INTERNAL ASSIGNMENT QUESTIONS M.A./M.Com./M.Sc. I YEAR ( )

Transcription

1 INTERNAL ASSIGNMENT QUESTIONS M.A.M.Com.M.Sc. I YEAR (04-05) 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. H.VENKATESHWARLU Hyderabad 7, Telagaa State

2 PROF.G.RAM REDDY CENTRE FOR DISTANCE EDUCATION OSMANIA UNIVERSITY, HYDERABAD Dear Studets, Every studet of M.A.M.Com.M.Sc. I year has to write ad submit Assigmet for each paper compulsorily. Each assigmet carries 0 marks. Uiversity Examiatios will be held for 80 marks. The cocered faculty evaluates these assigmet scripts. The marks awarded to you will be forwarded to the Cotroller of Examiatio, OU for iclusio i the Uiversity Examiatio marks. 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. The assigmet marks will ot be accepted after the stipulated date, You are required to pay Rs.300- fee towards Iteral Assigmet marks through DD (i favour of Director, PGRRCDE, OU) ad submit the same alog with assigmet at the cocered couter o or before 3 rd May, 05 ad obtai proper submissio receipt. ASSIGNMENT WITHOUT THE DD 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:. 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. (0.30 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 a. NAME OF THE STUDENT : b. ENROLLMENT NUMBER : c. Subject M.COM. M.Sc M.A. : D.D Detials.(Bak Name,DD No.Date,Rs).: d. NAME OF THE PAPER : e. DATE OF SUBMISSION : 6. Write the above said details clearly o every subject assigmets paper, otherwise your paper will ot be valued. 7. Tag all the assigmets paper wise ad submit assigmet umber wise. 8. Submit the assigmets o or before at the cocered couter at PGRRCDE, OU o ay workig day ad obtai receipt. Prof.B.APPA RAO JOINT DIRECTOR Prof.H.VENKATESHWARLU DIRECTOR

3 P G R R C D E Osmaia uiversity Assigmet M SC Mathematics (Previous) Paper title: ALGEBRA Paper: I Max.marks:0. Note: The assigmet cosists of two parts PART-A ad PART-B carryig 0marks each. Aswer all the questios from both PART-A ad PART-B. PART-A (5X=0 Marks) (50 Words). Show that every fiite group has a compositio series.. State ad prove Cayley s theorem. 3. State ad prove Gauss lemma. 4. Show that the polyomial is irreducible over Q. 5. S State ad prove Eisestei criterio. Part-B (X5=0 Marks) (50 Words) 6. State ad prove Cauchy s theorem for abelia groups. 7. If R is a commutative rig with uity ad M is a ideal of R the show that M is a maximal ideal of R if ad oly if RM is a field.

4 PGRRCDE, OSMANIA UNIVERSITY, HYDERABAD 7 M.sc ( Mathematics) PREVIOUS, INTERNAL ASSIGNMENT PAPER II ( REAL ANALYSIS) Aswer All Questios Max marks: 0 PART A 5X=0M ) Give a example to show that the arbitrary itersectio of ope sets eed ot be ope i a Metric space. ) Give a example to show that a cotiuous fuctio eed ot be uiformly cotiuous. 3) State the Fudametal Theorem of Itegral Calculus. 4) State Weierstass s M test. 5) State Iverse Fuctio Theorem. PART B X5=0M ) Let ( X, d ) ad ( Y, ρ) be ay two Metric spaces. The show that a mappig f : X Y is cotiuous if ad oly if f ( V ) set V i Y. is ope i X for every ope ) Let γ be a curve o [ a, b] such that γ is cotiuous o [ b] γ dt. that γ is rectifiable ad ( ) = γ ( t) b a a,. The show

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6 PGRRCDE, OSMANIA UNIVERSITY, HYDERABAD 7 M.sc ( Mathematics) PREVIOUS, INTERNAL ASSIGNMENT PAPER IV ( Elemetary Number Theory) Aswer All Questios Max marks: 0 PART A 5X=0M ) Express GC D of 56, 7 as 56 x + 7 y where x, y are itegers. ) Fid φ ( 600), σ ( 600). 3) Solve the cogruece 5x ( mod 6). 9 4) Evaluate ( ) ) Fid p(), p() where p() deotes the umber of partitios of. PART B X5=0M ) a) Let be a fixed positive iteger ad S be the set of all positive itegers which are less tha ad relatively prime to it. What is the sum of all the elemets of S. b) If p is a prime ad if r Ν is such that r < p, the show that p divides p C. r ) Show that the quadratic cogruece x + 0( mod p) where p is a odd prime has a solutio if ad oly p ( mod 4).

7 P G R R C D E Osmaia uiversity Assigmet M SC Mathematics (Previous) Paper title: Mathematical methods Paper: V Max.marks:0. Note: The assigmet cosists of two parts PART-A ad PART-B carryig 0marks each. Aswer all the questios from both PART-A ad PART-B. PART-A (5X=0 Marks) (50 Words). Explai F robeius method of series solutio.. Defie Regular,Sigular,ad Ordiary poits ad hece the same to the equatio 0 3. Show that (i) cos (ii) si. 4. Show that (i) (ii). 5. Express i terms of Legedre polyomials. Part-B (X5=0 Marks) (50 Words) 6. State ad prove orthogoal property of Legedre polyomials. 7. State ad prove Rod rigues formula for Legedre polyomials.

8 FACULTY OF SCIENCE M.Sc. (Previous) CDE : ASSIGNMENT - 05 SUBJECT : STATISTICS PAPER- III: DISTRIBUTION THEORY& MULTIVARIATE ANALYSIS MAX. MARKS: 0 Name of the Studet : Roll No: SECTION-A (0X ½ = 5 Marks) Give the correct choice of the aswer like a or b etc i the brackets provided agaist the questio.. Match the distributio with the property a) log ormal i) M.D. = 45 σ b) Pareto ii) (c)! c) Normal iii) Mea = ak(a-) d) Weibul iv) Media = exp (µ) 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) Noe of these [ ]. Match the distributio with the property a) log ormal i) Mea < Media > Mode b) Pareto ii) Mea > Media > Mode c) Normal iii) Mea < Media< Mode d) Weibul iv) Mea = Media = Mode a) a-iii, b-ii, c-iv, d-i b) a-ii, b-i, c-iv, d-iii c) a-i, b-ii, c-iv, d-iii d) Noe of these [ ] 3. The distributios belog to the expoetial family amog : i) Biomial ii) Negative Biomial iii)hyper- Geometric iv) Uiform v) Gamma vi) Cauchy vii) Normal viii) Beta, is a) i, ii, iii, iv, v b) i, ii, iv, v, vii c) i, ii, v, vii,viii d) Noe of these [ ]3 4. Which of the followig distributio is said to be the distributio of icome (a) Pareto (b) log ormal (c) Normal (d) Noe of these [ ]4 5. If X, & X are idepedet ad follows Logormal with ( μ,σ ) & ( μ,σ ) the X X (a) LN ( μ + μ, σ + σ ) (b) N ( μ + μ, σ + σ ) (c) LN ( μ μ, σ + σ ) (d) N ( μ μ, σ + σ ) [ ]5 6. Idetify the correct MGF for the distributio a) Biomial i) (-qe it ) - b) Gamma ii) (q + p e it ) c) Cauchy iii) exp( - t ) d) Laplace iv) (+t ) - 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) Noe of these [ ]6 7.. If A W P (Σ, m) B W P (Σ, ) (m, p) are idepedet the A A+B follows a) Normal b) W P (Σ, m+) c) Wilks Λ d) Noe of these [ ]7 8. If X N P ( μ, Σ) the the distributio of sample mea vector follows a) N P (μ, Σ) b) N P (μ, Σ) c) N P (μ, Σ) d) Noe of these [ ]8 9. If X N P ( μ, Σ), ad cosider a liear trasformatio Y () =X () +M.X (), Y () =X () with Covariace (Y (), Y () ) the the value for M is. a) Σ Σ - Σ b) -Σ Σ - Σ c) -Σ Σ - d) Noe of these [ ]9 0. If A W P (Σ, ) the E[A] = a) Σ b) Σ c) Σ d) Noe of these [ ] SECTION-B (0 x ½ = 5) Fill i the blaks.. If X follows Stadard Normal the X follows variate.. The sum of two idepedet Expoetial variates is. 3. If X & Y follows Chi-square with ad d.f. idepedetly the the ratio of two chi-squares XY is variate. ( x+ y) 4. If f (x,y) = e ; x 0, y 0 the the margial desity of y is f(y) = 5. The If the Normal desity fuctio f(x)= ½exp{-½{(x -) + (x -) }} the mea vector μ ad covariace matrix Σ of the distributio are 6. I CGF of two-variate ormal distributio, the coefficiets of (t!) ad (t!)are. 7. The proportio of the total variace explaied by the i th pricipal compoet is 8. The mai purpose of caoical correlatio aalysis is to study. 9. I MDS, the measure stress is defied as 0. The Multi Dimesioal Scalig is defied as

9 SECTION-C (0x=0) Aswer the followig questios i brief.. Derive the MGF of stadard Weibul distributio.. If X follows Logormal distributio the derive the distributio of X. 3. State the properties of Laplace distributio. 4. State the importace of Cauchy distributio 5. Defie the Mixture distributio ad illustrate with a suitable example. 6. State the applicatios of Wishart distributio 7. Explai the steps ivolved i the costructio of Path diagram 8. State the Assumptios o Orthogoal Factor Model. 9. How the factor model ca be obtaied from Pricipal compoets model 0. Draw the Dedogram for the followig distace matrix after clusterig usig Sigle likage method D =

10 FACULTY OF SCIENCE M.Sc. (Previous) CDE : ASSIGNMENT - 05 SUBJECT : STATISTICS Paper-I : MATHEMATICAL ANALYSIS & LINEAR ALGEBRA Name of the Studet : Roll No: 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.. If f(z) = 0, the z is kow as of f(z) a) Sigular poit b) Aalytical poit c) Saddle poit d) Pole ( ). 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 ). k = a) < b) > c) < d) > ( ) 3. 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) ( ) 4. I bivariate case, if partial derivatives of a fuctio f(x, y) with respect to x ad y exist at (a, b), the f(x, y) at (a, b). a) is cotiuous b) is discotiuous c) eed ot be cotiuous d) eed ot be discotiuous ( ) 5. A series of the form a o + a (z a) + a (z a) = of (z a) = 0 a ( z a) is called series a) Tailor b) Power c) Lauret d) Aalytical ( ) 6. For ay matrix A, (A A) + is a) (A ) + A + b) A + (A ) + c) A A + d) A + A ( ) 7 For a characteristic root of a matrix there correspods a) Oe characteristic vector b) two characteristic vectors c) Differet characteristic vector d) oe of the above ( ) 8. If the system AX = b is cosistet the the solutio X o = A + b is uique if a) A + = A - b) AA + = b c) A + A = b d) A + A = I ( ) 9. For ay geeralized iverse A - of a mx matrix A ad the matrices A - A ad A A - are a) Idempotet b) orthogoal c) I d) oe of the above ( ) 0. If A is ull matrix of order mx the A + is a ull matrix of order a) mx b) xm c) mxm d) x ( )

11 II. Fill i the blaks. Each questio carries half Mark.. 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.. If a fuctio f is mootoic o [a, b] the f is a fuctio of. 3. Cotour itegral is also kow as itegral. 4. If partial derivatives of a fuctio exist, the the fuctio cotiuous. 5. If f(z) is ot defied at z = a, but lim f ( z) exists, the z a is called sigularity. z a 6. The ecessary ad sufficiet coditio for a liear trasformatio X = PY to preserve legth is that 7. The MP iverse of the traspose of A is the traspose of the 8. I the Hermite form of a matrix, the diagoal elemets of the matrix are 9. A solutio to the system AX = b exists iff the rak of the coefficiet matrix A is equal to the rak of the 0. Let Abe a (x) matrix. The liear homogeous system AX = 0 has a solutio other tha X = 0, iff III. Write short aswers to the followig. Each questio carries TWO Marks.. State Cauchy Residue theorem.. Defie removable discotiuity. 3. Riema-Steiltjes (R-S) Itegral ad its liear properties 4. State additive property ad sufficiet coditios of a fuctio of bouded variatio. 5. Defie statioary poit, extreme poit, saddle poit. Derive them with a example. 6. Compute the orm of [3,-,-,]. 7. Fid a vector orthogoal to [,,0]. 8. Defie idex ad sigature of a real Quadratic form. 9. Defie algebraic ad geometric multiplicity of characteristic roots. 30. Defie Moore Perose iverse ad write ay two properties of the same.

12 FACULTY OF SCIENCE M.Sc. (Previous) CDE : ASSIGNMENT - 05 SUBJECT : STATISTICS PAPER- II: PROBABILITY THEORY Name of the Studet : Roll No: N.B.: Aswer all questios. (a) Give the correct choice of the aswer like a or b etc i the brackets provided agaist the questio, Each questio carries ½ mark:. If { ; } the the sequece { } X is a sequece of radom variables such that P[ X = ] = ad P[ X = 0 ] = - ; X coverges i ( ) (a) probability (b) quadratic mea (c) both (d) Noe. The distributio fuctio of a radom variable X is give by F(x)=0;otherwise.The, the probability desity fuctio x (a) ( + x) e x (b) x ) e e x + x ( (c) ( +) F x ( x) = ( + x) e ; for x 0 (d) ad x e ( ) 3. If A,B,C ζ ad A ad B imply C, the ( ) (a) p ( C ) p( A ) + P( B ) (b) p( C ) p( A ) P( B ) (c) p( C ) p( A ) P( B ) (d) p ( C ) p( A ) + P( B ) 4. The characteristic fuctio of Poisso distributio is ( ) (a) it ( e ) e λ (b) it ( e + ) e λ (c) it ( e ) e λ (d) it ( e + ) e λ 5. If β = E X < the for k we have ( ) k k + k k + k k + (a) β k β k + (b) β k β k + (c) β k β k + (d) β k β k + 6. If a sequece of idepedet radom variables is defied as P [ X = ] = ad P [ X = ] = ; the E [ X ] is ( ) (a) (b) + (c) 0 (d) 7. If X i ~ N ( μ, σ ); i =,,... ; idepedetly ad if S = i = X i the V ( S ) = ( ) (a) σ (b) σ (c) σ (d) σ 8. The radom variables for which Borels strog law of large umbers is defied are ( ) (a) i.i.d. biomial (b) i.i.d. Poisso (c) i.i.d. Beroulli (d) i.i.d. ormal

13 9. For the followig TPM, whe the state space is S={0,,}the the absorbed state is P = 0 0 ( ) 0 (a) 0 (b) (c) (d) oe ( 3 ) 0. The otatio p ij i the Markov chai represets ( ) (a) Probability of reachig the state j from state i i 3 steps (b) Probability of reachig the state i from state j i 3 steps (c) Probability of ot reachig the state j from state i i 3 steps (d) Probability of ot reachig the state i from state j i 3 steps (b) Fill up the blaks, Each questio carries ½ mark:. If φ X (t), t R is a characteristic fuctio of a radom variable X the the characteristic fuctio of Z = (X μ) σ is where μ ad σ are the mea ad variaces of X.. If a sequece of radom variables is coverget i probability the P( X X ε ), as. 3. I terms of Distributio fuctio, P(a X b) = 4. If { X ; } is a sequece of Beroulli radom variables such that P[ X = ] S P[ X = 0] = q the E = 5. If { ; } X is a sequece of large umbers such that S = X i, i= = p ad S p S E[ X i ] = μ i ad V ( X i ) = σ the E as is kow as WLLN s. 6. Accordig to Chebychev s iequality P( X μ k ) 7. Beroulli process is a Stochastic process i which state space is ad idex set is. 8. The SLLN s which is a particular case of Kolmogorov s SLLN s is. 9. A positive recurret ad aperiodic state is kow as state. 0. E X + Y E X + E Y is kow as iequality.

14 Each questio carries mark Aswer the followig questios. Defie distributio fuctio ad Probability desity fuctio.. State Lyapuovs iequality. 3. Defie the mode of covergece i law:- 4. Obtai the characteristic fuctio of Cauchy distributio. 5. State Slutzkys theorem ad Borel 0 law 6. State Kolmogorovs SLLN s ad Kolmogorovs iequality. 7. State Levy Lideberg CLT ad Khitchies WLLN s. 8. I the followig TPM, idetify the closed class, whe the state space is {0,,} = = ; k=,,3, the obtai V( X k ). 9. If P [ X k ± log k] 0. Defie Iitial distributio of a Markov chai P =

15 FACULTY OF SCIENCE M.Sc. (Previous) CDE : ASSIGNMENT - 05 SUBJECT : STATISTICS PAPER IV: SAMPLING THEORY ad ESTIMATION THEORY Name of the Studet : Roll No: Aswer all questios. (a) Give the correct choice of the aswer like a or b etc i the brackets provided agaist the questio, Each questio carries ½ mark:. SRSWOR is always efficiet tha SRSWR. ( ) a) less b) equally c) more d) Noe of the above. I optimum allocatio of stratified radom samplig, h is large if stratum variability ( ) S h is. a) large b) small c) zero d) Noe of the above 3. I ppswr, the ubiased estimator of V( Yˆ pps ) is ( ) yi a) ( ˆ Ypps ) b) yi ( Yˆ pps ) i= pi ( ) i= pi c) yi ( Yˆ pps ) d) Noe of the above i= pi 4. Bias of Rˆ is of the order ( ) a) b) b) d) Noe of the above 5. V( y lrs ) is miimum if b h = B h = ( ) S a) S yh xh S c) S yxh xh S b) S xh yh d) Noe of the above 6. If E( Statistic ) = Parameter the the statistic is kow as ( ) (a) Ubiased estimator (b) Sufficiet estimator (c) Efficiet estimator (d) Cosistet estimator 7. Rao-Blackwell theorem gives ( ) a) MVUE b) UMVUE c) Ubiased ad Cosistet estimator d) Sufficiet estimator

16 8. If X i ~ P ( λ ); i =,,... ; the the estimator of λ ( ) (a) X (b) S (c) s (d) Both (a) ad (b) 9. Let x, x,......,x be a radom sample from N(θ,σ ), both parameters ukow. The the joit sufficiet estimator for (θ,σ ) is i) (Σx i, Σx i ) ii) ( x, s ) iii) (Σx i, (Σx i -θ) ) iv) ( x, (Σx i - θ) ) a) (i) b) (i) ad (iii) c) both (i) ad (ii) d) both (i) ad (iv) ( ) 0. Fisher s iformatio of the radom sample X, X,..... X is writte as d log L d log L a) E[ ] b) E[ ] dθ dθ d log L d log L c) E[ ] = 0 d) [E ] dθ dθ (b) Fill up the blaks, Each questio carries ½ mark: ( ). I Stratified Radom Samplig, the populatio is.. I Systematic samplig, V( y sys ) =. 3. If the total sample size is large the V( Y ˆRC ) =. 4. I cluster samplig, the relatioship betwee S,. Sb ad S w is S = 5. I two-stage samplig, first stage uits ad m secod stage uits from each chose first stage uits are selected by SRSWOR the its samplig variace is give by V( y r ) =. 6. A specified value of the estimator based o the sample is. 7. If X i ~ N ( μ, σ ); i =,,... ; the amog the estimators x ad M d of μ, the more efficiet estimator is. 8. X i ~ b(, p); i =,,... ; the T= x i is a sufficiet statistic for. i = 9. Let x, x,......,x be a radom sample from G(α,β) populatio. The the sufficiet estimator for (α,β) is 0. Let x, x,......,x be a radom sample with pdf f(x, θ) ad T(x) be a ubiased estimate of τ(θ). The Cramer-Rao lower boud for variece of T uder some regularity coditios is

17 Each questio carries mark Aswer the followig questios. Defie ratio estimator ad regressio estimator.. Defie ppswor ad ppswr. 3. Defie cluster samplig ad two stage samplig. Give a example for each. 4. Defie o-samplig error. 5. Write about o-samplig bias. 6. Name the methods of costructig oparametric desity estimators. 7. Defie the Geeralized Jackkife estimator. 8. Defie a complete estimator. 9. State Rao Blackwell theorem. 0. Defie a cosistet estimator ad give a example.

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