MAT 363 Statistical Inference [Pentaabiran Statistik]

Transcription

1 UNIVERSITI SAINS MALAYSIA Frst Seester Exaato 009/00 Acadec Sesso Noveber 009 MAT 363 Statstcal Iferece [Petaabra Statstk] Durato : 3 hours [Masa : 3 ja] Please check that ths exaato paper cossts of EIGHT pages of prted aterals before you beg the exaato. [Sla pastka bahawa kertas peperksaa egadug LAPAN uka surat yag bercetak sebelu ada eulaka peperksaa.] Istructos: [Araha: Aswer all four [4] questos. Jawab seua epat [4] soala.] I the evet of ay dscrepaces, the Eglsh verso shall be used. [Sekraya terdapat sebarag percaggaha pada soala peperksaa, vers Bahasa Iggers hedaklah dgua paka]. /-

2 [MAT 363]. (a) Gve that the dstrbuto fucto 4 FY, ( x, y) k(4x y 5 xy ), 0 x,0 y, fd k ad the correspodg probablty desty fucto ad use t to calculate P(0, Y ). (b) Fd the varace of Y f the oet geeratg fucto t MY ( t) e /( t ). [0 arks] (c) If f, Y ( x, y), x 0, y 0, x y show that the codtoal probablty desty fucto of Y gve = x s fro a ufor dstrbuto. [0 arks] (d) Let f(x, y) = e -x-y, 0 < x <, 0 < y <, zero elsewhere, be the probablty desty fucto of ad Y. The f Z = + Y, fd P (Z z), for 0 < z <. What s the probablty desty fucto of Z?. (a) Let ad Y have jot probablty desty fucto f ( x, y ) Y, 0 xy, 0 x y 0, elsewhere Suppose A = /Y ad B = Y. Fd the jot probablty desty fucto (A, B) ad hece the argal probablty desty of A ad B. (b) Let Y, Y,..., Y be a rado saple fro the expoetal probablty y desty fucto fy ( y) e, y 0. What s the sallest for whch P ( Y 0.) 0.9? 3/-

3 3 [MAT 363]. (a) Dber bahawa fugs tabura 4 FY, ( x, y) k(4x y 5 xy ), 0 x,0 y, car k da fugs ketupata kebaragkala yag sepada da guakaya utuk egra P(0, Y ). [30 arkah] (b) Car varas bag Y jka fugs pejaa oe t MY ( t) e /( t ). [0 arkah] (c) Jka f, Y ( x, y), x 0, y 0, x y tujukka bahawa fugs ketupata kebaragkala bersyarat bag Y dber = x adalah dar tabura seraga. [0 arkah] (d) Barka f(x, y) = e -x-y, 0 < x <, 0 < y <, sfar selaya, adalah fugs ketupata kebaragkala bag da Y. Keuda jka Z = + Y, car P (Z z), utuk 0 < z <. Apakah fugs ketupata kebaragkala bag Z? [30 arkah]. (a) Barka da Y epuya fugs ketupata kebaragkala tercatu f Y, ( x, y) 0 xy, 0 x y 0, d tepat la Adaka A = /Y da B = Y. Car fugs ketupata kebaragkala tercatu (A, B) da seterusya fugs ketupata kebaragkala sut bag A da B. [30 arkah] (b) Barka Y, Y,..., Y sebaga sapel rawak dar fugs ketupata y kebaragkala ekspoe fy ( y) e, y 0. Apakah la yag terkecl bag P ( Y 0.) 0.9? [30 arkah] 4/-

4 4 [MAT 363] (c) If,,..., are rado saples fro the dstrbuto s defed as,, N,4, ad Fd the dstrbuto of the followg statstcs: () () ( ) ( ) ( ) () 4 (v) ( 4 ) [40 arks] 3. (a) Let,,, be a rado saple of sze fro the Posso k e k! dstrbuto p ( k; ), k 0,,,... Show that effcet estator for. ˆ s a (b) Show that ˆ Y s suffcet for saple fro a oral pdf wth μ = 0. f Y, Y,, Y s a rado [0 arks] (c) Let Y, Y,, Y be a rado saple of sze fro the pdf f Y ( y; ) y, 0 y. Show that W Y s a suffcet estator for θ. Is the axu lkelhood estator of θ a fucto of W? (d) If,,..., s a rado saple fro the Beroull dstrbuto (p), P 0 < p < ad s the saple ea, prove that p [0 arks] 5/-

5 5 [MAT 363] (c) Jka,,..., adalah sapel rawak darpada tabura da adalah dtakrfka sebaga,, N,4, Car tabura bag statstk berkut: () () ( ) ( ) ( ) () 4 (v) ( 4 ) [40 arkah] 3. (a) Barka,,, sebaga sapel rawak bersaz dar tabura k e k! Posso p ( k; ), k 0,,,.... Tujukka bahawa ˆ adalah pegaggar cekap bag. [30 arkah] (b) (c) Tujukka bahawa ˆ Y adalah cukup bag jka Y, Y,, Y adalah sapel rawak darpada tabura oral yag epuya fugs ketupata kebaragkala dega μ = 0. [0 arkah] Barka Y, Y,, Y sebaga sapel rawak bersaz darpada fugs ketupata kebaragkala ( y; ) y, 0 y. Tujukka bahawa W Y f Y adalah pegaggar cukup bag θ. Adakah pegaggar kebolehjada aksu bag θ suatu fugs W? [30 arkah] (d) Jka,,..., sapel rawak darpada tabura Beroull (p), P 0 < p < da alah sapel, buktka bahawa p [0 arkah] 6/-

6 6 [MAT 363] 4. (a) Assue that,,..., s a rado saple fro a oral dstrbuto, N(,4). Defe () Is a pvotal quatty? Expla. () Is. Aswer the followg questos below: a pvotal quatty? Expla. () Is a pvotal quatty? Expla. [0 arks] (b) Assue that,,..., s a rado saple fro the gaa dstrbuto G(, ). Based o the saple ea () approxate cofdece terval for whe s bg. () cofdece terval for whe s sall., costruct [40 arks] (c) Assue that s a sgle observato fro a dstrbuto wth probablty desty fucto f ( x; ) ( ) x I ( 0,) ( x), wth > -. () Fd the ost powerful test of sze α to test H 0 : 0 versus H :. () To test H 0 : 0 versus H : 0, the followg test s used: Reject H 0 f 3 4. Fd the power fucto ad sze for the test. [40 arks] 7/-

7 7 [MAT 363] 4. (a) Adaka,,..., sapel rawak darpada tabura oral, N(,4). Takrfka. Jawab setap soala d bawah: () Adakah suatu kuatt pagsa? Jelaska. () Adakah suatu kuatt pagsa? Jelaska. () Adakah suatu kuatt pagsa? Jelaska. [0 arkah] (b) Adaka,,..., suatu sapel rawak darpada tabura G(, ). Berdasarka sapel, terbtka () selag keyaka hapra bag apabla besar. () selag keyaka bag apabla kecl. [40 arkah] (c) Adaka suatu cerapa tuggal darpada tabura dega fugs ketupata f ( x; ) ( ) x I ( 0,) ( x), dega > -. () Car uja palg berkuasa bersaz utuk eguj H 0 : 0 lawa H :. () Utuk eguj H 0 : 0 lawa H : 0, uja berkut dguaka: Tolak H 0 jka 3 4. Car fugs kuasa da saz bag uja tersebut. [40 arkah] 8/-

8 8 [MAT 363] APPENDI / LAMPIRAN - ooo O ooo -