STAT 514 - HW 4 Solution Key - 1 23’ 341 a{X(i)lx(j(u!V =...

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Unformatted text preview: 1;. 23’ . 341 a) {X (i)lx(j)(u!V) = fXfi)’ thz, ,6v)/fx (v). Consider two cases, depending on which of 1 orj G.) is greater. Using the formulas from Theorems 5.5.2 and 5.5.3, and after cancellation, we obtain the following. i)ifi<j, f (”IV)— wr—Q—Dwa (u)FE1(u){ F (v)w F (u)]j'i*1F1X-§(v) X(i)JXO) (1*E)EU"H1*1)! X {X X ifoM Mn) 1* we) 5+1 ”(1 })EU— 1 i)! FX(V) FXCV) Fxbr) a “<V. Note this interpretation. This 15 the pdzf of the ith order statistic from a. sample of size j-l, from a population with pdf given by the truncated distribution, f(u) r: fX(u)/Fx(v), u < v ,1 t '_ , , 3. w. {finixmw '5 (n 1)!}______:)1 1‘): ‘XM [1* PM] n” 1%)? Fxm} “'3 [1 ~ Fx(v)}’"“ _ (eel-)2 {X00 ' FXmH‘Xm H“ affiumxw) ”“ "(a~j—13.!(n—i)11—Fx(v) l—Fx(v) H‘xm ’ u>v. This is the pdf of the (i—j)th order statistic from a eempie of size n-j, see a population with pa; given by the truncated distribution,- r(u) ee' fx(u)/(I~Fx{v)}, u > v. 5M L(A[x1,...,xn)= {fig—2:513...“ ”5“ ' . .l | : 1:1 X]. [hi , 5 (”1(3ng ")1 nA+mlegA KW:— +P§=n(2-A)/A 8’ a WIDE-L“ -* "3'2"” => ”ES-FIDEL 2 §, - Thus 8 a 1%“ _n(i~—A)/A __ i __ 5‘ E62 “flogL :in/A A/n , E69 - Problem From Notes X1...X,l are it‘d Derivethe joint distribution of Itwo order statistics Y; and Yj. 52 _ F(y+Ay,z+Az)—F(y,z) fY-Y. (ye?) - {3sz (y 2) §§8—_MmW—n—WA})AZ _ PmmUXi’s S y,0neXt- E (y,y+Ay],(j —i-1)Xt-'s e (y+ Ay,z],oneXi 6 (2,2 +dz],(n —j)X]-'s E (z + Az,oo]) m Ayilo AyAz fizz—+0 _ [I'm ”1 XFJ} 1(y)><{Fx 0’ +40?) FX(y)]X[FX(Z) Fx(y+fi})}jfl 1><[Jf"x(2’+é?z)- Fx(z)]>< {1- Ire/(2)1“ Wdue—936wl)!1!0'—1'-1)!H(n-j)! Ayedz 432—) '—i-—1 : n! XF;X1(_}7)XW[1 FAQ} "a; X hm {Fxflv+Ay)“FX(Y)]X[Fx(Z)“Fx(y+fiy)}“’ XEFx(2+Az)—Fx(2)] (i»—1)!(j—1'-1)!(n-j)! fiifié’ AM: I . . ._:._ ”‘ x Fe“ (y)><[1 — FX(Z}]"_J m (y>><[FX(z> —Fx<y)]J ‘ x Lee) ...
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