MST 561 Statistical Inference [Pentaabiran Statistik]

Transcription

1 UNIVERSITI SAINS MALAYSIA Secod Semester Eamato 03/04 Academc Sesso Jue 04 MST 56 Statstcal Iferece [Petaabra Statstk] Durato : 3 hours [Masa : 3 jam] Please check that ths eamato paper cossts of NINE pages of prted materals before you beg the eamato [Sla pastka bahawa kertas peperksaa megadug SEMBILAN muka surat yag bercetak sebelum ada memulaka peperksaa ] Istructos: [Araha: Aswer all fve [5] questos Jawab semua lma [5] 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 -- [MST56] (a) A co s tossed three tmes Let H ad T deote head ad tal, respectvely Evet B s defed as a evet where at least two heads are obtaed () Fd the set of all possble outcomes, Ω for evet B () From (), fd the σ-feld, S of ths epermet for evet B [40 marks] (b) (c) Assume that C ad D are two evets defed over the same sample space wth probabltes P( C ) = ad P( D ) = Show that P( C D) 4 [0 marks] The radom varables ad Y are uformly dstrbuted wth the followg jot probablty desty fucto (pdf):, f + y fy, ( y, ) = π 0, otherwse Fd the margal probablty desty fuctos of ad Y [0 marks] State at least oe property of the cumulatve dstrbuto fucto (cdf) as F below s ot a vald cdf to why the fucto 0, < F ( ) =,, > [0 marks] (a) Suatu dut sylg dlambugka tga kal Barka H da T mewakl kepala da buga, masg-masg Perstwa B dtakrfka sebaga perstwa sekurag-kuragya dua kepala dperoleh () Car set bag semua kesudaha yag mugk, Ω utuk perstwa B () Darpada (), car meda-σ, S utuk eksperme bag perstwa B [40 markah] (b) Adaka bahawa C da D adalah dua perstwa yag dtakrfka pada ruag sampel yag sama dega kebaragkala P( C ) = da P( D ) = Tujukka bahawa P( C D) 4 [0 markah] 3/-

3 -3- [MST56] (c) Pembolehubah rawak da Y adalah bertabura seragam dega fugs ketumpata kebaragkala (fkk) tercatum berkut:, jka + y fy, ( y, ) = π 0, sebalkya Car fugs ketumpata kebaragkala sut utuk da Y [0 markah] Nyataka sekurag-kuragya satu sfat fugs tabura loggoka (ftl) F d bawah buka ftl yag sah keapa fugs 0, < F ( ) =,, > [0 markah] (a) The momet geeratg fucto of the radom varable s M ( t) = t + p pe, t + p where e < Fd E() ad Var() p [40 marks] (b) Let ad be cotuous radom varables wth the jot pdf f,,, < <, =, Let Y = + ad Y = () Fd the jot pdf of Y ad Y, e fy, Y y, y () Fd the margal pdf of Y, e fy ( y ) (c) Let Y < Y < < Y represet the order statstcs of a radom sample of 3 f =, 0 < < θ, zero elsewhere θ sze from a dstrbuto wth pdf Y Fd P c< < θ, where 0 < c < 3 4/-

4 -4- [MST56] (a) Fugs pejaa mome utuk pembolehubah rawak alah M ( t) = t + p pe, t + p yag maa e < Car E() da Var() p [40 markah] (b) Barka da sebaga pembolehubah rawak selajar dega fkk tercatum f,,, < <, =, Barka Y = + da Y = () Car fkk tercatum bag Y da Y, atu fy, Y y, y () Car fkk sut bag Y, atu fy ( y ) (c) Barka Y < Y < < Y mewakl statstk tertb utuk sampel rawak saz 3 f =, 0 < < θ, sfar d tempat la θ darpada tabura dega fkk Y Car P c< < θ, yag maa 0 < c < 3 3 (a) Let,,, be depedet ad detcally dstrbuted radom varables from a (, ) = () () N µσ dstrbuto The sample mea s defed as Fd the dstrbuto of the followg statstcs: µ σ µ ( µ ) (b) Let the probablty mass fucto of Y be f ( y ) =, for y =, ad zero elsewhere Show that Y does ot have a lmtg dstrbuto [0 marks] 5/-

5 -5- [MST56] (c) Show that the product of the sample observatos s a suffcet statstc for θ (θ > 0) f the radom sample s take from a gamma dstrbuto wth G αλ,, the parameters α = θ ad λ = 6 Note that f λ f e α λ α =, for > 0 Γ α Let be a radom varable from a bomal, b(, θ) dstrbuto Show that s ot a ubased estmator of θ [0 marks] 3 (a) Barka,,, sebaga pembolehubah rawak tak bersadar da bertabura secama darpada tabura N ( µσ, ) sebaga = () () µ σ µ ( µ ) M sampel dtakrfka Car tabura utuk statstk-statstk berkut: (b) (c) Barka fugs jsm kebaragkala utuk Y sebaga f ( y ) =, utuk y =, da sfar d tempat la Tujukka bahawa Y tdak mempuya tabura peghad [0 markah] Tujukka bahawa hasl darab cerapa-cerapa sampel adalah statstk cukup utuk θ (θ > 0) jka sampel rawak dambl darpada tabura gama G αλ,, dega parameter α = θ da λ = 6 Perhatka bahawa jka maka λ f e α λ α =, utuk > 0 Γ α Barka sebaga pembolehubah rawak darpada tabura bomal, b(, θ) Tujukka bahawa buka pegaggar saksama utuk θ [0 markah] 6/-

6 -6- [MST56] 4 (a) Twety motors were put o test uder a hgh temperature settg The lfetmes hours of these motors uder ths temperature settg are as follows: Suppose that the lfetme of a motor uder ths temperature settg has a gamma, G, θ dstrbuto () Fd the mamum lkelhood estmator of θ () From (), fd the mamum lkelhood estmate of θ (b) Assume that,,, s a radom sample from a ormal, N(α, ) dstrbuto () Fd the Cramer Rao s lower boud for the varace of ubased estmators of α () () Fd the effcecy of the estmator T = Is T a asymptotcally effcet estmator of α? Epla [50 marks] (c) If has a Posso dstrbuto wth parameter θ, e P θ ad the pror dstrbuto of Θ s gamma, G(α, λ), α ad λ are kow, fd the posteror dstrbuto of Θ gve = [0 marks] 4 (a) Dua puluh buah motor duj d bawah keadaa suhu yag tgg Masa hayat dalam jam utuk motor-motor d bawah keadaa suhu adalah sepert berkut: Adaka bahawa masa hayat suatu motor d bawah keadaa suhu mempuya tabura gama, G, θ () Car pegaggar kebolehjada maksmum utuk θ () Darpada (), car aggara kebolehjada maksmum utuk θ (b) Adaka bahawa,,, alah suatu sampel rawak darpada tabura ormal, N(α, ) () Car batas bawah Rao Cramer utuk varas pegaggarpegaggar saksama α () Car kecekapa pegaggar T = o 7/-

7 -7- [MST56] () Adakah T suatu pegaggar cekap berasmptot utuk α? Jelaska [50 markah] (c) Jka mempuya tabura Posso dega parameter θ, atu P ( θ ) da tabura pror utuk Θ alah gama, G(α, λ), α da λ adalah dketahu, car tabura posteror utuk Θ dber = [0 markah] o 5 (a) Assume that a certa radom varable has a bomal dstrbuto If we desre a 90% cofdece terval for p that s at most 00 legth, fd the sample sze, (Ht: Note that y y ) [0 marks] (b) Let,,, be a radom sample from a dstrbuto wth pdf ( θ) f ( ; θ ) = e, >θ, <θ< The followg test s used to test H0: θ=θ 0 versus H: θ=θ : Reject H 0 f ad oly f Y = m (,,, ) > c Fd the power fucto of ths test [0 marks] (c) Assume that,,, Po θ dstrbuto havg probablty mass fucto (pmf) e ( ; ) θ θ f θ =, = 0,,, ; θ > 0! Fd the uformly most powerful test of sze α = 0 for testg H : θ= 0 versus H : θ> Let = 30 ad use the cetral lmt theorem s a radom sample from a Posso, Let be a sgle observato from a dstrbuto wth pdf f ( ; θ ) =θ θ, 0 < <, θ > 0 Fd the lkelhood rato test of sze-α for testg H 0 : θ= versus H : θ

8 -8- [MST56] 8/- 5 (a) Adaka bahawa suatu pembolehubah rawak tertetu mempuya tabura bomal Jka kta g suatu selag keyaka 90% utuk p dega pajag sebayak-bayakya 00, car saz sampel, (Petua: Perhatka bahawa y y ) [0 markah] (b) Barka,,, sebaga suatu sampel rawak darpada tabura dega fkk ( θ) f ( ; θ ) = e, >θ, <θ< Uja berkut dguaka utuk meguj H0: θ=θ 0 lawa H: θ=θ : Tolak H 0 jka da haya jka Y = m (,,, ) > c Car fugs kuasa uja [0 markah] (c) Adaka bahawa,,, alah suatu sampel rawak darpada tabura Posso, Po ( θ ) yag mempuya fugs jsm kebaragkala (fjk) e ( ; ) θ θ f θ =, = 0,,, ; θ > 0! Car uja palg berkuasa secara seragam saz α = 0 bag meguj H : θ= lawa 0 H : θ> Barka = 30 da guaka teorem had memusat Barka sebaga cerapa tuggal darpada tabura dega fkk f ( ; θ ) =θ θ, 0 < <, θ > 0 Car uja sbah kebolehjada saz-α utuk meguj H 0 : θ= lawa H : θ

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

10 -0- [MST56] - ooo O ooo -