INTERNAL ASSIGNMENT QUESTIONS M.Sc. STATISTICS PREVIOUS. ANNUAL EXAMINATIONS June / July 2018

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1 INTERNAL ASSIGNMENT QUESTIONS M.Sc. STATISTICS PREVIOUS ANNUAL EXAMINATIONS Jue / July 018 PROF. G. RAM REDDY CENTRE FOR DISTANCE EDUCATION (RECOGNISED BY THE DISTANCE EDUCATION BUREAU, UGC, NEW DELHI) OSMANIA UNIVERSITY (A Uiversity with Potetial for Excellece ad Re-Accredited by NAAC with A + Grade) DIRECTOR Prof. C. GANESH Hyderabad 7 Telagaa State

2 PROF.G.RAM REDDY CENTRE FOR DISTANCE EDUCATION OSMANIA UNIVERSITY, HYDERABAD Dear Studets, Every studet of M.Sc. Statistics Previous Year has to write ad submit Assigmet for each paper compulsorily. Each assigmet carries 0 marks. The marks awarded to you will be forwarded to the Cotroller of Examiatio, OU for iclusio i the Uiversity Examiatio marks. The cadidates have to pay the examiatio fee ad submit the Iteral Assigmet i the same academic year. If a cadidate fails to submit the Iteral Assigmet after paymet of the examiatio fee he will ot be give a opportuity to submit the Iteral Assigmet afterwards, if you fail to submit Iteral Assigmets before the stipulated date the Iteral marks will ot be added to Uiversity examiatio marks uder ay circumstaces. You are required to submit Iteral Assigmet Aswer Script alog with Examiatio Fee Receipt at the cocered couter o or before 15 th Jue, 018 ASSIGNMENT WITHOUT THE FEE RECEIPT WILL NOT BE ACCEPTED Assigmets o Prited / Photocopy / Typed papers will ot be accepted ad will ot be valued at ay cost. Oly had writte Assigmets will be accepted ad valued. Methodology for writig the Assigmets: 1. First read the subject matter i the course material that is supplied to you.. If possible read the subject matter i the books suggested for further readig. 3. You are welcome to use the PGRRCDE Library o all workig days icludig Suday for collectig iformatio o the topic of your assigmets. (10.30 am to 5.00 pm). 4. Give a fial readig to the aswer you have writte ad see whether you ca delete uimportat or repetitive words. 5. The cover page of the each theory assigmets must have iformatio as give i FORMAT below. FORMAT 1 NAME OF THE COURSE :. NAME OF THE STUDENT : 3. ENROLLMENT NUMBER : 4. NAME OF THE PAPER : 5. DATE OF SUBMISSION : 6. Write the above said details clearly o every assigmets paper, otherwise your paper will ot be valued. 7. Tag all the assigmets paper-wise ad submit 8. Submit the assigmets o or before 15 th Jue, 018 at the cocered couter at PGRRCDE, OU o ay workig day ad obtai receipt. Prof. C. GANESH DIRECTOR

3 M.Sc. STATISTICS - PREVIOUS CDE - ASSIGNMENT Paper-I : MATHEMATICAL ANALYSIS & LINEAR ALGEBRA I. Give the correct choice of the Aswer like a or b etc. i the brackets provided agaist the questio. Each questio carries half Mark. 1. f(x) = 1 if x 0 has at x = 0. 0 if x = 0 a) cotiuity b) removable discotiuity c) irremovable discotiuity d) discotiuity ( ). Derivability of a fuctio at a poit implies cotiuity of the fuctio at that poit. Its coverse is a) ot true b) true c) caot be proved d) true subject to some coditios ( ) 3. A fuctio f is said to be a fuctio of Bouded variatio if there exists a positive umber M such that Δf k M for all partitios o [a, b], where Δf k = f ( xk f k 1 ). k = 1 a) < b) > c) < d) > ( ) 4. If f, g, (f+g), (f-g) ad f.g are fuctios of Bouded variatio o [a, b], the V fg (a,b) A V f (a,b) + B V g (a,b). a) > b) > c) < d) < ( ) 5. Which of the followig coditios are equivalet to Riema Stieltjes a) f ε R(α) o [a,b] b) I ( f, α) = I( f, α) c) either (a) or (b) d) both (a) ad (b) ( ) 6. If A + is Moore - Perose iverse of A, it should satisfy coditios. a) AA + ad A + A should be symmetric b) AA + A = A ad A + AA + =A + c) both (a) ad (b) d) either (a) or (b) ( ) 7. If A is a mx matrix, its geeralized iverse is of order a) xm b) mx c) mxm d) x ( ) 8. A vector X is said to be orthogoal to oother vector Ý if the ier product of X ad Y is a) positive b) egative c) zero d) All the above three ( )

4 9. If V (F) is a vector space ad the set S = {X 1, X,....., X } is a fiite set of vectors i V (F), the the set S is called a basis of subspace if a) S is liearly idepedet b) S spas V (F) c) both (a) ad (b) d) either (a) or (b) ( ) 10. A system of liear equatios AX = b is said to be cosistet iff a) AA + b = b b) AA b = b c) either (a) or (b) d) either (a) or (b) ( ) II. Fill i the blaks. Each questio carries half Mark. 11. A fuctio f is said to ted to a limit Ç as x teds to h if the left limit ad right limit exist ad are. 1. If F : [a, b] ε R is a fuctio ad a = x o < x 1 < x < <x = b, the the set of poits P = { x o, x 1, x,....., x } is called. 13. If a fuctio f is mootoic o [a, b] the f is a fuctio of. 14. If V f (a,b) = Sup{ΣΡ, P ε P[a,b]} is kow as of the fuctio f o [a,b]. 15. The fuctio f is said to satisfy coditio with respect to α, if for every > 0 there exists a partitio P such that for every fier partitio P P, we have 0 < U(P, f, α) L(P, f, α) <. 16. Every Moore - Perose iverse is also iverse. But, the coverse eed ot be true. 17. Rak of Moore - Perose iverse of a matrix A is the rak of matrix A. 18. If A is a mx matrix, A or A c is called its geeralized iverse if. 19. The umber of liearly idepedet vectors i ay basis of a sub-space S is kow as of the sub-space. 0. A system of liear equatios AX= b is said to be cosistet iff where B is the augmeted matrix.

5 III. Write short aswers to the followig. Each questio carries ONE Mark. 1. Defie removable discotiuity.. State Secod Mea Value theorem. 3. State the properties of trace of a matrix. 4. State the properties of geeralized iverse. 5. Defie legth of a vector of elemets. 6. Defie ier product of two vectors X ad Y of order x1. 7. Defie homogeeous ad No-Homogeeous system of equatios. 8. State the properties of trace of a matrix. 9. State the properties of geeralized iverse. 30. Defie Algebraic ad Geometric multiplicity of characteristic root.

6 M.Sc. STATISTICS - PREVIOUS CDE - ASSIGNMENT PAPER- II : PROBABILITY THEORY I. Give the correct choice of the Aswer like a or b etc. i the brackets provided agaist the questio. Each questio carries half Mark. 1. If the cumulative distributio fuctio of a radom variable X is F(x) = 0 if x < k ad F(x) = 1; if x k. The F(x) is called distributio. a) Uiform b) Beroulli c) Degeerate d) Discrete Uiform ( ). If A ad B are two idepedet evets such that P(A c ) = 0.7, P(B c ) = x ad P(A B) = 0.8 the x = a) 0.1 b) /7 c) 5/7 d) 1/3 ( ) 3. A experimet is said to be a radom experimet if a) All possible outcomes are kow i advace b) Exact outcome is kow i advace c) Exact outcome is ot kow i advace d) Both (a) ad (c) ( ) 4. If the occurrece of oe evet prevets the occurrece of all other evets the such a evet is kow as evet. a) Mutually exclusive b) Idepedet c) Equally likely d) Favorable ( ) 5. If A ad B ξ ad A B the P(A) P(B) a) > b) < c) d) ( ) 6. If A ad B are two idepedet evets the a) A ad B c are also idepedet b) A c ad B are also idepedet c) A c ad B c are also idepedet d) All the above ( ) 7. Match the followig iequalities a) Chebychev s iequality i) E( XY ) E( X ) E( Y ) b) Markov s iequality ii) φ (E (X)) E (φ (X)) c) Jese s iequality iii) P [ X ε] E(X p )/ ε p d) Schwartz iequality iv) P [ X ε] E(X )/ ε a) a-iii, b-iv, c-ii, d-i b) a-iv, b-iii, c-ii, d-i c) a-iv, b-iii, c-i, d-ii d) a-iii, b-iv, c-i, d-ii ( )

7 8. Match the followig iequalities a) Holder s iequality i) E( X+Y p ) 1/p (E X p ) 1/p (E Y p ) 1/p b) Liapuov s iequality ii) E( XY ) (E X p ) 1/p (E Y q ) 1/q, p >1 c) Triagular iequality iii) E( X r ) 1/r (E X p ) 1/p, r > p > 0 d) Mikowski s iequality iv) E( X+Y ) 1/ (E X ) 1/ + (E Y ) 1/ a) a-iii, b-iv, c-ii, d-i b) a-iv, b-iii, c-ii, d-i c) a-iv, b-iii, c-i, d-ii d) a-iii, b-iv, c-i, d-ii ( ) 9. Match the followig Characteristic fuctios a) Biomial i) p(1-qe it ) -1 b) Geometric ii) (q + p e it ) c) Cauchy iii) exp( - t ) d) Laplace iv) (1+t ) -1 a) a-i, b-ii, c-iii, d-iv b) a-ii, b-i, c-iv, d-iii c) a-i, b-ii, c-iv, d-iii d) a-ii, b-i, c-iii, d-iv ( ) 10. If X follows U(0,1) the P[ X-6 >4] a) 0.75 b) c) 0.5 d) Noe of these ( ) II. Fill i the blaks. Each questio carries half Mark. 1. If the umber of items produced durig a week is a radom variable with mea 00. The probability for weeks productio will be at least 50 is. The joit p.d.f. of (X,Y) is give by f(x, y) = 0 < y < x the the f ( y / X=x) is 3. For ay characteristic fuctio φ x (t), the real part of (1-φ x (t)) 4. Borels SLLN is defied for radom variables. 5. The WLLN s defied for Beroulli radom variables is kow as 6. Demoivre s Laplace CLT is defied for radom variables. 7. SLLN s is a particular case of Kolmogorov s SLLN s. 8. Bocher s stated that, the ecessary ad sufficiet coditio for φ x (t) to be characteristic fuctio is. 9. The variace of X U(5,9) is 10. If f(x) is a covex fuctio ad E(X) is fiite the f [E(X)] E[f (X)] this is kow as iequality.

8 III. Write short aswers to the followig. Each questio carries ONE Mark. 1. Give the Kolmogorov s defiitio to the probability.. State Iversio theorem of characteristic fuctio. 3. State the Uiqueess ad iversio theorems for characteristic fuctio. 4. State Holder s iequality. 5. Defie Weak ad Strog Law of Large umbers. 6. Defie covergece i Probability ad Covergece i Quadratic mea. 7. Show that covergece i probability implies covergece i law. 8. State the Levy cotiuity theorem ad give its applicatio. 9. State Liapuov Cetral Limit Theorem. 10. State Lidberg Feller Cetral Limit Theorem.

9 FACULTY OF SCIENCE M.Sc. (STATISTICS) CDE PREVIOUS, INTERNAL ASSESMENT, MAY 018 PAPER-III : DISTRIBUTION THEORY & MULTIVARIATE ANALYSIS Time: 60 Mi Max. Marks:0 Name of the Studet Roll No: Note: Aswer Sectio-A & B o the Questio paper by takig prit. Aswer the questios i Sectio C i the order o white papers. SECTION-A ( Multiple Choice : 10 x ½ = 5 Marks) 1. Whe 1 =1, = ad F = t the F distributio teds to. (a) χ distributio (b) t distributio (c) F (,1) distributio (d) Noe [ ]. The ratio of No-cetral χ variate to the cetral χ variate divided by their respective degrees of freedom is defied as (a) No-cetral χ (b) No-cetral t (c) No-cetral F (d) Noe [ ] 3. Distributio fuctio of miimum order statistics is. (a) [F(x)] (b) 1-[1-F(x)] (c) [1-F(x)] (d) 1+[1-F(x)] [ ] 4. The Distributio of Quadratic forms is (a) χ distributio (b) t distributio (c) F distributio (d) Noe [ ] 5. The coditioal desity fuctio of Multi-omial P[X 1 =u / X =v]= a) -v C u [ p 1 /(1-p ) ] u [ (1-p -p 1 )/(1-p ) ] -u-v b) (π) -p/ Σ -½ 1 -½ (X-μ) Σ e (X-μ) c) -v C u [ p 1 /(1-p 1 -p ) ] u -v C u [ p /(1-p 1 -p ) ] -u d) Noe of these [ ] 6. The Probability desity fuctio of Wishart distributio is a) (π) -p/ Σ -½ 1 -½ (X-μ) Σ e (X-μ) b) (π) -p/ Σ -½ 1 -½ (X-μ) Σ e (X-μ) c) (π) -p/ Σ -/ 1 -½ (X-μ) Σ e (X-μ) d) Noe of these [ ] 7. The Characteristic Fuctio of Wishart Distributio is a) [ Σ / Σ-it ] / b) [ Σ -1 / Σ -1 +it ] / c) [ Σ -1 / Σ -1 -it ] / d) Noe of these [ ] 8. If X N P ( μ, Σ), ad Y (1) =X (1) +M.X (), Y () =X () be the a liear trasformatio such that Y (1), Y () are idepedet the the value of M is. a) Σ 1 Σ -1 Σ 1 b) -Σ 1 Σ -1 Σ 1 c) -Σ 1 Σ -1 d) Noe of these [ ] 9 If X N P (0, I p ), cosider the trasformatio Y= BX, the Bartlett s decompositio matrix (B), elemets b ii follows distributio a) Normal b) Wishart c) Chi-square d) F [ ] 10 The correlatio betwee the i th Pricipal Compoet (Y i ) ad the k th variable(x k ) is a) 0 b) 1 c) 1/ d) Noe of these [ ] P.T.O.

10 SECTION-B (Fill i the Blaks: 10 x ½ = 5 Marks) 1. Whe =, t- Distributio teds to distributio.. (+λ) is the mea of distributio. 3. If X i ~ N(μ i, 1); i =1,,3,, μ i 0 idepedetly the X i ~ distributio. 4. If X 1, X, X 3 ~ exp(1) the the distributio fuctio of Maximum ordered statistics is. 5. The Correlatio coefficiet betwee the two-variates of Multiomial is 6. I case of ull distributio, probability desity fuctio for simple sample correlatio coefficiet (r ij ) is f (r ij ) =. 7. I case of ull distributio, the probability desity fuctio for Multiple correlatio coefficiet R is f(r ) =. 8. The Geeralized Variace S is defied as 9. If X N P (μ, Σ) the the distributio of sample mea vector f(x) = 10. If X N P (μ, Σ), ad cosider a liear trasformatio Y (1) = X (1) +M.X (), Y () =X () with Covariace (Y (1), Y () ) the the variace of Y (1) is i= 1 SECTION-C (5x1=5 Marks) (Aswer the followig questios i the order oly) 1. Defie order statistics ad give its applicatios. Defie o-cetral t- ad F- distributios 3. Fid the distributio of ratio of two chi-square variates i the form X/(X+Y) 4. State the physical coditios of Multi-omial distributio 5. Obtai the Margial distributio of Mutiomial Variate. 6. State the applicatios of distributio of Regressio coefficiet. 7. State the Properties of Wishart distributio. 8. Obtai the Covariace betwee two multi-ormal variates from its CGF. 9. Defie Caoical variables ad caoical correlatios 10. Explai the procedure for obtaiig the Pricipal compoets. P.T.O.

11 M.Sc. STATISTICS - PREVIOUS CDE - ASSIGNMENT : SAMPLING THEORY & THEORY OF ESTIMATION I. Give the correct choice of the Aswer like a or b etc. i the brackets provided agaist the questio. Each questio carries half Mark. 1. I stratified radom samplig with optimum allocatio, h is large if stratum variability S h is a) zero b) small c) large d) oe of the above ( ). I two stage samplig with equal first stage uits, variace of sample mea per secod stage uits is give by 1 f a) 1 1 f + m S b b) S b S (1 f 1 ) + w m c) (1 f 1 ) + (1 f ) d) Noe of the above ( ) 3. I PPSWOR, Horwitz Thomso estimator of populatio total Y is defied by N Yi yi a) b) c) i= 1 Pi i= 1 Pi i= 1 π i 4. PPSWOR is efficiet tha PPSWR. y i (1 f ) d) Noe of the above ( ) a) more b) less c) equally d) Noe of the above ( ) 5. The errors arisig at the stages of ascertaimet ad processig of data are termed as a) Samplig errors b) group A errors c) No samplig errors d) Noe ( ) 6. If T 1 (x) ad T (x) be two ubiased estimators of θ, E(T 1 (x)) < ad E(T (x)) <, the efficiecy of T 1 (x) relative to T (x) is deoted by e ad is defied as V ( T ( x)) V ( T1( x)) a) e = b) e = V ( T ( x)) V ( T ( x)) 1 c) e = V T ( x)) V ( T ( )) d) e = V T ( x)) + V ( T ( )) ( ) ( 1 x 7. I Cramer Rao iequality Var(T(x)). a) ( ψ ( θ )) I x ( θ ) b) 1 ( ψ ( θ )) I x ( θ ) ( 1 x I c) x ( θ ) I d) x ( θ ) 1 ( ) ( ψ ( θ )) ( ψ ( θ )) 8. Let x 1, x,,x be a radom sample from P(λ) populatio. Method of momet estimator of λ is x a) b) x c) x d) x + ( )

12 9. Cofidece iterval for chebychev s iequality is give by T(x) ±.. a) E ( E( T ( x)) + θ ) b) E ( E( T ( x)) θ ) c) E ( E( T ( x) + θ ) d) E ( E( T ( x) θ ) ( ) 10. Roseblatt s aïve estimator, optimum bad width h is give by a) C1 4C 0 1/ 5 1/ 5 b) C1 4C 0 1/ 5 + 1/ 5 c) C0 4 C1 1/ 5 1/ 5 d) C0 4 C1 1/ 5 + 1/ 5 ( ) II. Fill i the blaks. Each questio carries half Mark. 11. SRSWOR is the techique of selectig a sample i such a way that each of the N C sample has a equal probability of beig selected. 1. The approximate variace of the ratio estimator of populatio total is give by. 13. The liear regressio estimate of populatio total is give by. 14. The relatioship betwee S, S b ad S w is give by S =. 15. I PPSWOR, the estimate of Yates ad Grudy form of variace of Y HT =. 16. Statistic is a fuctio of observatios. 17. A sufficiet statistics for θ i a U[0, θ ] distributio is. 18. Jackkife ad Bootstrap are kow as techiques. 19. MLE s are estimators. 0. {f (x) ; 1} is said to be asymptotically ubiased if, for every x ad f(x), Lim E f (x) f (x). III. Write short aswers to the followig. Each questio carries ONE Mark. 1. Defie ratio estimator i stratified radom samplig.. Defie regressio estimator i simple radom samplig. 3. Defie cluster samplig with a example. 4. Defie sub samplig with a example. 5. Defie PPSWOR ad PPSWR. 6. Explai Rao-Blackwellizatio. 7. Explai Bootstrap method. 8. State the properties of ML estimator. 9. Defie CAN ad BAN estimators. 30. Describe Shortest legth CI estimatio method. F