et 2 v'),

--..,

_ §.i -v F(2n, 1 Z,~)

(z;:2~c 9, y

Q = p(fl:::§(Zn,2n~) ( ~ .K_y~ ( f~(Zn,, Z(l.) \ ~ I Z.

~ P(-x~ t 1_r (Z.,,,zO'll) z ~e~ < _[ f,rll' (z"', /n,) - I y -2 1\ ~

5o q IOCoro C. .I \or ~ i-s

(% tli- (Zr1,,z~.J, _h t!il- (z.,, > zV1,) . e, )

Page 20: T So•~) J hreeder/5090/5090solutions.pdf · z5"A ' P (X,:~ i.x) = l-P( X,;., "7 x) ::: l-( \ - F(X) r ;: t 't x > e J l' -(' -[1 -ee{x-e1r] - t1s

b $I '

'?.h-'.)<:) -l =

( e;; (Xte) d -( 0 < ( Xt'e )d -1 <

[97\xte)n(e<lkte~ -\ ==

(e< (xre)LJ le>lXPe) )d = ( CX)ne) e / lx)'G )d ( ( if'e ) e )d -= ~;c_ I

( e > ( X) B )d = 1"" -I

•( Sf~W.'/ aJ"'df ~j\10")) S";rVVt.Q Jfr-.o a'1 + ~x)'Y)e I ( ;t).,e flAb

s~;q•.'J~/1 l.M()r~)oJ a~+ +"";;Js;;uddJ (X)..,.,G r (X) ,9 +"d1 h-z

Page 21: T So•~) J hreeder/5090/5090solutions.pdf · z5"A ' P (X,:~ i.x) = l-P( X,;., "7 x) ::: l-( \ - F(X) r ;: t 't x > e J l' -(' -[1 -ee{x-e1r] - t1s

z5"A ' P (X,:~ i.x) = l- P( X,;., "7 x) ::: l- ( \ - F(X) r

;: t 't x > e J l' -(' -[1 -ee{x-e1r] - t1s <[ I -V\ e&-e )J - J.(_x )6J - e ( ca II H1s G0; e))

25\S. Let kCe) be det';.., e J b 'J 1- <X : G(h(e); e)

5o +~c,+ 1-c..-.::: P(X.: ./: h(e )) •

The, -~ e(h(e }-e)

I - e = l -~

-) -V\e(h®)-e) - O('"=e

-=7 I oj (ex) ;: - 1'\ e[ keG )-e]

? lojCC(') +- e = h(G) -l'\e

:;-0 ~ '(121) ::: I + I 0 :) ( o() / 0 +c..- l«rJ e G ~ ' t'\e2.

ctvtd t.._ Q to\ SVt~~Q{( 6~

lhus hCe) is Vloi lo/loVto+ovtic,

Page 22: T So•~) J hreeder/5090/5090solutions.pdf · z5"A ' P (X,:~ i.x) = l-P( X,;., "7 x) ::: l-( \ - F(X) r ;: t 't x > e J l' -(' -[1 -ee{x-e1r] - t1s

l5"C • G - JoJ(e><) = S

h6

'Z. ) e -s e - (o~~) ::: 0 Y\

=-/ e ::= $ + ~SZ+'-f lo1~~

z.

A-ssv....,E S ~ / - '-lloj(~) so ·-1 hcr-f roo-Is ctre req /, Y\

t + + --- +- +--.. i- i-

e_ e-~--

S"o ~(e)>s to, 6(0, o' E>/e.,...

Ntiw pl'-lj '"' x.:" tor s Q"' d. o. 1-~ CoVI t':Je,ce re j loV\ i's :

[ e I h(e) :> s J =- (- co; e_) v ( e+ J cv).

No i e tl1o i th ;· s ,- s not Gf "' i Vl 1-~ r Vet I , -~iVlce Vv-e kvto"'-' G /0 ~c'(" OVI) di5tr;b~..~tioV\ il'\ tt,;s

~OIVVIi/'1) VL-e CaVl Wrife fl-,e 'ejiOV\ "'S (o 10_) u(e+.)CO).

Page 23: T So•~) J hreeder/5090/5090solutions.pdf · z5"A ' P (X,:~ i.x) = l-P( X,;., "7 x) ::: l-( \ - F(X) r ;: t 't x > e J l' -(' -[1 -ee{x-e1r] - t1s

·tr 0 = (. •1.,)~j--;:rp ( "'t)0j-1..:-

X>-1 :: \"b) ~ -t~t~

Page 24: T So•~) J hreeder/5090/5090solutions.pdf · z5"A ' P (X,:~ i.x) = l-P( X,;., "7 x) ::: l-( \ - F(X) r ;: t 't x > e J l' -(' -[1 -ee{x-e1r] - t1s

( 1 'o) N

"Z

1

~-11 ntJ-lj

-

-

Page 25: T So•~) J hreeder/5090/5090solutions.pdf · z5"A ' P (X,:~ i.x) = l-P( X,;., "7 x) ::: l-( \ - F(X) r ;: t 't x > e J l' -(' -[1 -ee{x-e1r] - t1s
rreeder
Sticky Note
This should be the square root of mu. The solution is messed up from here.
Page 26: T So•~) J hreeder/5090/5090solutions.pdf · z5"A ' P (X,:~ i.x) = l-P( X,;., "7 x) ::: l-( \ - F(X) r ;: t 't x > e J l' -(' -[1 -ee{x-e1r] - t1s

30C. -

X -e ~ N(o,t) 9

bj C.LT

~

We need e j '(e) +c no1 d~pev1d o.-, 6.

::J ( G ') ::: ( j (G) ,_ / II d.... -+ i.t<? -/ r 1 c k,

lo~(X)- lc;(G) __; ~o,t) g_L

e -~

S c X /\-"'-s 'JI'tl r -t.·hcc:. /IJ 1\1 (1 o' (e) , * J

Cl~'~d tht- hq.,St'.,i'rnq-/io., ,·5 VC\rl"f"'C@ 5-/.:,b;/;'iiYIJ

0-= P('f.!:l < ~X -t- 11 "'(~) < -c ¥\::: P{K + ~%- (./o3(e)< X+ r~ - ) ~ ""' JV\

5<:> <>- l DO~ Y. CI: ( qpfr"-1() .for G is

( r- ~ J [v ~,+r · ~ ., \ exnX + .t;. ~'4 , exp f\ + 2:" ..Gi J j

rreeder
Sticky Note
This should be log of x bar has this asymptotic distribution. The solution is incorrect from here.
Page 27: T So•~) J hreeder/5090/5090solutions.pdf · z5"A ' P (X,:~ i.x) = l-P( X,;., "7 x) ::: l-( \ - F(X) r ;: t 't x > e J l' -(' -[1 -ee{x-e1r] - t1s

+ oz = g (~

= s 0 !>

~&.S'bl ::

=.So·~= Sh9'1-

• ( nil )! = 02-D

~

(n~w \ oz~>o:; oz~j )d-(oz=wl ~;-1)d ::so·

o z = >Y : <)H

( t 'w) rv r? ·'"' ·'X . V I

Page 28: T So•~) J hreeder/5090/5090solutions.pdf · z5"A ' P (X,:~ i.x) = l-P( X,;., "7 x) ::: l-( \ - F(X) r ;: t 't x > e J l' -(' -[1 -ee{x-e1r] - t1s

.1J3

'lc -

p ( X ~ "' I ~ = z I) = p ( X~-2.1 ~ q - 2 I I I'{ :2 I) /L.f Yt.t

= ~(<=\- 2 I ) ::: l/...,

-~ ~, Z7- f- 10 ctnd {!> = 1- ~.z=r ~fD __ ,

~j_

So f~-te ~ e s i h Q s e:r./ Wl o~ t /11 o l";;Jo ~n.. ('' r

a..'j~ivts-t ih/s <=< tfer~ot~/Ve, I+~ less

+hotvt +~ e s,·~E=' 0 { 1he 1t!='.s'"t ,...,hie. h /s

s; II) ,

p ( l ~ b I ,.F 2 I ) ~ P( i- 2 I ;> b - 2 I I ~ .:: 2 I )

Yt; v't

-- I - ~( b - 2 ' ) = . cry, v'-~

ctvtd f?J = I -. cr Cf 1 ~ • oo ~

If fl-!e q { fe r VtCi -1 i ve 1$ c ~CttA.J ed fo /C( I

t~e~ fo = f-~.27 flO -~ tor tt-te re je c ~ioll'l

re J ioVI ix \ x ~ bj aV\d .Ooct tor fl1P o i~er ovre,

]:V\ o-ther ~~ds-..~ t~e rr:~(..(( ts ar~ jLtst

Swrtc~~d.

Page 29: T So•~) J hreeder/5090/5090solutions.pdf · z5"A ' P (X,:~ i.x) = l-P( X,;., "7 x) ::: l-( \ - F(X) r ;: t 't x > e J l' -(' -[1 -ee{x-e1r] - t1s

£l,oo·::

I \ ~-,~'~ 4;, 11/t

lIt~ 171 l z-q ') l"T-]) l£-1? d ==

!;fdiMIM{5 ~q ( )l :no' Cf / _)( / 0 )d .::

(1 c:-!Qz-~ \ 9 7] / b)d = ?J '3[

01 ·o ~

[( o;~~) _f-(o~~~)~]-\ ~

( 0 2 ~ w I o h: _ q 7 :;~ 1 ) 0 > b d -\ =

( oz,...,) <f > 'jt o )d-\ ·aT

Page 30: T So•~) J hreeder/5090/5090solutions.pdf · z5"A ' P (X,:~ i.x) = l-P( X,;., "7 x) ::: l-( \ - F(X) r ;: t 't x > e J l' -(' -[1 -ee{x-e1r] - t1s

9 (e-h)e ::

( z::e \ M M n 8B )d :: (?. =G ) p[azl) d -= >o ·~2 i h If_ I \ ( o1 -:;1 (3 ..\OJ

(

le-h)e h h h 1-7

-e-h e + e . e-h .::

Page 31: T So•~) J hreeder/5090/5090solutions.pdf · z5"A ' P (X,:~ i.x) = l-P( X,;., "7 x) ::: l-( \ - F(X) r ;: t 't x > e J l' -(' -[1 -ee{x-e1r] - t1s

9b 'OJ :::

21 -b

~ ~ (21~w I ;r_o / Zt~] )d ~

(zt::w)o>J)J =(w[o>J)d ~d~; =10'

/0' =)c;

2 I '"? w ·. DH

Zl~w ;oN fsd_l

\ I = ~ z 2 ::-z...O

0""2='-f

( ~.o'w) N p;: -v :x V~

Page 32: T So•~) J hreeder/5090/5090solutions.pdf · z5"A ' P (X,:~ i.x) = l-P( X,;., "7 x) ::: l-( \ - F(X) r ;: t 't x > e J l' -(' -[1 -ee{x-e1r] - t1s

~ (" ·~) ::: l ~ 1 4-.. ( ,;:'r) h + " 5 · J) ~ 1..=­

" ( ~ '?) :o 7. ( S '0/-lo•~ ' ~ + Z I ) ~ (-

_,. "= --W-b (=

h•.f. ~ + ll .=b

: lA h .!oJf~~ ~0 -'Oj-,\o11 at~J~:>)b:)aeJ ,:)~

Page 33: T So•~) J hreeder/5090/5090solutions.pdf · z5"A ' P (X,:~ i.x) = l-P( X,;., "7 x) ::: l-( \ - F(X) r ;: t 't x > e J l' -(' -[1 -ee{x-e1r] - t1s

3D

==7 ~

lIs Y\ -'-\ +- §_ ~•0\ ~

l1 t-(l.o,)~-(t,~}~ = Q

==/ II E ~ /. q£1 j 2_3. 1'-lj

5,-vt c..e n =- Z 0 ~a .s ,·,.., s '-t f{/c ; e,., t a~ d 1'::> o...,.. ~ r

i Vl c ,e a s e ~ ~ ,· f 11 n' l--v' e 5 e e + l-u~ i

V\ = 2 '1 is n, "t"' ired -\'o, + t..E> +est -1-o

h_a ve

.ot==

o< ::: • 0 ' C( Vf J (:> ( /0,)) !: ' l 0.

s~ (I l? ( X- l < < q_ ,c.( )

I< ~\7.. s -ao

"' p ( [ -_1'2 < a. I .14. = I L

lzo

=) ~ = t ( l 9 \ ::: - z . 5 ~ .o I J I

S ,· vt c e X - \ '2.. = It- \ ~ = - I . I '2. > -Z, 5 L.f s '1 - -5o ~

"'-'e do ~oi re jec f Ho.

'

Page 34: T So•~) J hreeder/5090/5090solutions.pdf · z5"A ' P (X,:~ i.x) = l-P( X,;., "7 x) ::: l-( \ - F(X) r ;: t 't x > e J l' -(' -[1 -ee{x-e1r] - t1s

. 0 l :: SIA p '2. L 9 CJ-

PI 0 s~ > ~ 4

'2.. cr

3E

::: P't"tS" /4 ,a-2=1

.;; CA.;: ~2.0 '\) ;:_ 36. l'\ ·Cf'l

SiV\ce \C{s~ = \ ~ (I&) :: 3 3. 1-'8 < 36, 19 ~ 9

we do vto -f r e je' c. i H ().

SF f_ecq(ct-1fqfe Q +'o("- or b;i-rar) r1 ',

q = ,X~"~ ( n- I J ""'; t l, 0 +es-t slq-llrf,c of tn-1)5' 9 t

. l 0 ;:: f-. 0 ~) ;:: p ( (YI- \; 5 2. !: l-~ 't'f (.,-,) I o-" = I ~)

= p ( (n - l) 5 ~ = ~ :<tq ( " -I ) o-z. : I ~ ') 15? 2

-) ~~?Cf(11-1) _ ~2. (n-\\ 'I 0 }

2.

-==:; V\ = 5 3 l.s re "£. ..,;re J So thqf o<' = , o I 1 (3(/S) ~ ./0,

Page 35: T So•~) J hreeder/5090/5090solutions.pdf · z5"A ' P (X,:~ i.x) = l-P( X,;., "7 x) ::: l-( \ - F(X) r ;: t 't x > e J l' -(' -[1 -ee{x-e1r] - t1s

·so·.:; <J ~+~;'VI

\ 0 , d ?-~ .$ _} (') .J-S .J t 'o ~ Q J p OJ j ~ n 1, Q)J ~,1 g -z ~ Vf ( 05

( -YI (}f) ~ (~ )d ozz-02 _ S~>b·~ > ozz. _ _l > oz-..s···? :::

( ozz =w Sbl,·~ > 0:;_ X > $···:z.) J = so·

( So·=(o-z-z)f}) J;;A-~od ~Slo -l0_:j

Oott :-w I ~

soo~) J - ...SJ,Jo,~ >

.5'2 ) = bh .

ooz-x oo-z~ ~ : \#\-\

oo-z -::-w :0\1 '5

Page 36: T So•~) J hreeder/5090/5090solutions.pdf · z5"A ' P (X,:~ i.x) = l-P( X,;., "7 x) ::: l-( \ - F(X) r ;: t 't x > e J l' -(' -[1 -ee{x-e1r] - t1s

It:;-;; T

\A-

o{t Vt _ ( x>)b0 -:: :::1 0

'7tLt+?1.A-= (><>)bol ~

~ < -:> .J l "tH?-) ) -11 = J \-;>)~ -I) =(-q·u):~ < ~<:'X) cJ ~*' Q~L?t ;YH

07L77t ·. OH

( u I' ) d X~ f! ~ -v 1' . v 9

Page 37: T So•~) J hreeder/5090/5090solutions.pdf · z5"A ' P (X,:~ i.x) = l-P( X,;., "7 x) ::: l-( \ - F(X) r ;: t 't x > e J l' -(' -[1 -ee{x-e1r] - t1s

\A = (ff-l)ol (: d ~ ;; 8 -\ (==­

( ol(-'lt)v

d _)o -l -:: ~ tt)ll -J ~ v . J 9 ( 07L-'zt)\A

C)><::>

( ozt~ tt)v

[ c·~-u) lA + ~>o_)'O I J Jxd -

U?-::J T

7L t.. ~ d ::: r u I :) < ":'X)d = Cu)li ·a 9 u\..4+ ':>\A-\

Page 38: T So•~) J hreeder/5090/5090solutions.pdf · z5"A ' P (X,:~ i.x) = l-P( X,;., "7 x) ::: l-( \ - F(X) r ;: t 't x > e J l' -(' -[1 -ee{x-e1r] - t1s

0/') ito'.::

~-J I ._. o

·~...,, b.A-d

~AoJJu.1_1 F) t"'~O-"Yi

Page 39: T So•~) J hreeder/5090/5090solutions.pdf · z5"A ' P (X,:~ i.x) = l-P( X,;., "7 x) ::: l-( \ - F(X) r ;: t 't x > e J l' -(' -[1 -ee{x-e1r] - t1s

;-7 .

• 9::; ~X~ y .l ~ Jd~~l tJ!_ s I j~Sat c}~j_ "Ji 0-z

I • / gs o . --

8t

Page 40: T So•~) J hreeder/5090/5090solutions.pdf · z5"A ' P (X,:~ i.x) = l-P( X,;., "7 x) ::: l-( \ - F(X) r ;: t 't x > e J l' -(' -[1 -ee{x-e1r] - t1s

CJr -"" Q I --_x /~-"?.::> /

A-)( )d 2 "tw ~ 'J?' : v t1

ZJr.:'}?' ;QH

b = 7. U ( z !o 1 z;t) (\l f7 ~ -v J A b='t,A (~~~'n')N p~7-v7X ·v'o

Page 41: T So•~) J hreeder/5090/5090solutions.pdf · z5"A ' P (X,:~ i.x) = l-P( X,;., "7 x) ::: l-( \ - F(X) r ;: t 't x > e J l' -(' -[1 -ee{x-e1r] - t1s

o1·/ [ot,o·o =

•(<a)i-j-o Jp~ d~-f-,, -~ d J d lA I"V) _.... i

\ z)d -1 :: ( ~) =l -I -

M J?J Qs o=v h :: d~l>/1-d

9 ~-Y ' -- \ 0~ )?' : vt1 0 ~w : !)t-1

( ~.o'n-) N rJ? '""' ~a "dW YlSS\f .)b

Page 42: T So•~) J hreeder/5090/5090solutions.pdf · z5"A ' P (X,:~ i.x) = l-P( X,;., "7 x) ::: l-( \ - F(X) r ;: t 't x > e J l' -(' -[1 -ee{x-e1r] - t1s

so· < g~·o

dJd1~ (~)::d-I \

'

"'l...o :7£ 7:'5 d - -

_,.; -- -2.~ -.Sh

<: ?..z~

~~ Jf ..,.s-)d --< t~ .Sh ~~

I ~

17

--

~~ l7~

::= an) oJ. -J 1E.

d 1-1$

"Z t_o

___...

:"H ~ 'Z

' oz....O

·. OH 'Qb .---"%:.0

Page 43: T So•~) J hreeder/5090/5090solutions.pdf · z5"A ' P (X,:~ i.x) = l-P( X,;., "7 x) ::: l-( \ - F(X) r ;: t 't x > e J l' -(' -[1 -ee{x-e1r] - t1s

( 2.'2-)-1 ~ ::l dJd~/'1

([·(t'g)~b'J)_j -l =

I

U'J:.~~($/s)_s~,}< :~ d=(H'I)Jl

Page 44: T So•~) J hreeder/5090/5090solutions.pdf · z5"A ' P (X,:~ i.x) = l-P( X,;., "7 x) ::: l-( \ - F(X) r ;: t 't x > e J l' -(' -[1 -ee{x-e1r] - t1s

Oo-z. Of?2 t (5o')J;b" + (~·

0 o--z Orrz - 0 b 1 -08,1

'c11='a :vrl

cJ :: •d ·. o ~~ I) oz..

::_f -( cd)~ 12 F:: ~A

Oh I ooz=ziA

Ooz

( 8)?J~8 /Vx ·o1 -::: .X Oo-z ::'\.l f:! 0& 1 ~

Page 45: T So•~) J hreeder/5090/5090solutions.pdf · z5"A ' P (X,:~ i.x) = l-P( X,;., "7 x) ::: l-( \ - F(X) r ;: t 't x > e J l' -(' -[1 -ee{x-e1r] - t1s

Sb' _ 7Z ( T-

(f-1) [I< Jrv =

foc)]T(I :e/ ~7 T)d::

(I ::e) "? ¥) d :: (I :: e / :J 7 ('X/() d ~ so ·

so·=.>o -z:::e:vt-1 \ -=e :')M

I =u

Page 46: T So•~) J hreeder/5090/5090solutions.pdf · z5"A ' P (X,:~ i.x) = l-P( X,;., "7 x) ::: l-( \ - F(X) r ;: t 't x > e J l' -(' -[1 -ee{x-e1r] - t1s

so·"' (1 =e ? ( r~) d a!Jy~ I .~ 1 .. J .' oH }Jd('01..J '..)bJhJ!rj/Od \A..T

SJ.·

51: bO-::: t.(Sb ~)-) -xf X l I~ -

Page 47: T So•~) J hreeder/5090/5090solutions.pdf · z5"A ' P (X,:~ i.x) = l-P( X,;., "7 x) ::: l-( \ - F(X) r ;: t 't x > e J l' -(' -[1 -ee{x-e1r] - t1s

t ~ 1 ~.

('w) ~·' 0 t-1 + ?.;! ( J_j C)f"""1 'os • t-7 -;;-d --

1x ± (?' (0JY-'w)v v

-,{

\ •x'~~~~-j}l tw d '>f)( -:z 'n""'--

( . ' ~' \•w ~ ><)} -1 •x

''•. »' • .,. -d il d ----- -

l ?~~ 93 o;,.r oy,; U-

:' )( -:? (. \A J-0 .;J6_1S djJIMb$ 1'0 I-f PM ~~ Z1

# ((I )7) IOcd j-o j-p? J'1 /-s_1 (o#~ ·)j aJ~( M (""" ~x)..:J-} = ~h/I:JA-d : an;~all _J ~'1+ jnj

·( ow)Iod Jo ~~ 'tkd71J.Jd z;-,.>oOO/ < >< j ·' ot/ +-JaraJ '7-saf

_..~o ,;;)~.'5 o JG_J • ~bjo! S 1 )( j.l J-;),}~aJ JI!'Vf d("\

('n-~X )J -( 017': X)j-

~ = 1;1

(on-c:: 'w) 'w = w : vH ~~ = Y/ : o!i

\. rl) j, o d f ~ ~ /V ~x v -z '

Page 48: T So•~) J hreeder/5090/5090solutions.pdf · z5"A ' P (X,:~ i.x) = l-P( X,;., "7 x) ::: l-( \ - F(X) r ;: t 't x > e J l' -(' -[1 -ee{x-e1r] - t1s

",Jo f ,) J .' S c> p

·~{.Jto/ SJ

/c;;.' 1-s.a+

·~

. i 'e) 'e~>f ~ X-d ---

/X_?( o~-'~) lA \ 19 (3)

()o//x- ' £

'e 11 --z 'lfx ~

oe -JJ ~ik.f_

(:e ~ .k? _t -~

( ?9 ~ :i<<J

. oe < 'e +dl

of3( e :vH ()8 =-G : 'll

Page 49: T So•~) J hreeder/5090/5090solutions.pdf · z5"A ' P (X,:~ i.x) = l-P( X,;., "7 x) ::: l-( \ - F(X) r ;: t 't x > e J l' -(' -[1 -ee{x-e1r] - t1s

C~))O-~;.~)~-'­

(~I ( ~) .. _~;. ~ ( ~X2 )cJ ~ ( ...D t (~<)"'-~-;t .. :P( ~~? )d = (_c)Jl 811

•( ")-~1(2.:0 ::! O> '(");X "' J~)_z ( 01-/ ).Jf'Wfl

')o folJ_I..SC>p d~~

V7dSOL[-::> Sl ? ·-:;C 2!X7 j"! ofi o + .s .1 7-s .;;f j r j J .:J ""-1 od :j s; o w J/ w J~ ~ £-~ n

('

-1-db ot ~ '.Js d..l

)::) 'os;

(~:X)j

-\ ~: X)j-

~<...o :~\1

Q-D; _o ·. <lt\

( ""t ...0 Ia )N r~:,__, ~x v-t l

Page 50: T So•~) J hreeder/5090/5090solutions.pdf · z5"A ' P (X,:~ i.x) = l-P( X,;., "7 x) ::: l-( \ - F(X) r ;: t 't x > e J l' -(' -[1 -ee{x-e1r] - t1s

?Z9h. ~

( (o-z)-9:~~ )~-I = (z=--<>)Jl -.aH

..s 00 • (.) :: ,>.q

I=~

o z = u "Jt 1

Page 51: T So•~) J hreeder/5090/5090solutions.pdf · z5"A ' P (X,:~ i.x) = l-P( X,;., "7 x) ::: l-( \ - F(X) r ;: t 't x > e J l' -(' -[1 -ee{x-e1r] - t1s

\ o x~ ""~id u N~ F ( o, 0; l~o: G?. G~ \-\1\: G> < eo

Let 0 ~ 8, '-..Go ..

t(p.) e(;) ~(f; e;)

(k J ~ iixn:r1 (_ G~ ~

(~J 11 x.:- < e,J

(e,~"' e 0 I )<fl ~ n < <9 I ' CX) 6,( xl'l~l'l ( eG '

DN F 8(3 ( Xl"l~n •

So a. M os1 pawer f"' I fest of' 11 ~ e -·eo lo

v t:;, H" ~ e ::- e l ,· s 1 o rf'jf'c i jf Xh:n

1 s /es s The,., cJ ~--vhe,e c. IS c h ose t-1

to 'jf' 1 7 he- des ,·red c<.

mQx L C Ei = L. < EJ -~ - C ( I ) ( ( )l"l

"(= $?G, p ln:n - p Xn:n - I eo) - eo

==-) k Y\ c._c< ~) C = E)o o( Yn

• 60

s,- .. '\C:_e the MPT o"'lj reriA;red 8,(.EJa' th,'s IS 0..

U.,A-'1 p test of ~\(): G ~eo V'>. Hll: e L.... Go.

Page 52: T So•~) J hreeder/5090/5090solutions.pdf · z5"A ' P (X,:~ i.x) = l-P( X,;., "7 x) ::: l-( \ - F(X) r ;: t 't x > e J l' -(' -[1 -ee{x-e1r] - t1s

~ • ><>-ll-+ 0)7 < '! j-!

5/

I 0

j-•rr< w; V/j 'S/1 ''-" =itl' :oH

I 11/ J-.' 0'

9 - -

I

j.l

d { (o '>"-,, ) 2""' z-[, '#-, ·~ u) i-

d ( ".k'•n-Z -t , 0>-r--" ~X'-'X'>T(-"'H+_,? x)? "f

G ('!r-".-\/'...? t +-,. ( •;,;_ ;'~ f-

l l ; 1 ' ! __._

d -

Page 53: T So•~) J hreeder/5090/5090solutions.pdf · z5"A ' P (X,:~ i.x) = l-P( X,;., "7 x) ::: l-( \ - F(X) r ;: t 't x > e J l' -(' -[1 -ee{x-e1r] - t1s

•o}T'~ ~ :~fl ( ~ /1,1;-.blA J ,:;!/ b (

r.:-:dpl s { . o~f d'1+ .J o;t ~ 4Ji-_t5 J';:) 7-s df d Wh ou Si d-.J d'1 +

((8kJ+-vbl iA i iAd~S S'O) \tcs:-, ~ J.--0 d'vlj bl\ '-..)

att-1-j-0 V<0,/1-~L1hj-b $'1 _lc] hi d'vJ-f d-;) )/! -'5 • J h1 .

Page 54: T So•~) J hreeder/5090/5090solutions.pdf · z5"A ' P (X,:~ i.x) = l-P( X,;., "7 x) ::: l-( \ - F(X) r ;: t 't x > e J l' -(' -[1 -ee{x-e1r] - t1s

·)0 == (·e= e I (X>):/ <:: J) d <>J d'f/VI

_, ~ a[Q)J C..J.... Sl Jo ~ ¢ i r~ L<> -..,.. I . . --...J

( ( /O)J < ~ t ! ('ll ,

J..:>~r d w n ll 05

I +'G I' +'0~JJ \ "'x\ '0 /

oe< e ~VH Qe~e ~(\H

Jz

Page 55: T So•~) J hreeder/5090/5090solutions.pdf · z5"A ' P (X,:~ i.x) = l-P( X,;., "7 x) ::: l-( \ - F(X) r ;: t 't x > e J l' -(' -[1 -ee{x-e1r] - t1s

s i

h 1 J-z • -= ( ~ BJ~k;x)~ -I =o

(z,·e/'z-)~b;~t< ~~7)d =

(z-=e/ (8)~,\X< J.]z)cJ =(z)JJ

\ 0 I =/to

1 _oG I -

d t!V ()

oe<G :~'H

oe:;e : ()H

lA ~OVI >f S_l /( ' y

rreeder
Sticky Note
The degrees of freedom should actually be 2nk.
rreeder
Sticky Note
Part C is not correct, because the degrees of freedom should have been 16.
Page 56: T So•~) J hreeder/5090/5090solutions.pdf · z5"A ' P (X,:~ i.x) = l-P( X,;., "7 x) ::: l-( \ - F(X) r ;: t 't x > e J l' -(' -[1 -ee{x-e1r] - t1s

• .)o-== (·?(-=?II c )Qp < :x JJ )d ;) Ja1"' (\>c)? < :x Jj t.' r-; f ~cYrdj 0f s:' ?o e>-:t:s J-(.) 7-sJf dwn v

( (:),/ :) J1 \ f3/0<-/-~~ I

""'''"'"" ' ..... , ... -~-'. o~)J e~ (' )1 ~ x)j·

(. . ' ·-···· d . 0 :x _Ll -( ::;1 ~ ~j-' &9./7 1-~ \. v/ ' )( ....,.

. (1;)1 ( ~ ~ 0 l

~<~ :vrt Q~ 7 :>j ~ ott

LA /1-1 OM ~ S'.' e r z z

Page 57: T So•~) J hreeder/5090/5090solutions.pdf · z5"A ' P (X,:~ i.x) = l-P( X,;., "7 x) ::: l-( \ - F(X) r ;: t 't x > e J l' -(' -[1 -ee{x-e1r] - t1s

j-.' r .:;_;;f d w r;

OH +-':l .;;; ~ ;;l J

b r2't'ZI

r b r 5 :J:.i ,,:j J ,) lA I iA b) ' I

oe> e :!Itt

09< e ·. (JH

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