# MAT 263 Probability Theory [Teori Kebarangkalian]

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1 UNIVERSITI SAINS MALAYSIA Second Semester Examination 2016/2017 Academic Session June 2017 MAT 263 Probability Theory [Teori Kebarangkalian] Duration : 3 hours [Masa : 3 jam] 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.] Instructions: [Arahan: Answer all seven [7] questions. Jawab semua tujuh [7] 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 1. Let X be a function defined as follows, f X (x) = { (x + 2 x+3 x ) (1 2 ) x = 0,1,2, 0 otherwise. a. Find M X [ln(1.2)]. b. Find the second moment of this distribution. [4 marks] c. Given the following events, C and D are both sets of non-negative integer numbers, x, such that, C = {x Z x 2} and D = {x Z x < 4}. FromFind i. P(C D). ii. P(C D). iii. P[(C D) C]. [13 marks] 1. Biarkan X sebagai suatu fungsi yang ditakrifkan seperti berikut, f X (x) = { (x + 2 x+3 x ) (1 2 ) x = 0,1,2, 0 selainnya. a. Cari M X [ln(1.2)]. b. Cari momen kedua bagi taburan ini. [4 markah] c. Diberi peristiwa berikut, C dan D kedua-duanya merupakan set-set integer bukan negatif, x, iaitu, C = {x Z x 2} dan D = {x Z x < 4}. Cari i. Kb(C D). ii. Kb(C D). iii. Kb[(C D) C]. [13 markah]

3 2. Let A C = C, and B C, where A and B are independent. Also given that, P(A) = 0.5, P(B) = 0.4, P(C) = 0.3 and P(B C) = 0.6. Find a. P(C A). b. P(B C). c. P[(B C) A]. [4 marks] d. P[(A B C) ]. 2. Biar A C = C dan B C, yang mana A dan B adalah tidak bersandaran. Juga diberi, Kb(A) = 0.5, Kb(B) = 0.4, Kb(C) = 0.3 dan Kb(B C) = 0.6. Cari a. Kb(C A). b. Kb(B C). c. Kb[(B C) A]. [4 markah] d. Kb[(A B C) ].

4 3. Random variable, A has the moment generating function as follows, M A (t) = 1 12 e2t e2.5t e4t e7.5t e8t. a. By using the equation above, find the mean and variance of A. [8 marks] b. State the distribution of A. c. Using Markov s inequality, find the highest value of P [A 10 Var(A)]. 3. Pembolehubah rawak, A mempunyai fungsi penjana momen seperti berikut, M A (t) = 1 12 e2t e2.5t e4t e7.5t e8t. a. Dengan menggunakan persamaan di atas, cari min dan varians bagi A. [8 markah] b. Nyatakan taburan bagi pemboleh ubah rawak A. c. Dengan menggunakan ketaksamaan Markov, cari nilai tertinggi bagi Kb [A 10 Var(A)].

5 4. Consider the jointly random variables, X and Y, where, X has Binomial distribution with parameters of n = 2 and p = 3/4, whereas, Y X = x is a Binomial distribution with the parameters of n = 2 x and p = 4/5. a. Determine the conditional probability of Y given X = x, f Y x (y x). b. Compute, i. The expectation, E(XY), ii. The mean and variance of X and Y iii. The covariance between X and Y, Cov(X, Y). c. If Z = 3X + Y, then, find Var(Z). [7 marks] [11 marks] 4. Pertimbangkan pemboleh ubah rawak tergabung, X dan Y, yang mana, X adalah taburan Binomial dengan parameter n = 2 dan p = 3/4, manakala, Y X = x adalah taburan Binomial dengan parameter n = 2 x dan p = 4/5. a. Tentukan kebarangkalian bersyarat bagi Y diberi X = x, f Y x (y x). b. Kira i. Jangkaan, E(XY), ii. iii. Min dan varians bagi X dan Y, c. Jika Z = 3X + Y, maka, cari Var(Z). Kovarians antara X dan Y, Kov(X, Y). [7 markah] [11 markah]

6 5. Find P[X E(X)] where the random variable X has the moment generating function as follows, 2 a. 5t (e5t e 2.5t ). b. ( e t ) 8. [6 marks] [5 marks] 5. Cari Kb[X E(X)] yang mana pembolehubah rawak X mempunyai fungsi penjana momen seperti berikut, 2 c. 5t (e5t e 2.5t ). d. ( e t ) 8. [6 markah] [5 markah] 6. Let X 1,, X 17 be independently normal distributed with mean μ, and variance σ 2. Consider the following information, x = 4.33 and x 2 = a. Estimate the mean and standard deviation of the distribution above. b. Calculate a 95% confidence interval for the mean of this random variable. [4 marks] c. Calculate a 95% confidence interval for the standard deviation of this random variable. [5 marks] 6. Biarkan X 1,, X 17 sebagai taburan tak bersandar normal dengan min μ, dan varians σ 2. Pertimbangkan maklumat seperti berikut, x = 4.33 dan x 2 = a. Anggarkan min dan sisihan piawai bagi taburan di atas. b. Kirakan 95% selang keyakinan min bagi pembolehubah rawak ini. [4 markah] c. Kirakan 95% selang keyakinan sisihan piawai bagi pembolehubah rawak ini. [5 markah]

7 7. Let X be a continuously uniform random variable under the domain of 0 and 1, i.e. x [0,1] Let Y = 3lnX. Then, determine E(Y + Y 2 ). [8 marks] 7. Biarkan X sebagai taburan selanjar seragam di bawah domain 0 dan 1, iaitu, x [0,1]. Biar Y = 3lnX. Maka, tentukan E(Y + Y 2 ). [8 markah]

8 Appendix Random Variable, X Probability distribution function, Mean, E(X) Variance, Var(X) Moment Generating f X (x) Function, M X (t) bin(n, p) ( n x ) px (1 p) x, x = 0,1,, n np np(1 p) (pe t + 1 p) n Poisson(λ) e λ λ x, x = 0,1,2, x! NB(r, p) x + r 1 ( ) p x r (1 p) x, x = r(1 p) p 0,1,2, 1 uniform(a, b), a < x < b a + b b a 2 N(μ, σ 2 ) 1 exp [ 1 2πσ 2 (x μ σ )2 ], < x < 1 exp(θ) Gamma(α, θ) 1 Γ(α) x α 1 λ λ e λ(et 1) r(1 p) p r [ p 2 1 (1 p)e t] (b a) 2 12 μ σ 2 e tb e ta t(b a) exp (μt + σ2 t 2 2 ) θ e x/θ, x > 0 θ θ 2 (1 θt) 1 θ α exp( x/θ), x > 0 αθ αθ 2 (1 θt) α (a + b) n = n x=0 n C x a x b n x P(X Y) = P(X Y ) g(x) = m=0 g(m) (x 0 )(x x 0 ) m m! Var(aX + by) = a 2 Var(X) + 2abCov(X, Y) + b 2 Var(Y)

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