MAT 363 Statistical Inference [Pentaabiran Statistik]

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1 UNIVERSITI SAINS MALAYSIA Peperiksaa Kursus Seasa Cuti Pajag Sidag Akadeik / Ogos MAT 363 Statistical Iferece [Petaabira Statistik] Duratio : 3 hours [Masa : 3 ja] Please check that this exaiatio paper cosists of SEVEN pages of prited aterial before you begi the exaiatio [Sila pastika bahawa kertas peperiksaa ii egadugi TUJUH uka surat yag bercetak sebelu ada eulaka peperiksaa ii] Istructios: Aswer all four [4] questios [Araha: Jawab seua epat [4] soala] I the evet of ay discrepacies, the Eglish versio shall be used [Sekiraya terdapat sebarag percaggaha pada soala peperiksaa, versi Bahasa Iggeris hedaklah digua pakai] /-

2 - - [MAT 363] (a) If,,, is a rado saple fro the oral distributio, ad is defied as i,, i fid the distributios of the followig statistics: (i) μ (ii) 3σ N μ, 3σ [4 arks] (b) The rado variables U ad V have a joit probability desity fuctio (pdf) f u, v uv, 4 u v Fid (i) the coditioal desity fuctio of U give V v (ii) the coditioal ea of U give V [4 arks] (c) Assue that,,, is a rado saple fro the Be θ distributio, where θ If deotes the saple ea, show that probability to θ coverges i [ arks] (a) Jika,,, adalah suatu sapel rawak daripada tabura oral, N μ, 3σ da ditakrifka sebagai i i,, cari tabura utuk statistik berikut: (i) μ (ii) 3σ [4 arkah] (b) Pebolehubah rawak U da V epuyai fugsi ketupata kebaragkalia (fkk) tercatu f u, v uv, 4 u v Cari (i) fugsi ketupata bersyarat U diberi V v (ii) i bersyarat U diberi V [4 arkah] 3/-

3 - 3 - [MAT 363] (c) Adaika bahawa,,, adalah suatu sapel rawak daripada tabura Be θ, yag aa θ Jika ewakili i sapel, tujukka bahawa eupu secara kebaragkalia kepada [ arkah] (a) If Y deotes the th ordered statistic of a rado saple fro the uifor distributio, U, λ, fid the liitig distributio of Z λ Y [3 arks] (b) Let,,, be a rado saple of size with probability ass x λ λ fuctio (pf), f x; λ e x! likelihood estiator (le) of λ, for x,,,; λ Fid the axiu [ arks] (c) Assue that,,, is a rado saple fro the oral distributio, N θ, θ, θ (i) Show that f x; θ is a expoetial faily (ii) Fro (i), fid a coplete ad sufficiet statistic (iii) Is a uiforly iiu variace of ubiased estiator (UMVUE) of θ? Explai [5 arks] (a) Jika Y ewakili statistik tertib ke- bagi suatu sapel rawak daripada tabura seraga, U, λ, cari tabura peghad utuk Z λ Y [3 arkah] (b) Biarka,,, sebagai suatu sapel rawak saiz dega fugsi jisi x λ λ kebaragkalia (fjk), f x; λ e, x! pegaggar kebolehjadia aksiu (pk) utuk λ utuk x,,,; λ Cari [ arkah] (c) Adaika bahawa,,, adalah suatu sapel rawak daripada tabura oral, N θ, θ, θ (i) Tujukka bahawa f x; θ adalah suatu faili ekspoe (ii) Daripada (i), cari suatu statistik cukup da legkap (iii) Adakah suatu pegaggar saksaa bervarias iiu secara seraga (PSVMS) utuk θ? Jelaska [5 arkah] 4/-

4 - 4 - [MAT 363] 3 (a) Assue that ad are rado variables fro the Poisso distributio, P θ Is a sufficiet statistic for θ? [3 arks] (b) Let be a sigle observatio fro a distributio with pdf f x; λ λ x I, λ x, λ λ If, is a cofidece iterval for, fid the cofidece coefficiet, d c i ters of c ad d [3 arks] (c) Let,,, be a rado saple fro the expoetial distributio with paraeter θ Based o the saple ea i i, derive (i) the % approxiate cofidece iterval for θ whe is large (ii) the % exact cofidece iterval for θ whe is sall [4 arks] 3 (a) Adaika bahawa da adalah pebolehubah rawak daripada tabura Poisso, P θ Adakah suatu statistik cukup bagi θ? [3 arkah] (b) Biarka sebagai suatu cerapa tuggal daripada tabura dega fkk f x; λ λ x I, λ x, λ λ Jika, adalah suatu selag keyakia bagi, cari pekali keyakia, d c dala sebuta c da d [3 arkah] (c) Biarka,,, sebagai suatu sapel rawak daripada tabura ekspoe dega paraeter θ Berdasarka i sapel i i, terbitka (i) selag keyakia hapira % bagi θ apabila adalah besar (ii) selag keyakia tepat % bagi θ apabila adalah kecil [4 arkah] 5/-

5 - 5 - [MAT 363] 4 (a) Let,,, deote a rado saple of size havig pdf θx f x; θ θ xe I x, (i) Fid the uiforly ost powerful critical regio to test H : θ versus H : θ (ii) For testig H : θ versus H : θ, the followig test is used: Reject H if ad oly if k By assuig that is sufficietly large, fid the approxiate value of k usig the Cetral Liit Theore so that α 5 [5 arks] (b) Assue that,,, is a rado saple of size fro a oral distributio, N, θ, where θ For testig H : θ versus H : θ, the followig critical regio is used: C x, x,, x : x i c i Fid the value of c if the size of the critical regio C is [3 arks] θ (c) Assue that is a sigle observatio havig pdf x; θ θ x I x, f, where θ For testig H : θ versus H : θ, the followig test is used: Reject H if ad oly if Fid the power fuctio of this test [ arks] 4 (a) Biarka,,, ewakili suatu sapel rawak saiz yag epuyai θx fkk f x; θ θ xe I x (i), Cari ratau getig palig berkuasa secara seraga utuk eguji H : θ lawa H : θ (ii) Utuk eguji H : θ lawa H : θ, ujia berikut diguaka: Tolak H jika da haya jika k Dega egadaika bahawa adalah besar secara cukup, cari ilai hapira k dega egguaka Teore Had Meusat supaya = 5 [5 arkah] (b) Adaika bahawa,,, adalah suatu sapel rawak saiz daripada tabura oral, N, θ, yag aa θ Utuk eguji H : θ lawa H : θ, ratau getig berikut diguaka: 6/-

6 - 6 - [MAT 363] C x, x,, x : x i c i Cari ilai c jika saiz ratau getig C adalah [3 arkah] (c) Adaika bahawa adalah cerapa tuggal yag epuyai fkk θ x; θ θ x I x, yag aa θ Utuk eguji H : θ f, lawa H : θ, ujia berikut diguaka: Tolak H jika da haya jika Cari fugsi kuasa ujia ii [ arkah] 7/-

7 - 7 - [MAT 363] APPENDI / LAMPIRAN - ooo O ooo -

MST 561 Statistical Inference [Pentaabiran Statistik]

MST 561 Statistical Inference [Pentaabiran Statistik] UNIVERSITI SAINS MALAYSIA Secod Semester Examiatio 05/06 Academic Sessio Jue 06 MST 56 Statistical Ierece [Petaabira Statistik] Duratio : 3 hours [Masa : 3 jam] Please check that this examiatio paper cosists