MST 565 Linear Model [Model Linear]

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1 UNIVERSITI SINS MLYSI Second Semester Examination 009/00 cademic Session pril/may 00 MST 565 Linear Model [Model Linear] Duration : 3 hours [Masa : 3 jam] Please check that this examination paper consists of NINE pages of printed material before you begin the examination [Sila pastikan bahawa kertas peperiksaan ini mengandungi SEMBILN muka surat yang bercetak sebelum anda memulakan peperiksaan ini] Instructions : nswer all four [4] questions [rahan : Jawab semua empat [4] soalan] In the event of any discrepancies, the English version shall be used [Sekiranya terdapat sebarang percanggahan pada soalan peperiksaan, versi Bahasa Inggeris hendaklah diguna pakai] /-

2 a in which a a (a) Consider the symmetric nonsingular matrix square matrix, a is a matrix, and a is a vector Then, if that the inverse of is given as: is a exists, show b a b a a where b a a a [5 marks] (b) Let Find the two generalized inverses of, denoted as and and verify that and [5 marks] (c) Using matrix in (b) and consider between tr, tr and rank , state the relationships [5 marks] (d) If is any 33 matrix with eigenvalues, 5 and 3 3, determine and tr [5 marks] 3/-

3 3 (a) Pertimbangkan satu matriks tak singular yang simetri, yang a a mana matriks segi-empat, a matriks x, dan a suatu vektor Seterusnya, jika wujud,tunjukkan bahawa songsangan bagi diberikan sebagai a b a b a a yang mana b a a a [5 markah] (b) Biar Cari dua songsangan teritlak bagi, yang diwakili sebagai dan dan tentusahkan bahawa dan [5 markah] (c) Menggunakan matriks dalam (b) dan pertimbangkan , nyatakan hubungan antara surihan, surihan dan pangkat [5 markah] (d) Jika suatu matriks 33 dengan nilai eigen, 5 dan 3 3, tentukan dan surihan [5 markah] 4/-

4 4 (a) Let y y, y, y3 be a random vector with mean vector and covariance matrix 0 μ, Σ (i) Let z y 3y y 3 Find Ez and Var z (ii) Let z y y y 3 and z 3y y y 3 Find E z and Var z where z z, z (b) Let the random vector v be N4 μσ, with μ, Σ [8 marks] v y, y, x, x If v is partitioned as (i) Determine μy, μx, Σyy, Σ yx and Σ xx (ii) Find the conditional distribution of yx (iii) Find matrix of partial correlations, P 34 [6 marks] v y x y, y, x, x (c) Let and covariance matrix given as in Question (b) Given constants, determine the expected value of xy be a partitioned random vector with mean vector 3 4, a matrix of [6 marks] 5/-

5 5 (a) Biar y y, y, y3 sebagai satu vektor rawak dengan vektor min dan matriks kovarians 0 μ, Σ (i) Biar z y 3y y 3 Cari Ezdan Var z (ii) Biar z y y y 3 dan z 3y y y 3 Cari E z dan Var z, yang mana z z, z (b) Biar vektor rawak v sebagai N4 μσ, dengan μ, Σ y, y, x, x Jika v terpetak sebagai v [8 markah] (i) Tentukan μy, μx, Σyy, Σ yx dan xx (ii) Cari taburan bersyarat untuk yx Σ (iii) Cari matriks korelasi separa, P 34 [6 markah] v y x y, y, x, x (c) Biar satu vektor rawak terpetak dengan vektor min dan matriks kovarians seperti diberikan dalam Soalan (b) Diberikan satu matriks pemalar, 3 4, tentukan nilai jangkaan bagi xy [6 markah] 6/-

6 6 3 (a) Suppose that yis N3 μ, I and let 3 μ,, 3 B 0 (i) Determine the distribution of yy (ii) Verify either yy and By are independent or not? (iii) What about y y and y y y 3, are they independent or not? [ marks] (b) ssuming that y is N4 μσ,, where μ, Σ Find a matrix such that yy is 4, μμ then determine the value of [6 marks] (c) Suppose that for a particular linear model, we found that 0 0 XX and Xy and the normal equations are b0 4 0 b 6 0 b 8 (i) Determine either the model is a full- rank or not (ii) Using M, find the generalized inverse of XX (iii) Check either 0 is estimable or not [7 marks] 7/-

7 7 3 (a) Katakan bahawa yadalah N3 μ, 3 μ,, 3 B I dan biar 0 (i) (ii) Tentukan taburan bagi yy Sahkan sama ada yy danby adalah tak bersandar atau tidak? y y y, adakah ia tak bersandar atau tidak? [ markah] (iii) Bagaimana dengan y y dan 3 (b) ndaikan bahawa y sebagai N4 μσ,, yang mana μ, Σ Cari satu matriks supaya yy merupakan tentukan nilai 4, μμ, kemudian [6 markah] (c) Katakan bahawa untuk satu model linear, kita dapati bahawa 0 0 XX dan Xy dan persamaan normal sebagai b0 4 0 b 6 0 b 8 (i) Tentukan sama ada model tersebut berpangkat penuh atau tidak (ii) Menggunakan M, dapatkan songsangan teritlak XX (iii) Semak sama ada 0 teranggarkan atau tidak? [7 markah] 8/-

8 8 4 (a) Consider the one-way classification model yij i ij; i,, j, (i) Write the model in matrix form y Xβ ε and determine the rank of X (ii) Identify the estimable functions, γ Uβ (iii) Determine Z by using a reparameterize technique to reduce the non-full rank model y Xβ ε to a full- rank model y Zβ ε [5 marks] (b) Weight gains in cows subjected to three different treatments are given in Table Table : Weight Gain of Cows Subjected to Three Treatments Treatment Test the hypothesis of equal mean treatment effects [0 marks] 9/-

9 9 4 (a) Pertimbangkan model sehala yij i ij; i,, j, (i) Tuliskan model tersebut dalam bentuk matriks y Xβ ε dan tentukan pangkat X (ii) Kenalpasti fungsi teranggarkan γ Uβ (iii) Tentukan Z menggunakan teknik parameterkan untuk turunkan model tak berpangkat penuh y Xβ ε kepada model berpangkat penuh y Zβ ε [5 markah] (b) Pertambahan berat pada lembu bergantung kepada tiga rawatan berlainan diberikan dalam Jadual Jadual : Pertambahan Berat Pada Lembu Bergantung Kepada Tiga Rawatan Rawatan Uji hipotesis kesamaan min kesan rawatan [0 markah] - ooo O ooo -

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