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