MAT363 Statistical Inference [Pentaabiran Statistik]

Transcription

1 UNIVERSITI SAINS MALAYSIA Frst Semester Examato Academc Sesso 05/06 Jauary 06 MAT363 Statstcal Iferece [Petaabra Statst] Durato : 3 hours [Masa : 3 jam] Please chec that ths examato paper cossts of SI pages of prted materal before you beg the examato [Sla pasta bahawa ertas pepersaa megadug ENAM mua surat yag berceta sebelum ada memulaa pepersaa ] Istructos: Aswer FOUR (4) questos [Araha: Jawab EMPAT (4) 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 - - (a) Let set be {A, B, C, D, E} ad set be {F, G, H, I, J, K} A alphabet wll be tae from each set What s the probablty of tag a B or a I? [ 0 mars ] Proof that E E E Y y for the dscrete case [ 0 mars ] (c) Assume that ad are depedet radom varables wth the probablty mass fucto (pmf), f x P x p p 0,,,, ad zero elsewhere () Fd P x, for x = () Fd the codtoal pmf of gve [ 40 mars ] Let be a postve radom varable wth dstrbuto fucto F ad desty fucto f Fd the desty fucto g(y) of radom varable Y [ 0 mars ] (a) Bara set sebaga {A, B, C, D, E} da set sebaga {F, G, H, I, J, K} Satu abjad aa dambl darpada setap set Apaah ebaragala utu megambl satu B atau satu I? [ 0 marah ] Buta bahawa E E E Y y utu es dsret [ 0 marah ] (c) Adaa bahawa da adalah pembolehubah rawa ta bersadar dega fugs jsm ebaragala (fj), f x P x p p utu x = 0,,,, da sfar d tempat la () Car P, x () Car fj bersyarat utu dber [ 40 marah ] Bara sebaga pembolehubah rawa postf dega fugs tabura F da fugs etumpata f Car fugs etumpata g(y) utu pembolehubah rawa Y [ 0 marah ] 3/-

3 - 3 - (a) If ad Y are depedet ad each has a expoetal dstrbuto wth parameter, fd the jot desty fucto of S ad T Y Are Y S ad T depedet? [ 30 mars ] Cosder a radom sample of sze from a populato wth the stadard ormal, N(0,) dstrbuto For <, defe ad Fd the dstrbutos of the followg statstcs by usg some theorems: () () [ 40 mars ] (c) If, are depedet radom varables ad each has a stadard ormal dstrbuto, show that V P [ 30 mars ] (a) Ja da Y adalah ta bersadar da setap satu mempuya tabura espoe dega parameter, car fugs etumpata tercatum utu S da T Y Adaah S da T ta bersadar? Y [ 30 marah ] Pertmbaga suatu sampel rawa saz darpada populas dega tabura ormal pawa, N(0,) Utu, tarfa da Car tabura utu statst berut dega megguaa teorem tertetu () () [ 40 marah ] 4/-

4 - 4 - (c) Ja, adalah pembolehubah rawa ta bersadar da setap satu mempuya tabura ormal pawa, tujua bahawa V P [ 30 marah ] 3 (a) Let, be a radom sample from a dstrbuto wth desty fucto θ estmator of f x;θ θ x, x > 0 ; > Fd the method of momets [0 mars ] Let, be a radom sample from a dstrbuto havg the desty fucto ;α,β β xα f x βe I x α, suffcet statstcs for (α, ) usg the factorzato theorem, > 0 Fd the jot [ 40 mars ] (c) Let, be a radom sample wth desty fucto f x;θ θe x I x ; > 0 Fd the Cramer Rao s lower boud for the 0, varace of ubased estmators of [ 0 mars ] Defe the pvotal quatty [ 0 mars ] 3 (a) Bara, sebaga suatu sampel rawa darpada tabura dega fugs etumpata θ pegaggar aedah mome bag f x;θ θ x, x > 0 ; > Car [ 0 marah ] Bara, sebaga suatu sampel rawa darpada tabura yag mempuya fugs etumpata ;α,β β xα f x βe I x α,, > 0 Car statst cuup tercatum utu (α, ) dega megguaa teorem pemfatora [ 40 marah ] (c) Bara, sebaga suatu sampel rawa dega fugs f x;θ θe x I x ; > 0 Car batas bawah Cramer Rao etumpata 0, utu varas pegaggar-pegaggar sasama [ 0 marah ] Tarfa uatt pagsa [0 marah] 5/-

5 - 5-4 (a) Cosder the desty fucto ;θ xθ f x e I x θ, ; < < For estmatg, a radom sample, s tae from the dstrbuto wth the above desty fucto f(x; ) Let Y m,,, () Show that W Y θ has a gamma, G(,) dstrbuto () Fd a 00 percet cofdece terval for by usg W [ 30 mars ] Assume that s a radom varable from a dstrbuto wth desty fucto f Fd the most powerful test of sze-α to test H : G (,) 0 versus H : N (,) [ 30 mars ] (c) Assume that, s a radom sample from the ormal, N(, 9) dstrbuto Fd the lelhood rato test for testg H0: θ θ0 versus H: θ θ0 [ 40 mars ] 4 (a) Pertmbaga fugs etumpata ;θ xθ f x e I x θ, ; < < Utu megaggara, suatu sampel rawa, dambl darpada tabura dega fugs etumpata f(x; ) d atas Bara Y m,,, () Tujua bahawa W Y θ mempuya tabura gama, G(,) () Car selag eyaa 00 peratus utu dega megguaa W [ 30 marah ] Adaa bahawa alah suatu pembolehubah rawa darpada tabura dega fugs etumpata f Car uja palg beruasa saz-α utu meguj H : G (,) lawa H : N (,) 0 [ 30 marah ] (c) Adaa bahawa, alah suatu sampel rawa darpada tabura ormal, N(, 9) Car uja sbah ebolehjada utu meguj H0: θ θ0 lawa H: θ θ0 [ 40 marah ] 6/-

6 - 6 - APPENDI / LAMPIRAN - ooo 0 ooo -