MAT 363 Statistical Inference [Pentaabiran Statistik]

Transcription

1 UNIVERSITI SAINS MALAYSIA Frst Semester Eamato Academc Sesso / Jauary MAT 363 Statstcal Iferece [Petaabra Statstk] Durato 3 hours [Masa 3 jam] Please check that ths eamato paper cossts of EIGHT pages of prted materal before you beg the eamato [Sla pastka bahawa kertas peperksaa megadug LAPAN muka surat yag bercetak sebelum ada memulaka peperksaa ] Istructos Aswer all four [] questos [Araha Jawab semua empat [] 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) Fd the 5 th percetle of the dstrbuto havg probablty desty fucto (pdf) f, < < [ marks] (b) Let have the pdf f, < <, that s symmetrc at = Why s E() ot equal to zero? [ marks] ( y) (c) Let ad Y have the jot pdf f, y e, < < y < Fd () the codtoal pdf of Y gve =, e y fy the codtoal mea of Y gve =, e E(Y = ) (d) Let ad have the jot pdf h, e, jot pdf of Y ad Y [ marks] Fd the [ marks] (a) Car persetl ke-5 utuk tabura yag mempuya fugs ketumpata kebaragkala (fkk) f, < < [ markah] (b) Barka mempuya fkk f, < <, atu bersmetr pada = Keapa E() buka bersamaa dega sfar? [ markah] (c) Barka da Y mempuya fkk tercatum f, y () fkk bersyarat Y dber =, atu y fy m bersyarat Y dber =, atu E(Y = ) e ( y), < < y < Car [ markah] (d) Barka da mempuya fkk tercatum h e,, Car fkk tercatum utuk Y da Y [ markah] 3/-

3 3 [MAT 363] (a) If T has a t dstrbuto wth r degrees of freedom, fd the dstrbuto of T usg the relevat theorems [ marks] (b) If has a F dstrbuto wth v ad v degrees of freedom, fd the dstrbuto of usg the relevat theorems [ marks] (c) Let,,,, represet a sequece of radom varables such that has the dstrbuto fucto F ad desty fucto f Fd the lmtg dstrbuto of radom varable, where ~ N, [3 marks] (d) Let be a radom varable havg a uform dstrbuto wth pdf f() =, < < A radom sample of sze s take from a populato havg ths dstrbuto () Let Y Y Y3 Y represets the correspodg order statstcs of ths radom sample Fd the jot pdf of Y ad Y By usg the jot pdf of Y ad Y (), fd the probablty that the rage of ths radom sample s less tha 5, e P Y Y 5 [3 marks] (a) Jka T mempuya tabura t dega darjah kebebasa r, car tabura utuk T dega megguaka teorem-teorem berkata [ markah] (b) Jka mempuya tabura F dega darjah kebebasa v da v, car tabura utuk dega megguaka teorem-teorem berkata [ markah] (c) Barka,,,, mewakl suatu jujuka pembolehubah rawak sedemka sehgga mempuya fugs tabura F da fugs ketumpata f Car tabura peghad utuk pembolehubah rawak, yag maa ~ N, [3 markah] /-

4 [MAT 363] (d) Barka sebaga suatu pembolehubah rawak yag mempuya tabura seragam dega fkk f() =, < < Suatu sampel rawak saz dambl darpada populas yag mempuya tabura () Barka Y Y Y3 Y mewakl statstk tertb yag sepada utuk sampel rawak Car fkk tercatum utuk Y da Y Dega megguaka fkk tercatum utuk Y da Y dalam (), car kebaragkala bahawa julat utuk sampel rawak adalah kurag darpada 5, atu P Y Y 5 [3 markah] 3 (a) Cosder the followg pdf 3 f e, < <, < < 6 () Let,,, represet a radom sample take from a populato wth the above pdf Fd a complete ad suffcet statstc for Fd the uformly mmum varace of ubased estmator (UMVUE) for usg the Lehma Scheffe's theorem Ht e d [3 marks] (b) Assume that a radom sample of sze s draw from a populato havg probablty mass fucto (pmf) mamum lkelhood estmator of (c) Let,, f e!, =,,, Fd the [ marks], represet a radom sample of sze from the Posso e dstrbuto wth pmf f, =,,,! () Fd the Cramer-Rao lower boud for the varace of a ubased estmator of Show that ˆ s a effcet estmator for [3 marks] (d) Let,,, represets a radom sample of sze from a ormal dstrbuto () Show that s a pvotal quatty N, 5/-

5 5 [MAT 363] Costruct a % cofdece terval for () usg the pvotal quatty [ marks] 3 (a) Pertmbagka fkk berkut 3 f e, < <, < < 6 () Barka,,, mewakl suatu sampel rawak yag dambl darpada populas dega fkk d atas Car statstk cukup da legkap utuk Car pegaggar saksama bervaras mmum secara seragam (PSVMS) utuk dega megguaka teorem Lehma Scheffe Petua e d [3 markah] (b) Adaka bahawa suatu sampel rawak saz dambl darpada suatu populas e yag mempuya fugs jsm kebaragkala (fjk) f, =,,!, Car pegaggar kebolehjada maksmum utuk [ markah] (c) Barka,,, mewakl suatu sampel rawak saz darpada tabura e Posso dega fjk f, =,,,! () Car batas bawah Cramer-Rao utuk varas pegaggar saksama (d) Barka ormal Tujukka bahawa ˆ adalah pegaggar cekap utuk [3 markah],,, mewakl suatu sampel rawak saz darpada tabura N, () Tujukka bahawa adalah suatu kuatt pagsa Ba suatu selag keyaka % utuk kuatt pagsa dalam () dega megguaka [ markah] 6/-

6 6 [MAT 363] (a) If,,, s a radom sample from the gamma dstrbuto, e G(, ) ad the sample mea s for whe s large (b) Cosder a dstrbuto havg a pmf, fd a appromate % cofdece terval f, =, Let [3 marks] ad H () Show that the uformly most powerful (UMP) test of sze for testg (c) Let H versus H s gve by reject H f ad oly f Use the cetral lmt theorem to fd a relatoshp betwee the costat c () ad sample sze of a radom sample so that a UMP test of H versus H has a power fucto ( ), wth a appromate value of,, = 5 c H [ marks], be a radom sample of sze from the ormal dstrbuto, H versus H N(, 9) Fd the lkelhood rato test for testg [3 marks] (a) Jka,,, alah suatu sampel rawak darpada tabura gama, atu G(, ) da m sampel alah apabla adalah besar, car selag keyaka % hampra utuk [3 markah] (b) Pertmbagka suatu tabura yag mempuya fjk f, =, Barka H da H () Tujukka bahawa uja palg berkuasa secara seragam (PBS) saz utuk meguj H lawa H dber oleh tolak H jka da haya jka c 7/-

7 7 [MAT 363] Guaka teorem had memusat utuk mecar suatu hubuga atara pemalar c dalam () da saz sampel supaya uja PBS utuk H lawa H mempuya fugs kuasa ( ), dega la hampra = 5 [ markah] (c) Barka,,, sebaga suatu sampel rawak saz darpada tabura H mormal, N(, 9) Car uja sbah kebolehjada utuk meguj lawa H [3 markah] 8/-

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