# lnstructions: Answer all ten [10] questions. Franan:- Jawab semua sepuluh [10] soalan.l MGM Probability Theory fteori Kebara ngka I ian]

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1 UNIVERSITI SAINS MALAYSIA Second Semester Examination Academic Session April/May 2009 MGM Probability Theory fteori Kebara ngka I ian] Duration : 3 hours [Masa : 3 jamj Please check that this examination paper consists of EIGHT pages of printed material before you begin the examination. [Sila pastikan bahawa kertas peperiksaan ini mengandungi LAPAN muka surat yang bercetak sebelum anda memulakan peperiksaan ini.l lnstructions: Answer all ten [10] questions. Franan:- Jawab semua sepuluh [10] soalan.l...2t-

2 2 IMGM 5621 t. 2. A box contains 6 black balls and r red balls. One of the balls is drawn at random, but when it is put back in the box, c additional balls of the same color are put in with it. Now, suppose that we draw another ball. Show that the probability that the first ball drawn was black given that the second ball drawn was red it 3. b+r+c Consider two boxes, one containing one black and one white marble, the other, two blacks and one white marble. A box is selected at random and a marble is drawn from the selected box. (a) What is the probability that the marble is black? (b) What is the probability that the first box was the one selected given the marble is white? 3. Let the density for the continuous random variable X be given by t-e-* Ir.r > 0 (a) Showthat 00 I f(x)ax=t. l1 'f (x) = 1'!"' x<o t2 (b) Showthat (c) Use u x Q) to find E(X). uxu)=;(* *) -1</<1 4. Let the probability density function of X be given by c (-.\ -2*') o < x <2 l"(t/=t' o otherwise what is pr(!.x. 1) r \2 2)...3/-

3 IMGM s62l l. Suatu katak mengandungi b bola hitam dan r bola merah. Satu daripada bola di dalam kotak dipilih secara ranak, tetapi apabila dimasukkan semula lce dalam kotak itu, sebanyak c bola yang sarna warna dengan yang diambil, ditambahkan lre dalam kotak tersebut. Sekarang, kita memilih semula sebiji lagi bola secara rawak dari dalam kotak tersebut. Tunjukkan kzbarangkalian bahawa bola pertama adalah hitam bersyaratkan bola kedua adalah merah, io1o1, - b. ' b+r+c flo markahl 2. Pertimbangkan dua kotak, yang pertama mengandungi satu gtli hitam dan satu guli putih, dan kotak lcedua mengandungi dua guli hitam dan satu guli putih. Salah satu katak dipilih dan satu guli dipilih secara rawak daripada kotak itu. (o) Apakah kebarangkalian bahawa guli itu hitam? (b) Apakah kebarangkalian bahawa latak pertama terpilih diberi bahawa guli yang terpilih adalah putih? 3. Diberi taburan untuk pembolehubah selanjar, X adalah l!r-..r > o t1 f(x)=1' -\/ lix<o [10 markahj (a) Tunjukknn bahawa f 7p1a, =t. (b) TunjuA*an bahav,,a un(t\=l(l- t ') ^\' 2[l+l t-l) (c) Dengan menggunakan Ux Q), cart E(X). -l</<l flo markahl 4. Diberifungsi ketumpatan kebarangkalian bagi X sebagai Apakah prl.1 -\22). x.1) z -r,') o < x <2 L 0 selainnya flo markahj..,4t-

4 IMGM Suppose the density of X is given by -, \ fl.*r-'''.x>o Jx lx) = 1aL 0 otherwise Calculate the moment generating function, E(X) andvar(x). Then, calculate / o\ Elsx +4x'). 6. The joint density * trto lris given by '[ 8 ).f (*,y\=ly-:*- lr-t, o1y<*, -y<x<y '.Y.f( t r\ -l Show that t ilryv' bw= 1. Also, show that n(x ly : y) = 0. o-yl\ I ) ) 7. Consider the following questions: [0 marks] (a) Let X be exponential with rn"un l, that is A, o<xco I o Find E(x lx'1) (b) Let X beuniforrnover(0,1).find Z(XlX <ll2). otherwise 8. In a study case over acertain period, a small firm is suffering a loss following an exponential distribution, that is, X - exp(). = ). The probability density function (PDF) of loss is shown as follows ^, \ l),r-1' o<x<oo Jx\x)=l o otherwise (a) Find Pr(7500 < X < 12000) using directly from the above PDF. (b) BV using Normal approximation, compute Pr(7500 < X < 12000). [0 marks]...5/-

5 r IMGM Diberitaburanbagi X sebagai ll rr-"rz x>o t 0 selainrrya Kira fungsi penjana momen, E(X) dan Var(X) Seterusnya, kira n(sx +qx2\ \/ 6. Taburan gabungan Utf, : olirt diberikan seperti berilut.f (r,y)--l'- '[ ='- 8 l"-,, o<y<*, -y<x<y ) Tunjuktran bahowa ]'[e+)urf* 6'-'rL\. 8 ) j s(xly=y)=0. 7. Pertimbangkan soalan-soalan berikut: p0 markahj =,. Juga, buktitran bahawa (a) Diberi X adalah taburan eksponen denganjangkao, l,,ouu /, o<x<o.fx ij)=[^'-^' I o selainnya Cari E(X lx > l) (b) Diberi X adalah taburan seragam pada (0,1). Cari n(x I X <t/2). flo markahl flo markahj 8. Dalam suatu kes kajian pada suatu masa tertentu, sebuah firma kecil sedang mengalami suatu kerugian mengihtti taburan eksponen, iaitu, X - exp()" = ). Taburan kerugian ditunjukknn seperti berihtt (' -)'x o<x<oo n". L 0 selainnya (a) Cari Pr(7500 < X < 12000) dengan menggunakan perkiraan secara terus daripada taburan di atas. (b) Dengan menggunakan taburan hampiran Normal, kira Pr(7soo < x <t2ooo). [10 marlcahj...6/-

6 \ IMGM A manuscript is sent to a typing firm consisting of typists A, B and C. If it is typed by A,then the number of enors made is apoisson random variable with mean2.6; if typed by B, then the number of errors is a Poisson random variable with mean 3; and if typed by C, then it is a Poisson random variable with mean 3.4. Let X denote the number of errors in the typed manuscript. Assume that each typist is equally likely to do the work. Find E (X) and Var (X). 10.For discrete numbers of x and.y, such that, I <x< y and y=1,2,3 the probability mass function ofjoint distribution of X and I is given as fn(x,y)= cx (a) Find c. (b) Compute /(x) and /(y) respectively for x =1,2,3 and y =1,2,3 (c) Find n(x),var(x), E(r) and var(v) [0 marks]...7t-

7 IMGM Suatu manuslvip dihontar kepada sebuah.fir*a perkhidmatan menaip, yang mempunyai tulmng taip A, B dan C. Jika manuslcrip itu ditaip oleh A, nombor lresalahan menaip tertabur secara Poisson dengan jangkaan 2.6; jifu ia ditaip oleh B, nombor kesalahan adalah pembolehubah rawak Poisson fungan jangkaan 3; dan jika ditaip oleh C, nombor kesalahan adalah pembolehubah rawak Poisson dengan jangkaan 3.4. Katakan X adalah nombor kesalahan menaip bagi manuslvip tersebut. Andaikan bahawa setiap penaip adalah sama dalam menjalanknn kerja mereka. Cari E(X) dan Var(X). flo markahj I0.Untuk nombor-nombor disleret x dan y yang mana I<x1! dan y=1,2,3, fungsi jisim kebarangkalian bagi gabungan X dan Y dinyatalmn seperti berihtt.f n(r,y)= (a) "* Cari c. (b) Kira f(x) aan f(1,,) masing-masing untuk x =1,2,3 dan y =1,2,3 (c) Cari n(x), Var(X), n(r) aan Var(r). p0 markahj...8/-

8 8 IMGM 5621 APPEIIDIX - FORMTJLA e) \$i\ *l'(')a (")] = o' (*)b (*) + a (x) b' (x) (iii) j a' (x) b (x) dx = o (*) b (*) -! <iv> = *, then,!!dx= ln(x) +c a (x) b' (x) dx + c!*nlt(")l o> = #, tnen! ffi a* = t"[/(')] + " (ri) *lf (x)]'=,f (*)"-' f '(*) (vii)ifx-exp(2),then a. f *(r)= i ll"-u L0 I b. n(x)= ; L c. d. (viii) a. b.,*8)=+ M.Q)=* If X -U(a,b) = fr tr;" E(x)=+ x>0 otherwise a<x<b otherwise c. d. axq) -ooooooo-

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