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Order sttistics:

et Y F2 - X n independent lertcal dstbuted voaiablog ench w: Cumulo ttve dstr:utTib funetton F(x) ond pfflx) f se ,ables ann oMoned inolendi rq oYesHes Maqmtuole and wt:Ter Xt 4ta) - -4Yn le rles Stot:st:CS (:1,2,., X) s a indeperodent ecaue neual ty ootion tm. Hee 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 strbulion urction Smallest olot Shtste q.von by F,) P * -Pxx; :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 . qet

nJ7

-fr-Fl] 1fTn, Ft)()F -F F) 3fit (ontinupus 79P 'hon an e ortten as, ) I F n)

wou, b Epfo, ) t-t) olt n Lompube th furcton A-t

his shouos 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 ) 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 + 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 ) 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: 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)

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

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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)-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: iven by