Problem6 (p5 in this solution)

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Solubon o HW 3 CD Vioblem 4, The density funckimot E(a,b) is X-a §e0= -L— expd - e (1 (=0, Sp Hhg jont densihy of L, Xn Vs n . A )= ‘lfie*”{‘%'%é,’(fig 1(7((,)70«) b Urhece Xey= mindx, -, Xa§ . Mowover, the sunuvad finch of Xy n S0 = Py (X, 7% )= Li- Pxen] x> a) = Qxyp g~%(’x-—a) %4(1‘)&)' (t.2) For Gny @, <an, Hhe ‘Qi\m\,‘e\w folo with @ Common (Rnivn) b is (% $a %) _ QKP{ fn(az—m§ A 28] (1.%) {ql(z") b 'l (7((|)7fit> (l) FO(IOWl's\‘& Hu_ e‘mt %fUQV\ in Ahiy '{3)’0\019(&(/ Wwe (owndey d '('Yavh/b,o( mabm -a Yizeqi{~ 20 28 (y(0,6), wit b=e " b kaown. Nolke 4€ H., a=0ao vs K. axay, <= H. 8=0, ¥° K, 0%0,, whey 9, = e-qo/‘” Accordia T Hw 2 problem 4 We know that thoe Rxisks a W\ifiwz UMP test of Ha follewiy Gy m . o4y 5{ L, B »e, or Upy €0 U v o, vtherwire Becuwnr 4 = Q-V”/ bhvo, 15 o decrtdid “ramfanate. FANN, wie o bitn L W< 8, 0f Ky ok lyg@) (14) { 0, of/w which 15 Ho lewel & UMP for H: a=do vs k. a¥4,.

‘ (h) When bis knewn we knod S Pavt (i) that the powen frnching of the UMP&POY' . G=ho versug K o= Q) <o S %(&b>_l—(ua>mm{ (aqui scale (74"4'“'/"" b. we hvdh & combact oo (ew teat 43’(7“(:) Fhal 1s f)—%df £ O(q\w( a UMy whe accodiy b (1.4) He fert ¢Cx) a(,e/{p-emh o1 He When b i's \M\hnmunl pavinafn b hut ach ifues the pamt pove abose v ovder Ve, Defiwe $(§3: g 11 Xy £Qo 0{, %(_I)} ’J-{o’ Lfsse S30 is ao ( )(Cl) fao) + o qu(x(l) >6\0) = £, — - — N Q, no datn¢ g, 1, aldata g, undey H Under H @(fi, (4 {” P ()%) Lbo) t+ L Pc‘ (x(‘)>0\o () - e~ L (o a4 kel L geman = e~ 2 (gan Now, e wesd 4o sh & ; ¥ o that CP(zg) 15 inde2d o UMp et d={z.~ ey <a, f B For any Lovel o oot B(x), iEis swhlcent b C—vhy;/ M(al',b) ; . N P 5¢ (@, b), ow equ.vzden%) (_:al§ q)(%) - @(f)SL >0 NolR Vhat Ao )4P'= {2(\' ey >a°&‘

On the et fi P, (x) and Py ) hawe Hu 0awe SepPt Reaune o datm o ey hebween A, dﬂd a.o on qf] Ex %70109 Thus 5 q’(l)— l()(zi)% APa,(’E’ - S §6|>(>}\)—k()(p% &Pa,(}) {7,(:: Im‘/;a,,fi d $7£= 7C(n’/t?(‘,(r — r {5 tamanf J {00k yeof d 7o, Thi) fant t:wﬂ,uain'ix) Dolcly becaure 0 < E{e)-veof= |+ ) [ar- ¢ f dfa) A A€ /(\ 3n Aum may - e. 5& 15 o QWU’W/H.C @d) (@ h) > 64) (a4, b), §ur aney 2. CF'(I)IS the MP et with pawen - (vt ) ((u) faccord MR Hhe P“’b‘e"“, W hat H: B=0Gp «» K. a=a,< Qo (b Unknswn) O bt nodaboctur < a, Under P / unhuoun b0, ~ n(ao-q')/fo and the beat ie 1 X - Qo P d)(l): { ) Sxexgy = o2C, ©, Ohu _ oy wWhie €, Gad Ce Grs nov ot chng Hg pon which 15 & Shoun an |~ (=) e Fiese, wR Shavdhat (1) X, (5 & Conplete and S\,IVk.u‘evxt' syehshe §for o i) 7= Z,_ (¥; =% ) 15 an antillar sYahohc vk . Detenlg ave Omiveek durl. Thew, boy Basn'sThm Xg) Te .m‘\ege,\\&&/\\' &-g_ Iz, = 2_‘: (X‘--- \(-(u).AW anSB 70, How/ we "IM’“Q A= Pao(X(., \C'Zf'ao) ‘1’ a° CX(.\ =0, 2+ o ) {’1(X(.)\C?+Q)’/Z¢2=+ Ea B {i(X(,/C 2+Q,) I'ZE’ (+ ka,,g‘[ exp(— nC,fa}&__ Ee, {e"i’(' “Cz )} (\.LT) = [+ E&{QKP(" nC\ - E ieﬂyt—nCIZ)% i\ Gny o B.o,crwo‘l pARRS awcgl Gr«a @

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On tha O/U;U" Q\C\V\$, Ho powen 'S . @ @q) CQ() = Pq‘(X(,)SC,'Z-{- as) + PQ,(X(,) 26,2 (-Qo) ': |+ (l"o() QX\’[l- ’% (fio'ql)g (N) gm& C‘ and C?- Qe chosen gﬂ-u D'é a., and b} ac(md.\,S 't{, \arEs () and (((()/ tho given Lent P(x) achreusr e mes powli for 01y (@, b) with O < Qo e e o\ Nows, tt Ol nazel 1y S Hudt fhis kept achiesta it inee penen dov ey (6, b) with O >Ro pr{- Nn(Ge~ad) tlg mor Qo w-‘fiaun;ﬂn b, e 2 , , Nole Ahat for On 70\0/ Po\ << P&o/ (L?e Pa, O & Qaviet ‘Ju.«zpd’)orf a; (S(Q\): S‘{/l ()((.) €6, Zta0) + 1 (X(l) 7 CZ'Z-QQO}g APC“Q‘) A&Qu&q) 3Pago x = g {1(X(l)scfz+qfi)+i(&‘)\7CZ}ZJ{QO)} X (n3) - QKP%— Db(m’;a()§ g {'t (X(I)SC;Z'\'O\.,)‘f' ’l()((.)bCz’ZHla)gC(Pfi‘q _ N{ae-a1) ((_ - = QKJ g - b §{L-%P (X(,, 2C 2rag) t t%P(Y(.fi/CZ’Zf 40)10 — o(e:w&-’%g_ (V) H: Q:O‘O,b:(f)o = K; 0<fiu, b<bo, \:01' Or\") Ay <o anid b‘< bU/ wi haue %’1(2‘\.’ he P L (h')e”“{((—bu-t)fx\‘““(&’-g-‘)} 1 (e >a) bo b' e — 1’ (x(')‘f ao) c\‘; i Meteomdsy & Ha Ny - Jearsne Lemma, o Level o Up t0st i b4 ) , 0/[,\) U} >ay 1 ((! J|(ZJ . , d’(l):i ’ L >« <= (0= [l 1, }q,¢%e<a, 0nTX > K tnd o, 9y

C LMY Y C‘;;O,'VQM_IV ! mm,ma(\ wpugy PrIeviy —'V”? Cfianwu D'qn.go@wbat/v Tm Y S WY vy Y vem G‘“’HNO FHO L R weqmd T woygeid f0 arva “upd U Mwu’)( Sy O DyTIeARd YIS ey i GRS C“’m‘“‘c‘*\ ~rnesvd = O Toagm %l‘\“ﬂ) (0‘\\\‘/3'\ \?WUQH 'fiWAﬂd-—nm4 s Gl 3 om0 Caalnd e “Spam o - ol AT ot 4 qeyers bopavre pew susus J%".'%S ?/7/‘9“«?7 >, 4,, vuboy0 177 , fo Per > sapiaed s Gyt oot e T O sy sy (9) {9 puo v Saeuvand vgoa 25 Win vy e fr (Y) Sov ol *“’VfV“‘O"".‘ omy fo 1. f"\"}-\““_'ﬂ(sz\to TN miodra Qn ) 2y w1 00 O Y BjaY TV (»wcmwd B g A 7)) (M] TS @ah o Ing 531 PUb i q@1d Pl D% L T Duwvant 4 pvugp T U e fvin v Buge : N Ve .%“’Wz/.mdzﬂnw-l vy (1 (%) Tl nymo () (PU)y aovy = N , / . N ot Gy 2 ( °9 /10) Y vatwno e 9D~ ey (°9%¢) w:”‘”@ H & )9 -} Mam (ovw 217 (o292 8g = (> x% )y ) o e} = " M 233N umyya P cu J (Qy< (l)x 5% >‘ -? “x\:-jz AO n’_\g’: 0o ’ w-< v > X3'p ) C{ =R NI 3 0 A T o

. 4 Prblems (a) Comider H, 0<0 «» K 024, Giew a “fi“e”@%{”‘““”mg;} - - L. Xi, - Xn, - from @ (x )4(3 L, (), we bave Hy LR jet statishc / N 7*/9-~tx~ . R (xw) = Tie, £, 20 _ e Y A (x> O, tf o<, <t " OO M T e T ] vy AT ' e/ ¢ 'X(-”?i, We chooye wo Cuto fte A dnd B Nuch thae O<A <R <o t conr hruet e lfij{c\dm yu‘QQ ¢ 'LG X(l) <4 , Rl“ -0 < /_\’ W a((’evt N cnd 3w tlhe eY\\QQ. " IF X vt * Gy > , w/e ey ? C\(C.Q()t' 8] W'Puvw \?r\l e <A or ] Yeject I Wiy N ’ aud e, - Wma eihet &wp\mr\\, W shrp 4he dval Ol—t\orum,e' wie. Combhnue Araw %Cwefilus.; Hm'q c'ncanc(us\‘aQ cane Covmpmc\s L{/ A<Ry<B o logA <N < log@ CeYb\\“'Q‘g N;fl/ N0 O<A < i chesen | Obv\'&‘f'&o, whin Newplig w\hm\esj 0= q Ene that- " 2 B must *’A“W"/ he e : : e Y\al oy 0 rivp suL@Lj v m oy 2096. Tha ipper Yaoun & i M Amallent puviow kg of g, 0 4 The tyne T ey of the SPRT teat i o ,= A (X >1 and Ny L@(g)) - e‘@?j@]'fl) o ol Lyt = g The *le L ¥vor B4 H#a SPST dert is obvimdy 2010, AR Undey K, Hs fi“’cmno\\n’& wall by m/iecbevl '8%( AL () To Bad B (0) | we nowa that under k. 9=t , Xey»1 for att m, 50 weneuer CLCCLJ?E H:O:Q go/El (N)= {Qo‘r)(}’]'f'i = (- 0050/() 7 To find B, (1), we know Hhat

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Po (nal slops ak Rt draw) p ( in (¥ & = m 0 \, /yk~|)>/l/ )(u<‘>/ l‘(-i[gojfijﬂ' A r—r “M Ao / k< m= Log(g1+ : f‘é““év“t Lo Thus/ ORI of o = (e—t)(nm\) Ul l’\_C\/U'Q- / m-\ t‘(N]: A?j'l ,-) (\‘é“)(é‘)h-‘.{_ m(e-l)m‘\ - Fltmupr |, wu v (¢) In a &,xecl-sowtﬂ,fi_a ‘QMES’"; i He oAk ans & collect M Sow@\fis. flwpea\—i% R AT AP \n Pc\y{'(a)/ we ’&M} 519 g N- D Lemm«) b= { L, X >4 ) o umg Ror vl Spe sahtdly 0 r 0< Xy X1, -h > - A, k0¢(&()_ e, So v n}-‘eobo(o oy N= [-ﬂogo(o]'i“ / in Ru wwganyem bl SPRT kW &ixed-swbﬂﬂ ump teat, under K, E (N) = [‘*eojd,,] > (g, T+l vo f\wa,(’_a SR AWty oCCurs, UnaerH) EQ(N): %&Q—(I-p)mgz—%{l~l+m?f’ (m”£+ 0(m3)f = /Mm - %(;hi)/p 0 (m?) wkp/ = MNey ~ (Ylﬂ)(n-n)lp )/ H ms= e I ‘\‘0(’]]2 E%BTfl=[,%y6]: -1 .Eo("’)'f‘f _ — = =5 (1) (n) P o) o .

]?Vblf/bm4 : (61) First, Ha LR s¥afhishe is 9/sén bg @ Fn A, %, K QZ_’/(‘ d’?n: l‘-‘-lf[ /3): (&) (fi) 2u >4 = /0 QI,, T‘.-n {"(/l(')?o) go ﬂ . —-770 ?% 70 (=) o, the inconclunice s 10 net wp an frllow's log A Lo b < Ry<B <=5 2L ofyn < 28 /033'1 = 4o Po YT 9/0 U — whew 0 < A < | < B G Awo Lw 01’&5. 2 ?_:IQ({—I)-H\] = ;%'z(xl--l) +1J N \\)\)R *b\‘\t XC é goll&f} \\\'e. Y ~\ C {“\ Og Ao ;g?(y{‘—‘) o%)ws wm“dm Wl thut never goes Up. 20-1) € -2 0(3 T oi‘l\MWUYcls Zfi_ Q(X(_"()“‘ N Canm eithin wolk down ra_?ld-tl‘/) or the tensth o .y S 1 we et wo\Lkw@ -stmNQoab c\nd}gacb\ move Lolen an nkgtn uw\lb WS, & wde that Ax: _l&{i] oy P)*: [‘fosfé ]-H, gt wAtl be Y Houna S ' £ wjeck R s UL randorma UJU&_\/L Can oyl gUY r%ec\n\m. TlrL Y \S} jec : ¥ RN‘:B*’~ oy aCCpt H 'f Rn = A", (tr B Past () for tha A2t i m (93 catolfs A and B 2k & M = Fo EZ = (2774)/&5_31 ) ”C‘V ) @’j T) (21" ). EZ 7 E. | V - fo /’0 f t%( "= (29%-1) 109_4_" {0(0 &3 ( l?; ) + (H’(O)—&j //L—'o_lo) o

(<) ‘70 d//iﬂ/i] Lemman 4 ( fase 8/ (ectune hufes ) we Chack +he FrR2ow ™y Cfn'wb\'h?,hj@ londition (if) Ep (2)= (a9 \%5 %U’_o £ 0 o1 Lonyas v4 L Camurned ) (lll) PT’(270)>0 on d Pe(2<o) RN , Ot ). (5 o Po‘)'ahk?‘vp]. Acum\i\j L Lewmma 4 thoul 2K A weal nuher = 1) £ 0 guch Hine _ 2h o \h 0\ k m@?wmb?inh/ t(’e =1 = T(L) +(!-'Y)(%) =14 T = Cem) (P Y (W \R 7 G () (?) L P =0 = (’]”0 W L im’oﬂtﬂ‘ﬂ'h& lecaune h¥0. Logllf- E} - { hop) = %»p - {‘/ P T/{)_ ; %5% -1, ’P:zo 67 T\"Q"‘, H’\,LG’)O'WQ\ @LMMW\\'5 hip) ~ - Bipry= A " T A 1;1,\01_)/‘5\»“@)“ B- =A i e "y P 18 IP-Z'O O \/‘CE k(y\QIZQ/MUY QU d jﬂjMEWW \’)-Q_ 0(0 G d &0’ TL\QV\ o( — ({P )Q ‘_/)” % - ( I"'A" o > ] = = = = |- (5 G-A ), 6 ‘3 %o) B—\—K\ Tt in eanmt pee that o A= and B= .|_—_P_o 1=y <, ' Which aae thi bounds in ein (2) on [n5e 6.

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Solulim o Rrblem 5 - g(a). e\o dﬂfh‘“\.\‘:m/ Uex;60) Ee‘u(x’.e” = 5 91‘6%) £(x;6,) dx R /360 - surs B U(X.5)=0 (e { fom- oo far, X Tt W § (o) - : 4 e dF jR@wRonle Sp. WCx,90) { £(x;8,) (9.-0‘,)\% 0 (Ifiyéo\)} dx Wt 9, = 3 (X> 64 Sa U$X;Go) ) ;__._Sja_.e_/_, )_ -;(_x}oo ) dx (91'90 )@ O(l)) n Sau(xi 0") u(x-)'Oo) ;(1;00) dx (0,~0) (\‘\’ 0(11) " E, U (X;00) (8-00) (11000 = UOH@-u) (iroty) o) By definihien _ &-(X 9.) _ . D|(90/°|)" JR o‘s {-(1 o) {'(1 e') dx = S&‘%L‘,o'){&s{{l}on)‘hfi{ujaﬂ Noke Yot | Aag F0;00 - Dagf0x; 00 = S W(X . 8,1£00,-60)) (6,-80) 4 E Q = S‘ d ﬂ.3'§~(159°+t(0|-@o$ 0 “hioin hecauns = N &5{'(7(} Oo+€(6:~0,) ) I ‘ Thus ‘ ° (S, 0))= S S U (2, 0,4 E(0~00)) (8,-06) 4 (X;6) dt dx bhini s TM S g u(x ' Qett (0-00)) £(X:0.Ydx & (640 )

‘o » [.uw - 20 U T ¢ 7(@&)1;‘;..+ (X)u y ‘ YWwan J-n my Y Cioy =9 :"H J73fer am ‘% Y vﬂw}‘s_\u@g A ey vuYSio YPWARU ".\\M&v\lgwx‘r\\;?%\\):‘yv\ {4 § by won } 1(\'3 ) ("% “)7 2 fmen ] (1g-%) (1)1 LN 0y (94 4 XP('O-XH on g - X ng ! \_{("0'*)‘} v (o.x)} .3' g -?( O.X)} (le‘fx)* qs. = (x)‘jnoi( s °Y{ 'Jw\-m)y\wd up . 0 (° M 14 / .H GOUHUN G (08T e T E N D () 40 \ - e Yo o (W-—()‘fg}l_:() up fi\ ‘ ‘419 PP T RS °a>'_l'-o “arly et (M) T 7 ey WYy ’ a_ - ((3’11)°9‘ ’ P OQ 15! w = on w = = (hy'v 5 2. Ja?! U (Qfi Yo} (0-'9) (e 1 2 0g _ o 9 = cnonviM) Jwor1 1 (*o-19) 43 (3-) fWortdeey (9)1 rnbaa fwos1l _(*6-) yp (3= (C0-'9)3+%9 )1 °g _— ) (o) Buiiddy . ©

1 - @ = 9,4 S Sinte By depinls o M, we apnly tla Lindehery- Teller Theowna ¢, establicl, TN LR (X,) Y NG, 5% . 5(&) Under Hl,@=e| M'-_-_ E Lk(é Y = j " '5'(!36.) pavt (b) © n X ONX — §2 Then 1 powen oy s N- P test Giewn in part Ce) i @“:p(%gcckl-lo \Hi)= P (accept Hy | 1, ) 13 = P( LR (Xn) > - 7',1(0,);;‘;-37, + 6 Jn 2. \H,) - P ( Lk(lh\-n/“l —_— . g r & e IO T 22 Mok Ht =% - 1% > : ° S, = Vowél(L(x\))- B, L(x) = A = _5_;3 16y (1tom) h 2 [ o- - U w— o C$eo ( L(X|)) - W I (00) ( [+ 0(!$ SO o N / Y] and o mza-v; ) 228, Ao T ) A -—(,:‘-; - N = m Tnorden v B4, we W uae G_‘mza-y._ 50 nandh ma-,‘{_’o > a<h Thus, whum A<y a3 Plz>-w)=4, r~wnon Thiy prves tat the N-P € is comnsiit.

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