BACHELOR'S DEGREE PROGRAMME Term-End Examination June, Time : 2 hours Maximum Marks : 50 (Weightage : 70%)

Transcription

1 No. of Printed Pages : BACHELOR'S DEGREE PROGRAMME Term-End Examination June, 2014 ELECTIVE COURSE : MATHEMATICS MTE-11 : PROBABILITY AND STATISTICS MTE-11 Time : 2 hours Maximum Marks : 50 (Weightage : 70%) Note : Question no. 7 is compulsory. Answer any four questions from questions no. 1 to 6. Calculators are not allowed. 1. (a) If R and S are the range and standard deviation for a set of n observations on a variable X, show that S R. (b) Two unbiased dice are thrown. Let Al : Odd number on the first die A2 : Odd number on the second die A3 : Odd sum of the numbers on first and second die. Comment on the independence of A1, A2 and A3. 3 MTE P.T.O.

2 (c) A random sample of size n is drawn from a 1 1 uniform population over [ Obtain maximum likelihood estimator of 0. Does a unique estimator exist? Give reasons (a) Let X be a random variable having uniform density over (0, 1). Find 4 (i) E (J), (ii) pdf of y = -%/TE, (iii) E(y) and (iv) check whether E(y) = E (IX) or not. (b) Develop a test for testing _1X2 H0 : f(x) = 1 e 2 co < x < 00 against Hi : f(x) = 1 2 exp( IxI), oo<x< oo based on a single observation drawn from f(x). Obtain expression for size and power of the test. MTE-1 1 2

3 3. (a) The random vector (X, Y) has the joint density function given by '1 x 2 + y 2 1 f(x, y) = 0, otherwise Then find the marginal probability density function of X and Y. Show that cov(x, Y) = 0 but X and Y are not independent. 6 (b) State Bayes theorem. There are three bags containing 2 white balls, 1 white and 1 black ball, and 2 black balls respectively. A bag is selected randomly and a ball is drawn. It is found to be white. What is the probability that remaining ball in the selected bag is white? 4 4. (a) The first three moments about the origin are given by m, = n (n + 1) (2n + 1), n (n + 1)2 m 2 = and m 3 = 6 4 Examine the skewness of the data. 5 MTE P.T.O.

4 (b) A taxi cab company has two types of cabs A and B. The number of cabs of type A and B are 12 and 8 respectively. If 5 of these taxi cabs are in the workshop for repairs and the time needed for repairing each of the types is same, what is the probability that (i) 3 of them are of type A and 2 of them are type B? (ii) at least 3 of them are of type A? (iii) all 5 are of same type? 5 5. (a) Let X be a random variable with pdf f x) = 0 e - ex; 0 > 0, x O. Find the moment generating function of X, and hence find first three moments about origin. 5 (b) The following table gives the number of aircraft accidents that occurred during six days of the week. Find whether the accidents are uniformly distributed over the week at 5% level of significance. 5 Days No. of accidents Mon. Tues. Wed. Thur. Fri. Sat. Total [You may like to use the following values : X5, 0.05 =11.070, 2 X 6, 0.05 = , x =14.067]. MTE-1 1 4

5 6. (a) For the data given below X Y find : 6 (i) the correlation coefficient between x and y. (ii) the regression equation of y on x. (b) If X is a random variate such that E(X) = 3 and E(X2) = 13, use Chebychev's inequality to determine a lower bound for P ( 2 < X < 8) Which of the following statements are true or false? Give reasons for your answer. 10 (a) For negatively skewed distribution, Pearson's coefficient of skewness is negative. (b) Variance is independent of change of origin and scale. (c) If A and B are independent events then A' (Complement of A) is also independent of B. MTE-11 5 P.T.O.

6 (d) X1, X2,... Xn is a random sample from a uniform distribution with probability density function f(x) = 1,. a < x < 13 R - a = 0, otherwise Then the maximum likelihood estimator of a is min (X1, X2,... Xn). (e) If X has F-distribution with n1, n2 degrees of freedom, then 1 also has F-distribution X with same n1, n2 degrees of freedoms. MTE-11 6

7 ,hilict) 341 chi 91)44 Wtrfrarr T4, 2014 'WOW 1.41cktistoi : Tara n.t.t.-11 : afrf : 2 ElvJ *: 50 (pi TT : 70%) :.V W frarif / 3/77 W A-47 V/?' 3& V / 4rignvicj IT T ch(4 3737:1"ft f 1. () R 3 S X n " kitzle4 cvm triter AK Jima) 14Aoq f, c tur flf S R. 3 t 3T-*-4ḏ trt4 t I.31-rq -4trA : tr-0 trt4 TR 1474 titai A2 : trt4 -NIErgrEilwr A3 : * trk kit9,4134 -PaTfq -1' Ail A2 * A3 * f-cild%1 1:R dutiu tri7ri 3 MTE-11 7 P.T.O.

8 (TT) [t till 14 3-TT/417 n zrr-qw* TAO 611 Ant 3Til-wd4 *Tram 3 -FT *TrĀ7I Twr altetzi atrw-ew 3Tfr alai? cnitut ti77 2. () TR R X, (0, 1) 1 h(-11-11,1 (4101 (i) E (J), 4 y = iii pdf, E(y) atit (iv) *11-4R E(y) = E 7TT (w) fkx)4 Tr7 hc-i 4-Tur ITT atrorita- 1 Hi : f(x) = 2 exp (-13c1), ' < X < 00 f0-4 Ho : f(x) =.577 1" t e 1 x 2 2, -00 < X < 00 -*T 1Z-1 7-* trctorr * trftuur ap:rrq atit 41- dt h wti c 6 MTE-1 1 8

9 3. () zfrqf kl (X, Y) ti3th 4-1R-Roci t : f (x, y) = x 2 + y P:filT T4 X 3 Y TT 3410 MI elcbc-11 't11cc 1:5"-eq Tff WA tur4r f cov(x, Y) = 0 TFT X * Y # I (w) ei -2.F tp4r 4tq 4-4 t P*14 2 ti4)4 ki4< aft i weir) 2 chic-11 t I 4-eT ellita9ell T 7TM # 3ki4 ij4q1"47-4t u11cf # I zrg q Pichorn # I qt4-1 -4;ErT ' f*-14 TrRi -4-4fi t? 4 4. (*) 10.- Gfrs * A 3reTri Alq 3 m' = n '...(n+1)(2n+1) at( m, = n(n+1) ITI f4r TIR # 13t+t w tsfizr 7rd *'tflri 5 MTE P.T.O.

10 (w) el 4o-4 trru SichR Alt 404 AATB# I A* B Nchlk Azi 4611 AI shlikl: 12 atit 8 #I 4-14 q*-71-ft 4 i-r44-ici Nchlt t't MIreichcll "ali 5 tt.4 afr{ -srft 1-R1-1-1c1 4 t WR (141, ffq (1) 4.1 A. 3, A Nchlt Azr 3 2, B 51chR Azr r44*w:1-14-w:r 3, A mcw Aztf? (iii) Lt-al? 5. () Trrq #11-A-R x, 1:11--GiRg1 pdf awn T* zrr-qr : f(x) = 0 e ex; 0 > 0, x O. X f lcb FOR. Ta. *zrftr 71- ck ito-refrq* 312T4 'erg 3TTEkai Atr-A-R Pli-ifZiRsio -a-rffewr t1hi * 'UF kh Ala *A. t I 5% TrrakaT * air* S'Efr4rq lc )t c1: IIrn Atr-A7 I fqq *T. Iiiio. T..1i-4 i. qjsb. kl. TO Ite-q-rall *tti [am 11-1R1Rsict 1TR1 r mei 1.4) t : x5, 0.05 = ' X6, 0.05 = , X2 7, 0.05 = ]. MTE-11 10

11 6. () TR 3T1-*--41* X Y orgi *AR : 6 (i) x31 yt W (ii) xl:fk yw. TP:TTWITIT ti&fichtui ZA X Zfr1%- E(X) = 3 E(X2) = 13, P ( 2 < X < 8)t f41:i f4tatff mta Pc-R 114r4).4 aturgwr whtt R d A. WaR F/Mr t ci, ? 3TO sil-r* cbitul Wd-14R I 10 () ttuiiciic i 1 ifug* tip4 Julia) ivilcigct) t (w) mk.rui To-RP-s* -4174*Trited4 k-c4ci1.101 t I (TI) A at B (-adi Ac (A tkr) B )411 MTE P.T.O.

12 X1, X2,... Xn mirtiondi 1Fica xḇf9' f(x)= 1 a < x < 13 - a = 0, 3 11T iia,( ēq zir-qf 3A-0 t ffq a fit atrt-*--dgr *Tram air*-f min (Xi, X2,... Xn)t I (g) TTRX Rsiv, n2 <COI F-4Zq 1 rs. _n t, x cr) ni, n2 c4ml F-4Zq I MTE ,000