Vnt- Order sttistics: et Y F2 - X be n independent lertcal dstbuted voaiablog ench w: Cumulo ttve dstr:utTib funetton F(x) ond pfflx) f se o,ables ann oMoned inolendi rq oYesHes Maqmtuole and wt:Ter Xt 4ta) - -4Yn le rles Stot:st:CS (:1,2,., Te X) s a indeperodent ecaue neual ty ootion tm. Hee u amorg Xt) Te Sma lest , Dstribution tunctien a sigle odee stat:st:cs: let . Fel) - , r'2, ,n deote t di strut functon oreles statstic K) hen a distbuton tuncton Fl) lasgest ores statistes )sqien E,)P s*] =pl%,s * :,a,..yn Px, xnxa Xo '3 0 P,] PIr]. PLXn Sice ,S nkperdent Fl)= [o Flx e strbulion urction Smallest olot Shtste q.von by F,) P * -Pxx; i:l,2,. .,nJ tp L(x,> c)n (ra >r) a. .. n (xh >r) plz,> *) I Lplx,sc). pl,>r). -. : p (X;> x) tT - Pl2, c)) )- Fl) dngenea p rles stat:stic, Ft) P Cx - P Attenst 9,s * i A) ) -Fl a snbimmia rokob: kty ole, Ta king TI in equation . we qet nJ7 -fr-Fl] 1fTn, Ft)()F -F F) 3fit (ontinupus 79P ue'hon an e ortten as, ) I F n) wou, b Epfo, ) t-t) olt n the Lompube th furcton A-t his shouos ot poae ty points n ordes obtonel o: the help in lormplete StatistiCS Con be Sing oroe statsties: We ohSume iidCentnuous Tarodom vou.able p, If e) lerwtes TRe påt t) lIFt(,nrd] O gtJt(tt t t (t-t)- J+tit: .F) la f-qo fl) Subst tutee in e qet (*) Flfl) B(ni) F Joint pof n two orle statistics; Xr) ard K¢s) Let uolenwte tha joint pdf g whun 14r4S^n by, An The erant F: (x44 x4 3z)nly£ *isY*oj) X 9s :x+d* oone *; 4 + ay rX; and :y+ay for [n-s) ; ubinBing mu lt:ommoad pobabi ky la, get pE) P xXy n EX*ar) lys Xs) Y48jJ plE) - S r- (r)pls-vi)!u tn-s) R Ps-n- whe PsP = X+Dx , £) Fy) -F(ta«) P Pa Ly, 4+dy)J = Flytay)- Fly) P-PLx:y+ayj = t-PLx; sy+ay] Subst:tute in @ ard usin oa qet Jrs 9 PlE) - nm +oe lc) Ffm Flx F (r-1)!ls-ri)!{n- Fy)SimF -Flxtsz) F(y+ay)- Fly) oTle soT:s tics joint pdf wo -Tr-I (r-1)!(s-r-D! FttFty)-Fi ty) j-Fl Dens urction er) he des it furctor ) FEoJ-[et] oblen Aor the exporentiadiste; b tron fx) e r o;show Cumulative d:str:hutron turcton X aa Toroom viable 9 s:2en. Flc)- C1-eHerce pove ak n Cof n) gn tende irritirg torm ep L-fexp(-)} - exeo] he Cdf , Ko ven by The cdf ,)f X, - kgn ven lby p [ t hgnj ( : 1 n tn(«) (.ep(-e) 2 Show Rat fror nom npe Si2e 2 N(oPopulotton. For na, te pRf h BCs-Flx fl)2 t-Fij4a) wheu ft) - aT e xNlo,s*) 00 f1-Ft ft) dae - We have = fto) - og (,)-2g2 D: ffeuntiatrq o,th Tespect toX we qet flc) J: flx) d* Jeot:rq pouts by aro wing oe aet Eo): D-Ftffto -00 -2 TFtJ dr TT j e dad 0o amples S2e fom vo,U Poplotto te ean ar Vasorte tre str:bution median a ard espectvel We hove ftc) os lz : p[*sx)» Jftu) du J1 u et n 2m loold), wha poS:T:ve in7eqre ro2odan ObseeVo-Tion Xtm) toki rq T- in med:an (mt) pd+ 9 YmX(m) w qivenqver y Ttm+1) () = -d Bmtl,m) B(m,m4) (-1-d D Pm42,mt) (mt1, m41) mtamt y 2mt 2 mt rn) am+3 mt mt mtI 2mt2 FLmJ =J:*fme )dar JBmal,mt) lm+9, m4) Bmt, m*1) 2(am 73) 22m+3)4 A (am+3 A (n+2) ALet, X2- be i.d X mn-neaetve ardom vasiable the totrmous type 6 polf ft) aro okist:buti on furction FL).o)T4 Elxlc o, shoo tAat Elx| co dte M Xn)= maT ( *a, .., X) show kat E (m)=(Mn-) + F ) F(r)J dcjn - 3, 3, Hence tre E Evouate (to) 4x, ,a p- .)Yn have lomen d:st; bton furcton F l«) =t j 02 r1 ronoom Vali a bles) n I n-r (r1)!Cn-v)!f)F[-Ft] d Hence elx|<0 f Elxlz o FM:mas0 fo)fn(*) dd* F(m:ro fl) de e qel Posts SrhearehN by FM.): n. m ). aFo-J "t dz a J[-F"(cc) de -a toF(o)1 ' Since E m) ec:sts by post ta) F(m-m T1-r t)Jdr -Fc) dr S-F ) Fl) d E () +jr^læ) f1- Flfde -®,Jrom f() =to r2 and Subst: tutnq in , e g Elm)-E(mn- ) J-) dz Chorgirqn v n-, n- n-2 , in e 9ekpet respecfively n Fn)lm) - Elm) Fm) =- Addirg anod Thea bove ar esuattos otirg That E[m) o e ge h Find tu po4 9 ' in randorm Sample Si 2e n fror di exponertial = str bution ft«) de o b) sho at X ard Wrs ¥ts)- Xtw) T4S n indepenesntty ohstbtel. Hence,f(c) = Plx<»¢)J«e du - e -O he ven by lr,nr) a)T-Fl) d) rc)= r) n-) e BYn-rH) os ln-r+ )),z>o e Tn-T+) * wrsXs) - Ky 9ien ou ardan 3 Xr joint pf bhe FSrSr- [Flr) ++s)- We)-Pn) Fc) flc) w) - Crs z, wre)]-r +Wpli-F(z+ f(c (r1)!Cn-r)! (s-T-!(n-S Ws) -*Wrs-TtI| e incleperoknt oist buteed a X ord wrs Xtir-) - Xr) Takirq, S T+Iir . p!f W -1) W t, n-) (n-v)en-T) .. 3W,20o ohstreu Which. shmos 1R w. has an eponentad powonebbt (n-r)" Joint Pdd o k-reles Stat:st: cs jointP 9 k-ovole statst:s Kn»a) ). ard X)hnl¢Ti 4. .. n 9iven b ITTa.. ,T,T2. ) n (rt!(-T-D!. (y-k-1-1) n fl,)x flc- ft]-* L- FlJ*J Joint f alln-orolet stot:stirs: Ln ta postcukos jolnnt pof a all tt n-ordes Storh st:cS s obtoin ontoking ken i Hence joint f n ovole Stot:st:es iven by .