MATHEMATICAL SCIENCES PAPER-II

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1 MATHEMATICAL SCIENCES PAPER-II. Let {x } ad {y } be two sequeces of real umbers. Prove or disprove each of the statemets :. If {x y } coverges, ad if {y } is coverget, the {x } is coverget.. {x + y } coverges to x+y if {x } coverges to x ad {y } coverges to y. 3. If{x /y } is coverget, the both {x } ad {y } are coverget. 4. If {x } is coverget ad {y } is diverget, the {x y } is diverget.. Let f : [a,b] be differetiable. (a) Prove that b f () t dt= 0iff f 0 o[ a, b].. a If b a f 3 () t dt= 0, the for some t 0 ε[a,b], either ' f ( t 0) = 0 or 0 f( t ) = Let {f } be a sequece of real-valued Lebesgue itegrable fuctios o such that f <. Prove that for ay α, e iα f( x) coverges a.e. to a fuctio = g α (x) ad that g α is Lebesgue itegrable. 4. Let a = (a, a, L, a ) x. a x. a. = ad let f(x) = e x.a where x = (x, x L, x ), ad = i i Compute the directioal derivative of f at a poit p i= directio of h. 5. (a) Let A = {(x,y) Prove that A is compact. x + y < 37} {(x,y) e x > y} i the Show that ay covex subset C of is coected. 6. Show that the set {log p p prime umber} is liearly idepedet over. 7. Let V be the vector space of all polyomial fuctios of degree <, >, ad let D : V V deote the derivative map P a P o V. Show that D is ilpotet ad that D is ot diagoalizable. 8. Let A ad B be two 3 3 complex matrices. Show that A ad B are similar if ad oly if χ A = χ B ad µ A = µ B, where χ A, χ B are characteristic polyomials of A, B, respectively ad µ A, µ B are miimal polyomials of A, B, respectively.

2 9. Determie whether the quadratic form q(x,x ) = 7X + 88X X + 88π degeerate or ot. X is 0. Let ϕ : Ω be a cotiuous fuctio ad suppose {z : z < } Ω. Prove that ϕ( w) the fuctio defied as f ( z) = dw is a aalytic fuctio o the ope uit w z w = disc.. Let f : be a etire fuctio ad that all such fuctios. f ' ( z) < f( z) for all z. Idetify. (a) Let f : be a aalytic fuctio ad g: a harmoic fuctio. Prove that gof is a harmoic fuctio. Prove that the fuctio xy uxy (, ) = e. cosx ( y), ( xy, ) is harmoic, ad fid the harmoic cojugate. 3. Compute z = e z e dz, where the circle is parametrized by t e it, 0 < t < π. 4. Let ϕ ad µ deote the Euler totiet fuctio ad Möbius fuctio respectively. ϕ ( ) µ ( k) Show that = ϕ( d). Hece show that =. k d/ k/ 5. Defie cojugacy class i a fiite group G ad show that the cardiality of ay cojugacy class divides the order of G. Use this to show that if p is a prime ad G is a group of order p, the the cetre of G cotais elemets other tha the idetity. 6. Let I, I, J be ideals i a commutative rig A. If I, J are comaximal, i.e., I + J = A ad I, J are comaximal, i.e., I + J = A, the show that I I ad J are also comaximal. 7. Let LK be a fiite field extesio of prime degree p. Show that L = K[α] for ay α L\K. 8. Solve the BVP by determiig the appropriate Gree s fuctio, expressig the solutio as a defiite itegral. -y = f(x), y(0) + y (0) = 0, y() + y () = 0.

3 9. Cosider the iitial value problem : y = f(x,y), y(0) = where f(x,y) : = xy + y, (x,y) D with D : = [-,] X [-,3]. Show that f (x,y) is bouded ad statisfies a Lipschitz coditio with respect to y o D ad determie a boud ad Lipschitz costat o D. Further, determie h, as required i the Picard s Theorem, for a uique solutio of the iitial value problem to exist o x < h. 0. Outlie briefly the three classes of itegrals of the o-liear first order partial z z differetial equatio f(x,y,z,p,q) = 0, where p=, q=. x y For the partial differetial equatio pqz = p (3p +qx) + q (py + 4q ), obtai oe of the itegrals ad idicate the procedure for determiig the remaiig two itegrals.. Classify ad reduce the secod order partial differetial equatio u xx 4x u yy = x u x ito caoical form ad hece, fid the geeral solutio.. Derive Simpso s a+ 3 rd rule to evaluate the itegral a h f ( xdx ). Estimate the error. 3. Fid the eigevalues ad the eigefuctios of the fuctioal ' J( y) = ( y + y ) dx subject to thecoditios y(0) = y() = 0, y dx= Fid the resolvet kerel for the itegral equatio ϕ() s = f() s + λ ( st+ s t ) ϕ() t dt 5. Show that the trasformatio q Q= ta, P= q + p p is caoical. Fid a geeratig fuctio Let X ad Y be two idepedet radom variables such that X is uiformly distributed o [0, ] ad Y has a discrete uiform distributio o {0,,,L, }, that is,, if k= 0,, L,, PY ( = k) = 0, otherwise. Defie Z = X + Y. Show that Z is uiformly distributed o [0, ].

4 7. Let M( g) deote the momet geeratig fuctio of the stadard ormal distributio. Let I(a) = sup{ta log M(t): t }. (i) Fid I( g) (ii) Express log M( g) i terms of I( g) 8. Usig the cetral limit theorem for appropriate Poisso radom variables show that j lim e =. j! j= 0 9. Let {X } be a Markov chai with trasitio probability matrix P give by / /4 /4 0 /3 /3 0 0 P =. 0 0 /5 4/5 0 0 / / Let p = P(X = j X =i). Fid lim p for all i,j. ij 0 ij 30. A coi with probability p for head is tossed. If a tail turs up, a radom umber of balls are added to a ur. (Assume that the ur is iitially empty). This procedure is repeated till a head appears at which stage it is stopped. Let N deote the umber of stages whe balls are added, ad X i = umber of balls added at i th stage. Assume that {X i } are i.i.d. Poisso (λ) radom variables, ad that N ad {X i } are idepedet. Fid the expected umber of balls i the ur whe the procedure termiates. 3. Let x, x L, x be the values of a variable x. Defie x max = max{x, L, x }, x mi = mi {x, L, x }, R = xmax xmi ad R R Show that s. 4 = i i= s ( x x) /. 3. Let T be the miimum variace ubiased estimator (MVUE) of θ. The prove that T K ( K a +ve iteger) is the MVUE for E(T K ) provided E(T K ) <. 33. Suppose (x, y ),L,(x, y ) represet a radom sample from N(0,0, σ, σ, ρ ). Suppose ρ = ρ0 (kow), the fid a cofidece iterval of σ/ σ with cofidece coefficiet ( α) that icorporates the iformatio that ρ = ρ Let X, X, L, X be i.i.d. with desity θ f( x, θ) =, x> θ, θ> 0. x (a) Fid MLE of θ Derive the likelihood ratio test for H 0 : θ= vs H : θ. (c) If =4 ad the observatios are X = 3., X = 4.0, X 3 =.0, X 4 = 5.6, fid the P-value of the test derived i.

5 35. Let X,L, X be idepedet radom variables with commo probability distributio fuctio 0 if x < 0 x α PX [ i x; α, β] = ( ) if 0 x β β if x > β where α, β > 0. (a) Fid a two dimesioal sufficiet statistic for (α, β) Fid a ubiased estimator of whe β=. α Cosider a regressio model Y i = θ 0 + θ x i + ε i, i=,, where if i =, L,, xi = 0, if i = +, L, ad ε i are ucorrelated radom errors with mea 0 ad commo variace σ. Let T ad T be the two estimators of θ give by T = Y - Y ad T = Y Y where Y = Y. (a) i = i Verify whether T ad T are ubiased ad fid their variaces. If possible, propose a ubiased estimator of θ, which has variace smaller tha that of T ad T, with justificatio. 37. Cosider a liear model Y = Xθ + ε where Y is a 4 vector of observatios, θ = ( θ, θ, θ ) T is a vector of ukow parameters, X 4 3= ad ε is a 4 vector of ucorrelated radom errors with mea 0 ad variace σ. (a) Verify whether the followig parametric fuctios are estimable (i) θ + θ, (ii) θ + θ + θ 3 Fid the best liear ubiased estimator(s) of the estimable parametric fuctio(s) i (a) above ad obtai the variace of the estimator(s).

6 x( p ) µ 38. Suppose x( p ) = ~ Np, with = x ( p ) µ > 0. Prove that the ecessary ad sufficiet coditio for x ad x to be idepedet is = 0. You may assume xi ~ Np ( µ, ),,. i i ii i= 39. Suppose the problem is to classify a observatio x ito oe of the populatios Pi, i=,. Suppose f i ( x ) deotes the desity of x correspodig to populatio P i. Also we attach the prior probability p i (i =, ) for a observatio x to belog to populatio P i. Fid the total probability of misclassificatio (TPM) ad prove that the classificatio rule miimizig TPM is give by: f( x) p for a x, if classify it as a observatio belogig to populatio P ad f( x) p otherwise belogig to populatio P. 40. A ukow umber N of taxis plyig i a tow are supposed to be serially umbered from to N. If the differet taxis you have come across i the tow ca be assumed to form a simple radom sample with replacemet, fid a ubiased estimator of the total umber of taxis i the tow. Also fid the variace of your estimator. 4. Show that i a radomized block desig the estimates of the elemetary block ad treatmet cotrasts are orthogoal observatioal cotrasts. 4. Suppose i a 5 factorial experimet with factors A, B, C, D ad E, a replicate is divided ito four blocks of size eight each. How may effects will be cofouded? Is it possible to cofoud the effects AB, BC ad ABC? Justify your aswer. 43. The daily demad for a commodity is approximately 00 uits. Every time a order is placed, a fixed cost of Rs.0,000/- is icurred. The daily holdig cost per uit ivetory is Rs./-. If the lead time is 5 days, determie the ecoomic lot size ad the reorder poit. Further suppose that the demad is actually a approximatio of a probabilistic distributio i which the daily demad is ormal with mea µ = 00 ad s.d. σ = 0. How would you determie the size of the buffer stock such that the probability of ruig out of stock durig lead time is at most 0.0?

7 44. Cosider the followig liear program (LP) max z = 4x + 4x Subject to x + 7x + x 3 = 7x + x + x 4 = x, x, x 3, x 4 0. Each of the followig cases provides a iverse matrix ad its correspodig basic variables for the LP above. Determie whether or ot each basic solutio is feasible. Iterpret these basic feasible solutios ad hece fid a optimal solutio. Is the optimal solutio uique? (a) (x, x 4 ); (x, x 4 ); (c) (x, x ); Cosider a M/M/c queuig system with parameters λ ad µ. Draw its statetrasitio rate diagram ad fid the steady-state probability distributio for umber of customers i the system.