# MGM 562 Probability Theory [Teori Kebarangkalian]

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1 UNIVERSITI SAINS MALAYSIA Firs Smsr Eaminaion 016/017 Acadmic Sssion Dcmbr 016 / January 017 MGM 56 Probabiliy Thory [Tori Kbarangkalian] Duraion : hours [Masa : jam] Plas chck ha his aminaion ar consiss of SIX ags of rind marial bfor you bgin h aminaion [Sila asikan bahawa kras riksaan ini mngandungi ENAM muka sura yang brcak sblum anda mmulakan riksaan ini] Insrucions: [Arahan: Answr all svn [7] qusions Jawab smua ujuh [7] soalan] In h vn of any discrancis, h English vrsion shall b usd [Skiranya rdaa sbarang rcanggahan ada soalan riksaan, vrsi Bahasa Inggris hndaklah diguna akai] /-

2 - - [MGM56] 1 L X b a funcion dfind as follows, whr, 0 a 1 and 0b 1 f X ab 0,1,, 0 ohrwis (a) (b) If X is a random variabl, find h rlaionshi bwn a and b, whr a and b ar boh consans Givn h following vns, C and D ar boh ss of non-ngaiv ingr numbrs such ha, C 6 and D 10 If h valu of a for h abov disribuion is 1 4, find (i) PC D (ii) PC D (iii) PC D C 1 Biar X ialah suau fungsi yang diakrifkan sri briku, yang mana, 0 a 1 f X dan 0b 1 ab 0,1,, 0 slainnya a Jika X adalah mbolhubah rawak, cari kaian anara a dan b yang mana a dan b kdua-duanya adalah malar b Dibri rnyaaan briku, C dan D kdua-duanya mruakan s-s ingr bukan ngaif, iaiu, C 6 dan D 10 Jika nilai a bagi aburan di aas adalah 1 4, cari (i) PC D (ii) PC D (iii) PC D C /-

3 - - [MGM56] L AC C, BC, and A and B ar indndn Also givn ha, P A 05, PB 04, PC 0 and PB C 06 Find (a) P A C (b) PB C (c) PB C A (d) P AB C' [ marks ] Biar AC C, BC, dan A dan B adalah idak brsandaran Juga dibri, P A 05, PB 04, PC 0 dan PB C 06 Cari (a) P A C (b) PB C (c) PB C A (d) P AB C' [ markah ] Random variabl, A has h momn gnraing funcion as follows, M A 15 (a) Find h man and varianc of A (b) Dfin h disribuion of h random variabl A (c) L BA, hn find P B E B [ 8 marks ] [ marks ] Pmbolhubah rawak, A mmunyai fungsi njana momn sri briku, M A 15 (a) Cari min dan varians bagi A [ 8 markah ] (b) Takrifkan aburan bagi mbolhubah rawak A [ markah ] (c) Biarkan BA, kmudian cari P B E B 4/-

4 - 4 - [MGM56] 4 Considr h joinly random variabls, X and Y, whr, X has Binomial disribuion wih aramrs of n and, whras, Y X is a Binomial wih h aramrs of n and 4 (a) Drmin h condiional robabiliy, (b) Comu, (i) Th caion, E XY, (ii) Th man of Y, EY, f y YX (iii) Th covarianc bwn X and Y, Cov X, Y (c) If Z X Y, hn, find Var Z [ marks ] [ 7 marks ] [ 6 marks ] 4 Primbangkan mbolhubah rawak rgabung, X dan Y, yang mana, X adalah aburan Binomial dngan aramr n dan, manakala, Y X adalah aburan Binomial dngan aramr n dan 4 f y (a) Tnukan kbarangkalian brsyara, YX (b) Kira (i) Jangkaan, E XY, (ii) Min bagi Y, EY, (iii) Kovarians anara X dan Y, Cov X, Y (c) Jika Z X Y, maka, cari Var Z [ markah ] [ 7 markah ] [ 6 markah ] 5/-

5 - 5 - [MGM56] 5 Find P X E X whr h random variabl X has h momn gnraing funcion as follows, (a) (b) 1 [ 7 marks ] [ 8 marks ] 5 Cari P X E X yang mana mbolhubah rawak X mmunyai fungsi njana momn sri briku, (a) (b) 1 [ 7 markah ] [ 8 markah ] 6 L X b h normal disribuion funcion having h man 7 and sandard dviaion L Y X 6 Find, (a) Th man of Y (b) PY 5X 0 [ 9 marks ] 6 Biarkan X sbagai fungsi aburan normal yang mmunyai min 7 dan sisihan iawai Biarkan Y X 6 Cari, (a) Min bagi Y (b) PY 5X 0 [ 9 markah ] 7 L X b a coninuously uniform random variabl undr h domain of 0 and 1, i 0,1 L Y ln X Thn, drmin E Y Y [ 10 marks ] 7 Biar X sbagai aburan slanjar sragam di bawah domain 0 dan 1, iaiu, 0,1 Biar Y ln X Maka, nukan E Y Y [ 10 markah ] 6/-

6 - 6 - [MGM56] Random Variabl, X bin n, Poisson NB r, Probabiliy disribuion funcion, fx n 1 n, 0,1,, n, 0,1,,! r 1 r 1 0,1,,, Andi Man, E X Varianc, Var X r Momn Gnraing Funcion, M X X E n n1 n r1 r 11 b a (b a) uniform(a, b), a < < b a + b (b a) b a 1 N, 1 1, 1 /, 0 Gamma, 1 1 /, 0 1 n a b n n n Ca b 0 P X Y P X Y g m m g 0 0 m0 m!, Var ax by a Var X abcov X Y b Var Y - ooo O ooo -

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