MST561 Statistical Inference [Pentaabiran Statistik]

Transcription

1 UNIVERSITI SAINS MALAYSIA Frst Semester Examato Academc Sesso 5/6 Jauary 6 MST56 Statstcal Iferece [Petaabra Statst] Durato : 3 hours [Masa : 3 jam] Please chec that ths examato paper cossts of NINE pages of prted materals before you beg the examato [Sla pasta bahawa ertas pepersaa megadug SEMBILAN mua surat yag berceta sebelum ada memulaa pepersaa ] Istructos: Aswer FIVE 5) questos [Araha: Jawab LIMA 5) soala] I the evet of ay dscrepaces, the Eglsh verso shall be used [Seraya terdapat sebarag percaggaha pada soala pepersaa, vers Bahasa Iggers hedalah dgua paa] /-

2 - - [MST56] a) If the radom varable has probablty fucto x) x, f x =,, 3,, fd the momet geeratg fucto, mea ad varace for [ 3 mars ] b) Let the jot desty fucto for ad be f x, x ) = x 3, x x ad zero elsewhere x ) Fd the codtoal desty fucto of, gve x ) Fd the codtoal mea of, gve x ) Fd the codtoal varace of, gve x [ 4 mars] x c) Gve the desty fucto for a Gamma dstrbuto as f x) xe, > ; < x < ad zero elsewhere ) If = s the mode for the dstrbuto, what s the value of the parameter? ) Hece, fd the value of P < 949) [ 3 mars ] a) Ja pembolehubah rawa mempuya fugs ebaragala x, f x) x =,, 3,, car fugs pejaa mome, m da varas bag [3 marah] b) Bara fugs etumpata tercatum bag ad sebaga f x, x ) = x 3, x x da sfar d tempat la x ) Car fugs etumpata bersyarat bag, dber x ) Car m bersyarat bag, dber x ) Car varas bersyarat bag, dber x [4 marah] 3/-

3 - 3 - [MST56] x c) Dber fugs etumpata bag tabura Gamma sebaga f x) xe, > ; < x < da sfar d tempat la ) Ja = adalah mod bag tabura tersebut, apaah la bag parameter? ) Seterusya, car la P < 949) [ 3 marah ] a) A radom sample of sze = 6 s tae from the probablty desty fucto f y) 3 y, y Fd the probablty of the order statstcs Y PY 5 75) [ 3 mars ] b) Use the fact that ) S s a ch square radom varable wth degree of freedom to prove that Var S ) [ mars ] c) Let Y, Y,, Y be a radom sample from a ormal dstrbuto Use the statemet b) to prove that S s cosstet for σ [ mars ] d) Let,,, represet a radom sample from the Be) dstrbuto If the pror dstrbuto of s gve by g θ I θ, fd the posteror Bayes estmator of ) wth respect to the pror pdf, g θ [ 3 mars ] a) Suatu sampel rawa bersaz = 6 dambl darpada fugs etumpata ebaragala f y) 3 y, y Car ebaragala statst tertb PY 5 75) Y [ 3 marah ] b) Guaa haat bahawa ) S adalah pembolehubah rawa h uasa dua dega darjah ebebasa utu membuta bahawa Var S ) [ marah ] 4/-

4 - 4 - [MST56] c) Bara Y, Y,, Y sebaga suatu sampel rawa darpada tabura ormal Guaa eyataa d b) utu membuta bahawa S adalah osste bag σ [ marah ] d) Bara,,, mewal suatu sampel rawa darpada tabura Be) Ja tabura pror bag dber oleh g θ I θ, car pegaggar Bayes posteror bag ) terhadap fugs etumpata ebaragala pror g θ, [ 3 marah ] 3 a) If the jot desty fucto for da s xx) e, x, x f x, x),, elsewhere fd the desty fucto for S Ht: Let T = ) [ 3 mars ] b) If,,, represet a radom sample from the dstrbuto N,) ad ad, fd the dstrbuto for the followg statstcs: ) ) ) [ 3 mars ] c) Assume,,, to be a radom sample from a dstrbuto wth desty fucto 3 x f x; ) x e, < x <, > 4 6 ) Fd a statstc whch s complete ad suffcet for ) Fd a maxmum lelhood estmate for [ 4 mars ] 5/-

5 - 5 - [MST56] 3 a) Ja fugs etumpata tercatum bag da x x ) alah e, x, x f x, x ), car fugs etumpata utu S, selaya Petuju: Bara T = ) [ 3 marah ] b) Ja,,, mewal sampel rawa darpada tabura N,) da da, car tabura utu statststatst berut: ) ) ) [ 3 marah ] c) Adaa,,, sebaga suatu sampel rawa darpada tabura dega fugs etumpata 3 x f x; ) x e, < x <, > 4 6 ) Car suatu statst yag cuup da legap bag ) Car suatu aggara ebolehjada masmum bag [ 4 marah ] 4 a) Gve the followg probablty desty fucto: x 3 f x; ) x e, < x <, < <, ad zero elsewhere 4 6 ) If,,, s a radom sample from ths dstrbuto, fd a complete suffcet statstc for ) Fd the uformly mmum varace ubased estmator UMVUE) for [ 3 mars ] 6/-

6 - 6 - [MST56] b) Let,,, be a radom sample wth desty fucto f x; ) x I x),) Fd a percetage cofdece terval for [ 3 mars ] c) Assume that,, s a radom sample from a dstrbuto whch has the desty fucto f x; ) x I x); >,) ) Fd the dstrbuto for Q log ) ad show that Q s a pvotal quatty gve that log has a expoetal dstrbuto wth parameter α ) Fd the 95 percet cofdece terval for whe = 3 ) Fd a approxmate 95 percet cofdece terval for whe = 3 v) Compare ad commet o the legth of both cofdece tervals ) ad ) above [ 4 mars ] 4 a) Dber fugs etumpata ebaragala berut: x 3 f x; ) x e, < x <, < <, da sfar d tempat la 4 6 ) Ja,,, alah suatu sampel rawa darpada tabura, car statst cuup legap bag ) Car pegaggar sasama bervaras mmum secara seragam PSVMS) bag [ 3 marah ] b) Bara,,, meada suatu sampel rawa dega fugs etumpata f x; ) x I x) Car suatu selag eyaa,) peratus bag [ 3 marah ] 7/-

7 - 7 - [MST56] c) Adaa,, suatu sampel rawa darpada tabura yag mempuya fugs etumpata f x; ) x I x); >,) ) Car tabura Q log ) da tujua bahawa Q alah uatt pagsa dber log mempuya tabura espoe dega parameter α ) Car suatu selag eyaa 95 peratus bag apabla = 3 ) v) Car suatu selag eyaa hampra 95 peratus bag apabla = 3 Badga da ome tetag pajag edua-dua selag eyaa dalam ) da ) d atas [ 4 marah ] 5 a) State the defto of a uformly most powerful UMP) test [ mars ] b) Let,,, represet a radom sample of sze from a G,) dstrbuto ) Fd the uformly most powerful test for testg H : vs H : ) The followg test s used for testg H : vs H : : Reject H f c Fd c so that the sze of the test s [Assume that s suffcetly large so that the cetral lmt theorem ca be used to fd a approxmate value of c] [ 5 mars ] c) Let,,, represet a radom sample from a N,) dstrbuto Fd the geeralzed lelhood-rato test for testg H : vs H : [ 4 mars ] 8/-

8 - 8 - [MST56] 5 a) Nyataa defs uja palg beruasa secara seragam UPBS) [ marah ] b) Bara,,, mewal suatu sampel rawa saz darpada tabura G,) ) Car uja palg beruasa secara seragam utu meguj H : lawa H : ) Uja berut dguaa utu meguj H : lawa H : : Tola H ja alah c Car c supaya saz uja [Adaa bahawa adalah cuup besar supaya teorem had memusat dapat dguaa utu mecar suatu la hampra bag c] [ 5 marah ] c) Bara,,, mewal suatu sampel rawa darpada tabura N,) Car uja sbah ebolehjada tertla bag meguj H : lawa H : [ 4 marah ] 9/-

9 - 9 - [MST56] APPENDI/LAMPIRAN - ooo O ooo -