MAT 263 Probability Theory [Teori Kebarangkalian]

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1 UIVERSITI SAIS MALAYSIA First Semester Eamiatio 4/5 Academic Sessio December 4/Jauary 5 MAT 63 Probability Theory [Teori Kebaragkalia] Duratio : 3 hours [Masa : 3 jam] Please check that this eamiatio aer cosists of TE ages of rited material before you begi the eamiatio [Sila astika bahawa kertas eeriksaa ii megadugi SEPULUH muka surat yag bercetak sebelum ada memulaka eeriksaa ii] Istructios: [Araha: Aswer all te [] questios Jawab semua seuluh [] soala] I the evet of ay discreacies, the Eglish versio shall be used [Sekiraya terdaat sebarag ercaggaha ada soala eeriksaa, versi Bahasa Iggeris hedaklah digua akai] /-

2 - - [MAT 63] Suose E, F ad G are evets i a samle sace ad E, F, G State whether the followig statemets are true or false Justify your aswers (a) If E ad F are mutually eclusive, the they are ideedet (b) If E c F, the F c E c (c) If E, F ad G are mutually ideedet, the E ad F G are also mutually ideedet [ marks] Adaika E, F ad G adalah eristiwa-eristiwa dalam ruag samel da E, F, G yataka sama ada eryataa yag berikut bear atau salah Tetusahka jawaa ada (a) Jika E da F salig eksklusif, maka E da F adalah salig tak bersadar (b) Jika E c F, maka F c E c (c) If E, F ad G are mutually ideedet, the E ad F G are also mutually ideedet [ markah] I good coditio, a machie roduces defective items oly % of the time I broke coditio, it roduces defective items 4% of the time Suose that the robability that the machie breaks dow is (a) (b) (c) What is the robability that a radomly selected item is defective? What is the robability that the machie is i good coditio, give that a radomly selected item is defective? Suose a samle of 6 items were selected ad were foud to be defective Was the machie i good coditio at the time? [ marks] Semasa dalam keadaa baik, sebuah mesi meghasilka baraga cacat haya % dariada masa Dalam keadaa rosak, mesi tersebut meghasilka baraga cacat 4% dariada masa Adaika bahawa kebaragkalia mesi itu rosak ialah (a) (b) (c) Aakah kebaragkalia bahawa suatu baraga yag diilih secara rawak adalah cacat? Aakah kebaragkalia bahawa mesi tersebut dalam keadaa baik, jika diketahui bahawa suatu baraga yag diilih secara rawak adalah cacat? Adaika suatu samel 6 baraga diilh secara rawak da dariadaya didaati cacat Adakah mesi tersebut dalam keadaa baik ketika itu? [ markah] 3/-

3 3 Let X be a geometric radom variable with arameter (a) Comute ( X ) (b) P > [MAT 63] Show that X has the memoryless roerty ie for ay itegers ad m, with > m, P( X > X > m) = P( X > - m) 3 Adaika X ialah suatu embohubah rawak geometrik dega arameter (a) Hitug ( X ) (b) P > [ marks] Tujukka bahawa X memuyai ciri tiada igata, iaitu bagi sebarag iteger da m, dega > m, P( X > X > m) = P( X > - m) [ markah] 4 X {,, } ad {, } Y are two discrete radom variables Their joit robability mass fuctio is give as follows: ( =, Y = ) = P( X Y ) ( =, Y = ) = P( X Y ) ( =, Y = ) = P( X Y ) =, = = 3 =, = = =, = = (a) What is the margial robability mass fuctio of Y? (b) Comute E[ XY ] (c) Comute the covariace of X ad Y (d) Are X ad Y ideedet radom variables? Justify your aswer 4 X {,, } da {, } [ marks] Y adalah dua embolehubah rawak diskret Fugsi jisim kebaragkalia tercatum mereka diberika seerti yag berikut: ( =, Y = ) = P( X Y ) ( =, Y = ) = P( X Y ) ( =, Y = ) = P( X Y ) =, = = 3 =, = = =, = = (a) Aakah fugsi jisim kebaragkalia sut Y? (b) Hitug E[ XY ] (c) Hitug kovarias X da Y (d) Adakah X da Y embolehubah rawak tak bersadar? Tetusahka jawaa ada [ markah] 4/-

4 - 4 - [MAT 63] 5 Suose that X ad Y are ideedet eoetial radom variables with arameter X λ If Z = X + Y (a) fid the distributio fuctio, F Z ( z), for < z < ; (b) fid the desity fuctio of Z ad ame the distributio [ marks] 5 Adaika X da Y adalah embolehubah rawak eoe dega arameter λ X Jika Z = X + Y (a) daatka fugsi tabura, F Z ( z), bagi < z < ; (b) daatka fugsi ketumata bagi Z da yataka ama tabura tersebut [ markah] 6 Let X ad Y be two ideedet radom variables, with resective robability desity fuctio f ( ) = e for > ad = y f y e for y > (a) Write the joit robability desity fuctio of X ad Y, f X, Y (, y) (b) Comute the robability P ( X Y > ) (c) Fid the distributio fuctio ( y) F X, Y, [ marks] 6 Fugsi ketumata tercatum X da Y diberika sebagai < < y < f (, y) = sebalikya (a) Daatka fugsi ketumata bersyarat X diberika Y = y, ( y) (b) Hitug P( X Y 3 ) < = 4 (c) Hitug E [ X Y = y] f X Y (d) Daatka E {[ X E( X Y = y)] Y = y} [ markah] 5/-

5 - 5 - [MAT 63] 7 The joit desity fuctio of X ad Y is give by < < y < f (, y) = otherwise (a) Fid the coditioal desity fuctio of X give Y = y, ( y) (b) Comute P( X Y 3 ) < = 4 (c) Comute E [ X Y = y] f X Y (d) Fid E {[ X E( X Y = y)] Y = y} [ marks] 7 Adaika X da Y adalah dua embolahubah rawak tak bersadar dega fugsi ketumata kebaragkalia masig-masig = bagi > f e da = y f y e bagi y > (a) Tuliska fugsi ketumata tercatum X da Y, f X, Y (, y) (b) Hitug kebaragkalia P ( X Y > ) (c) Daatka fugsi tabura ( y) F X, Y, [ markah] 8 Let X ad X be ideedet eoetial radom variables, each havig arameter λ (a) X Fid the joit desity fuctio of Y = X + X ad Y = X + X (b) Are Y ad Y ideedet? Justify your aswer [ marks] 8 Adaika X da X adalah embolehubah rawak eoe, setia satu dega arameter λ (a) Daatka fugsi ketumata tercatum bagi Y = X + X da Y = X X + X (b) Adakah Y da Y tak bersadar? Tetusahka jawaa ada [ markah] 6/-

6 - 6 - [MAT 63] 9 A biased coi is flied = times The robability of gettig a head (H) is a ukow value, Defie X i if heads aear = otherwise Let X = X i i= (a) Show that EX [ ] = ( - ) (b) Show that Var[ X ] = (c) Aroimate the robability that X (Hit: Use Cetral Limit Theorem) ( ) ( ) 5, + 5 [ marks] 9 Sekeig syilig tak saksama dilambug sebayak = times Kebaragkalia medaat suatu keala (K) ialah, suatu ilai yag tidak diketahui Takrifka X i = jika keala mucul sebalikya Adaika X = X i i= (a) Tujukka bahawa EX [ ] = ( - ) (b) Tujukka bahawa Var[ X ] = (c) Aggarka kebaragkalia bahawa X (Petujuk: Gua Teorem Had Memusat) ( ) ( ) 5, + 5 [ markah] 7/-

7 - 7 - [MAT 63] X i i= Let X = arameter, where the X i 's are ideedet eoetial radom variables of (a) Show that the momet geeratig fuctio of X is t ) E[ (b) Comute the momet geeratig fuctio of X, M ( t = e ] (c) Show that for ay a >, P( X > a) e M (t) (d) Give a uer boud of P(X > ) at tx [ marks] X i i= Adaika X = arameter, di maa X i adalah embolehubah rawak eoe dega (a) Tujukka bahawa fugsi ejaa mome bagi X ialah (b) Hitug fugsi ejaa mome bagi X, M ( t) = E[ e ] tx t (c) Tujukka bahawa bagi sebarag a >, P( X a) e M (t) (d) Berika suatu batas atas bagi P(X > ) at > [ markah]

8 - 8 - [MAT 63] APPEDIX / LAMPIRA Beroulli Biomial Geometric DISCRETE DISTRIBUTIOS t = + =, = ( ) f =, =, M t e µ σ egative Biomial Poisso! f ( ) = ( ), =,,,,!! ( ) t = ( + ) M t e, µ = σ = f =, =,,, M ( t) =, t < I t e ( ) µ, σ µ = = = ( + r )! ( r! ) ( ) t r ( ) e r( ) r( ) = =! r f =, =,,, r M t =, t < I µ, σ λ λ e f =, =,,,! t λ( e ) M t = e =, = µ λ σ λ Hiergeometric r f ( ) =, r,, r, t r r r µ =, σ = ( ) ( ) 9/-

9 - 9 - [MAT 63] Uiform Eoetial Gamma COTIUOUS DISTRIBUTIO f =, a b b a tb ta e e M( t) =, t, M( ) = t b ( a) ( b a) a+ b µ =, σ = θ f = e, θ M ( t) =, t < θ θt µ = θ, σ = θ α θ f e α M t Chi Square =, Γ α θ =, t < ( θt) α µ = αθ, σ = αθ r f r e, r M t ormal = Γ ( r ) =, t < r ( t ) µ = r σ =, µ t+ σ t = e, Var ( X ) r ( µ ) /σ f = e, < < σ π M t E X Beta = µ = σ = ( ),< < B α β f ( αβ, ) α αβ µ =, σ = α + β α + β + α + β /-

10 - - [MAT 63] FORMULA =! = = = = ar = e a =, r < r ba a b = + r = r r + k k w = ( w), w < k α ( α) e d ( α) ( α ) Γ =, Γ =! α ( αβ, ) = ( ) β B d B ( αβ, ) ( α) Γ( β) ( α β) = Γ Γ + 9 Polar coordiates: y = rcos θ z = rsi θ - 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