MGM 562 Probability Theory [Teori Kebarangkalian]
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1 UNIVERSITI SAINS MALAYSIA Second Semeser Examinaion 014/015 Academic Session June 015 MGM 56 Probabiliy Theory [Teori Kebarangkalian] Duraion : 3 hours [Masa : 3 jam] Please check ha his examinaion aer consiss of SEVEN ages of rined maerial before you begin he examinaion. [Sila asikan bahawa keras eeriksaan ini mengandungi TUJUH muka sura yang berceak sebelum anda memulakan eeriksaan ini.] Insrucions: [Arahan: Answer all seven [7] quesions. Jawab semua ujuh [7] soalan.] In he even of any discreancies, he English version shall be used. [Sekiranya erdaa sebarang ercanggahan ada soalan eeriksaan, versi Bahasa Inggeris hendaklah diguna akai]. /-
2 - - [MGM 56] 1. has he robabiliy mass funcion wih he domains of 0, 1,, 3 and 4. The momen generaing funcion of his disribuion is given as follows: M 3 4 ( = + e + e + e + e a Derive he mean and he variance of his disribuion b Hence, find he robabiliy mass funcion of his disribuion [10 marks] 1. memunyai fungsi jisim kebarangkalian dengan domain-domain 0, 1,, 3 dan 4. Fungsi enjana momen bagi aburan ini diberikan seeri beriku: M 3 4 ( = + e + e + e + e a Daakan min dan varians bagi aburan ini b Oleh iu, cari fungsi jisim kebarangkalian bagi aburan ini [10 markah]. Anis is considering in ariciaing wo consecuive weekly comeiions. Boh comeiions have rize worh RM,000 each. Losing each comeiion will gain nohing. She feels ha she is 70 ercen confiden o win he firs week comeiion. If she wins he firs comeiion, her confidence of winning he second week increases o 80 ercen. If she loses he firs, her confidence decreases o 50 ercen. a Consruc a ree diagram viewing he ossible oal rizes of Anis in enering he wo comeiions. b Find he firs momen and second momen of he oal rizes (in RM. [4 marks] c Using he Markov s inequaliy, find he highes robabiliy of her winning he oal rizes of more han or equal o RM 3,500 from boh comeiions. 3/-
3 - 3 - [MGM 56]. Anis sedang memerimbangkan unuk menyerai dua erandingan mingguan beruru-uru. Kedua-dua erandingan iu memunyai hadiah bernilai RM,000 seia sau. Kalah dalam seia erandingan idak akan menerima aa-aa hadiah. Beliau berasa bahawa beliau yakin 70 eraus memenangi erandingan ada minggu erama. Jika beliau menang erandingan yang erama, keyakinan beliau unuk memenangi erandingan ada minggu kedua meningka keada 80 eraus. Jika beliau kalah ada erandingan erama, keyakinannya menurun keada 50 eraus. a Bina suau gambarajah okok memerihalkan jumlah hadiah yang mungkin bagi Anis menyerai dua erandingan ersebu. b Cari momen erama dan momen kedua bagi hadiah (dalam RM. [4 markah] c Dengan menggunakan keaksamaan Markov, cari kebarangkalian eringgi bagi beliau memenangi jumlah hadiah lebih dariada RM 3,500 bagi kedua-dua erandingan ersebu. 5 x e 5 x! 3. Le f ( x =, for x = 0,1,,.... Le Y 5 9 = +. a Find he momen generaing funcion of Y. b Find E 3 ( Y. [9 marks] 5 x e 5 x! 3. Biarkan f ( x =, bagi x = 0,1,,.... Biarkan Y 5 9 = +. a Cari fungsi enjana momen bagi Y. b Cari E 3 ( Y. [9 markah] 4/-
4 - 4 - [MGM 56] 4. Find P( 3 < 6 when has he momen generaing funcion as follows: a ( ( 8 M = 0.5 e + 1 ( 4 8 b ( + M = e [6 marks] [6 marks] 4. Cari Kb( 3 < 6 aabila memunyai fungsi enjana momen seeri beriku: a ( ( 8 M = 0.5 e + 1 ( 4 8 b ( + M = e [6 markah] [6 markah] 5. Given 1,..., 0 are indeenden and idenically disribued, having exonenial disribuion wih mean 0.5. Also, le S = a Find he momen generaing funcion of S. b Calculae mean and variance of S. [5 marks] [6 marks] c Using Chebychev s inequaliy, find he highes robabiliy of P S E( S 10. [5 marks] 5. Diberi 1,..., 0 adalah erabur secara idak bersandar dan secaman dan memunyai aburan eksonen dengan min 0.5. Juga, biarkan S = a Cari fungsi enjana momen bagi S. b Kira min dan varians bagi S. [5 markah] [6 markah] c Dengan menggunakan keaksamaan Chebychev, cari kebarangkalian eringgi bagi Kb S E ( S 10. [5 markah] 5/-
5 - 5 - [MGM 56] 6. Le A, B and C be indeenden random variables and having he resecive momen generaing funcions as follows: ( ex( ( ex( ( ex( 3 3 MA = + MB = + MC = + Le = A+ B+ C, and skewness coefficien for, γ is defined as γ = E µ σ ( 3 / 3, where µ is he mean and σ is he sandard deviaion of, a By using momen generaing funcion, calculae E k ( for k = 1,, 3. [10 marks] b Show ha, γ = 0. c Deermine he disribuion of. [4 marks] [ marks] 6. Biarkan A, B dan C masing-masing meruakan embolehubah- embolehubah idak bersandar dan memunyai fungsi enjana momen seeri beriku: ( eks( ( eks( ( eks( 3 3 MA = + MB = + MC = + Biarkan = A+ B+ C, dan ekali keencongan bagi, γ diakrifkan sebagai γ = E ( µ 3 / σ 3, yang mana µ adalah min dan σ adalah sisihan iawai bagi. a Dengan menggunakan fungsi enjana momen, kira E k ( bagi k = 1,, 3. [10 markah] b Tunjukkan bahawa, γ = 0. c Tenukan aburan bagi. [4 markah] [ markah] 6/-
6 - 6 - [MGM 56] 7. Le he variable of and Y has a funcion of f ( xy, kxy x = 1,,3 and 1 y x = 0 oherwise. a Find k so ha he above funcion be a joinly random variable. b Find he marginal disribuion of and Y, resecively. c Wha is Cov(, Y? [9 marks] [9 marks] 7. Biarkan embolehubah dan Y memunyai fungsi f ( xy, kxy x = 1,,3 dan 1 y x = 0 selainnya. a Cari k suaya fungsi di aas menjadi suau aburan rawak ercanum. b Cari aburan su masing-masing bagi dan Y. c Aakah Kov(, Y? [9 markah] [9 markah] 7/-
7 - 7 - [MGM 56] Random Variable, bin( n, Poisson( λ NB( r, (, Probabiliy disribuion funcion, f ( x n x ( 1 n x, x = 0,1,..., n x e λ λ x, x = 0,1,,... x! x+ r 1 r ( 1 x x = 0,1,,... x, Aendix U ab 1, b a a < x< b a b (, 1 1 N µσ ex ( x µ σ, ( ex θ 1 e x/ θ, x > 0 θ 1 x α 1 ex ( x / θ Γ( α θ α Gamma ( aθ, Bea ( aβ, ( ( α Γ( β < x <, x > 0 Γ α + β x α 1 1 ( 1 x β, 0 x 1 Γ Mean, E( Variance, Var ( r Momen Generaing Funcion, M ( = E( e n n ( 1 ( e n + 1 λ λ ex λ ( e 1 ( 1 r( 1 + ( a b 1 µ σ θ αθ α α β αβ + ( α + β ( α + β + 1 r 1 ( 1 e e b e a b ( a σ ex µ + θ ( θ 1 1 αθ ( 1 θ α Binomial Exansion:- n ( a+ b n n x n x = Cxa b x= 0 Taylor s Series:- ( m ( 0 ( 0 m g x x x g( x = m= 0 m! - ooo0ooo -
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