MST 561 Statistical Inference [Pentaabiran Statistik]
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1 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 o NINE pages o prited material beore you begi the examiatio. [Sila pastika bahawa kertas peperiksaa ii megadugi SEMBILAN muka surat yag bercetak sebelum ada memulaka peperiksaa ii.] Istructios: Aswer FIVE (5) questios. [Araha: Jawab LIMA (5) soala.] I the evet o ay discrepacies, the Eglish versio shall be used. [Sekiraya terdapat sebarag percaggaha pada soala peperiksaa, versi Bahasa Iggeris hedaklah digua pakai.] /-
2 - - [MST56]. (a) A die desiged i such a way that each odd umber is two times more likely to occur tha each eve umber. (i) What is the probability that the umber obtaied rom a roll is a perect square? What is the probability that the umber obtaied rom a roll is a perect square give that it is greater tha 3? Let the radom variable have the probability desity uctio cx, 0 x 3 x c6 x, 3 x<6. 0, elsewhere (i) Fid the value o the costat c. Fid the cumulative distributio uctio o. [ 40 marks ] (c) The probability desity uctio o the radom variable is < x <. Fid the momet geeratig uctio o. x x e,. (a) Suatu dadu direkabetuk sedemikia sehigga setiap ombor gajil adalah dua kali lebih cederug utuk berlaku daripada setiap ombor geap. (i) Apakah kebaragkalia bahawa ombor yag diperoleh daripada suatu lambuga adalah suatu kuasa dua sempura? Apakah kebaragkalia bahawa ombor yag diperoleh daripada suatu lambuga adalah suatu kuasa dua sempura jika diberi ia adalah lebih daripada 3? 3/-
3 - 3 - [MST56] Biarka pembolehubah rawak mempuyai ugsi ketumpata kebaragkalia cx, 0 x 3 x c6 x, 3 x<6. 0, di tempat lai (i) Cari ilai pemalar c. Cari ugsi tabura loggoka utuk. [40 markah] (c) Fugsi ketumpata kebaragkalia bagi pembolehubah rawak ialah x x e, < x <. Cari ugsi pejaa mome bagi.. (a) Assume that the joit radom variables (, Y) have the joit desity uctio give below. For each case, determie whether ad Y are idepedet or otherwise. Explai your aswer. (i) y x y e, 0 < x < y, zero otherwise. Y,, x, y Y, xy, < x < ad < y <, zero otherwise. 4 [ 0 marks ] Let ad Y deote cotiuous radom variables. I ad Y are idepedet ad U = + Y, show that FU ( b) F b y Y y dy, where F U, F ad Y are the distributio uctio o U, margial distributio uctio o ad margial desity uctio o Y. 4/-
4 - 4 - [MST56] (c) I the joit probability mass uctio o ad Y is x y k xy x = 0, ad y =,, 3, ad zero otherwise, id (i) the value o k. the margial probability mass uctio o. (iii) the margial probability mass uctio o Y.,,, Y (iv) the coditioal expectatio o Y give =. [ 50 marks ]. (a) Adaika bahawa pembolehubah rawak tercatum (, Y) mempuyai ugsi ketumpata tercatum yag diberika di bawah. Utuk setiap kes, tetuka sama ada da Y adalah tak bersadar atau sebalikya. Jelaska jawapa ada. (i) y x y e, 0 < x < y, siar sebalikya. Y,, x, y Y, xy, < x < da < y <, siar sebalikya. 4 [ 0 markah ] Biarka da Y mewakili pembolehubah rawak selajar. Jika da Y adalah tak bersadar da U = + Y, tujukka bahawa FU ( b) F b y Y y dy, yag maa F U, F da Y adalah ugsi tabura utuk U, ugsi tabura sut utuk da ugsi ketumpata sut utuk Y. (c) Jika ugsi jisim kebaragkalia tercatum utuk da Y ialah Y, x, y k xy, x = 0, da y =,, 3, da siar sebalikya, cari (i) ilai k. ugsi jisim kebaragkalia sut utuk. (iii) ugsi jisim kebaragkalia sut utuk Y. (iv) jagkaa bersyarat Y diberi =. [50 markah] 5/-
5 - 5 - [MST56] 3. (a) Assume that is a positive radom variable that has the desity uctio x. Fid the desity uctio o the radom variable Y [ 0 marks ] Assume that, is a radom sample rom the stadard ormal, N(0, ) distributio. Fid the limitig distributio, i it exists, or the statistic Z. i i (c) Assume that, is a radom sample rom the uiorm U, distributio, where > 0. Fid (i) the method o momets estimator o. the maximum likelihood estimator o. (iii) the joit suiciet statistic or this distributio. [ 50 marks ] 3. (a) Adaika bahawa ialah suatu pembolehubah rawak positi yag mempuyai ugsi ketumpata x. Cari ugsi ketumpata utuk pembolehubah rawak Y. [ 0 markah ] Adaika bahawa, ialah suatu sampel rawak daripada tabura ormal piawai, N(0, ). Cari tabura peghad, jika ia wujud, utuk statistik Z. i i 6/-
6 - 6 - [MST56] (c) Adaika bahawa, ialah suatu sampel rawak daripada tabura seragam U,, yag maa > 0. Cari (i) pegaggar kaedah mome utuk. pegaggar kebolehjadia maksimum utuk. (iii) statistik cukup tercatum utuk tabura ii. [ 50 markah ] 4. (a) Let the radom variable have the Poisso, P0 distributio ad the prior distributio o is G(, ), where ad are kow. Fid the posterior distributio o give = x. Assume that is a sigle observatio rom the distributio with the desity uctio x; x, 0 < x < ; > 0. Deie Y log. Y (i) Fid the coidece coeiciet o the radom iterval, Y o. Fid a coidece iterval o which is better tha that i (i), i the orm (ay, by), where a. [ 50 marks ] (c) Let, be a radom sample rom the N(, ) distributio, where is ukow. Are the ollowig variables a pivotal quatity? Explai your aswer. (i).. [ 0 marks ] 7/-
7 - 7 - [MST56] 4. (a) Biarka pembolehubah rawak mempuyai tabura Poisso, P0 da tabura prior utuk ialah G(, ), yag maa da adalah diketahui. Cari tabura posterior utuk diberi = x. Adaika ialah suatu cerapa tuggal daripada tabura dega ugsi Y log. ketumpata x; x, 0 < x < ; > 0. Takrika Y (i) Cari pekali keyakia utuk selag rawak, Y bagi. Cari suatu selag keyakia utuk yag lebih baik daripada yag terdapat dalam (i), dalam betuk (ay, by), yag maa a. [ 50 markah ] (c) Biarka, sebagai suatu sampel rawak daripada tabura N(, ), yag maa tidak diketahui. Adakah pembolehubah-pembolehubah rawak berikut suatu kuatiti pagsia? Jelaska jawapa ada. (i).. [ 0 markah ] 5. (a) Assume that, 5 is a radom sample o size 5 rom the N(0, ) distributio, where > 0. For testig : 0 ollowig critical regio is used:. 5 C x, x x5 : xi c i Fid the value o c i the size o the critical regio C is /-
8 - 8 - [MST56] Based o a radom sample o size, id the most powerul test o size- or testig the hypothesis H : : N 0 0, versus H : : N,. (c) Let, be a radom sample rom a N(, ) distributio. Fid the likelihood ratio test or testig H0: 0 versus H: 0. [ 40 marks ] 5. (a) Adaika bahawa, ialah suatu sampel rawak saiz 5 daripada tabura N(0, ), yag maa > 0. Utuk meguji H : 0 lawa H :, ratau getig berikut diguaka:. 5 C x, x x5 : xi c i Cari ilai c jika saiz ratau getig C ialah Berdasarka suatu sampel rawak saiz, cari ujia palig berkuasa saiz- utuk meguji hipotesis H : : N 0 0, lawa H : : N,. (c) Biarka, sebagai suatu sampel rawak daripada tabura N(, ). Cari ujia isbah kebolehjadia utuk meguji H0: 0 lawa H :. 0 [ 40 markah ] 9/-
9 - 9 - [MST56] APPENDI / LAMPIRAN - ooo O ooo -
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