Mathematical Statistics - MS

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1 Paper Specific Istructios. The examiatio is of hours duratio. There are a total of 60 questios carryig 00 marks. The etire paper is divided ito three sectios, A, B ad C. All sectios are compulsory. Questios i each sectio are of differet types.. Sectio A cotais a total of 0 Multiple Choice Questios (MCQ). Each MCQ type questio has four choices out of which oly oe choice is the correct aswer. Questios Q. Q.0 belog to this sectio ad carry a total of 50 marks. Q. Q.0 carry mark each ad Questios Q. Q.0 carry marks each.. Sectio B cotais a total of 0 Multiple Select Questios (MSQ). Each MSQ type questio is similar to MCQ but with a differece that there may be oe or more tha oe choice(s) that are correct out of the four give choices. The cadidate gets full credit if he/she selects all the correct aswers oly ad o wrog aswers. Questios Q. Q.40 belog to this sectio ad carry marks each with a total of 0 marks. 4. Sectio C cotais a total of 0 Numerical Aswer Type (NAT) questios. For these NAT type questios, the aswer is a real umber which eeds to be etered usig the virtual keyboard o the moitor. No choices will be show for these type of questios. Questios Q.4 Q.60 belog to this sectio ad carry a total of 0 marks. Q.4 Q.50 carry mark each ad Questios Q.5 Q.60 carry marks each. 5. I all sectios, questios ot attempted will result i zero mark. I Sectio A (MCQ), wrog aswer will result i NEGATIVE marks. For all mark questios, / marks will be deducted for each wrog aswer. For all marks questios, / marks will be deducted for each wrog aswer. I Sectio B (MSQ), there is NO NEGATIVE ad NO PARTIAL markig provisios. There is NO NEGATIVE markig i Sectio C (NAT) as well. 6. Oly Virtual Scietific Calculator is allowed. Charts, graph sheets, tables, cellular phoe or other electroic gadgets are NOT allowed i the examiatio hall. 7. The Scribble Pad will be provided for rough work. MS /7

2 R R M T f P(E) E(X) All agles are i radia Set of all real umbers Special Istructios/Useful Data {(x, x,, x ): x i R, i } Traspose of the matrix M Derivative of the fuctio f Probability of the evet E Expectatio of the radom variable X Var(X) Variace of the radom variable X i.i.d. U(a, b) Φ(a) Γ(p)! Idepedetly ad idetically distributed Cotiuous uiform distributio o (a, b), < a < b < The gamma fuctio The factorial fuctio π a e x / dx Γ(p) = e t t p dt, p > 0 0! = ( ) MS /7

3 Q. Q.0 carry oe mark each. SECTION A MULTIPLE CHOICE QUESTIONS (MCQ) Q. Let {a } be a sequece of real umbers such that a = ad, for, The (A).5 a, for all atural umber a + = a + a +. (B) there exists a atural umber such that a > (C) there exists a atural umber such that a <.5 (D) there exists a atural umber such that a = + 5 Q. The value of is lim ( + ) e (A) e (B) e (C) e (D) e Q. Let {a } ad {b } be two coverget sequeces of real umbers. For, defie u = max{a, b } ad v = mi{a, b }. The (A) either {u } or {v } coverges (B) {u } coverges but {v } does ot coverge (C) {u } does ot coverge but {v } coverges (D) both {u } ad {v } coverge Q.4 Let M = [ ]. If I is the idetity matrix ad 0 is the zero matrix, the (A) 0 M M + 7 I = 0 (B) 0 M M 7 I = 0 (C) 0 M + M + 7 I = 0 (D) 0 M + M 7 I = 0 MS /7

4 Q.5 Let X be a radom variable with the probability desity fuctio α p f(x) = { Γ(p) e αx x p, x 0, α > 0, p > 0, 0, otherwise. If E(X) = 0 ad Var(X) = 0, the (α, p) is (A) (, 0) (B) (, 40) (C) (4, 0) (D) (4, 40) Q.6 Let X be a radom variable with the distributio fuctio 0, x < 0, 4x x F(x) = { +, 0 x <, 4 8, x. The P(X = 0) + P(X =.5) + P(X = ) + P( X ) equals (A) 8 (B) 5 8 (C) 7 8 (D) Q.7 Let X, X ad X be i.i.d. U(0, ) radom variables. The E ( X +X X +X +X ) equals (A) (B) (C) (D) 4 Q.8 Let x = 0, x =, x =, x 4 = ad x 5 = 0 be the observed values of a radom sample of size 5 from a discrete distributio with the probability mass fuctio θ, x = 0, θ f(x; θ) = P(X = x) =, x =, θ {, x =,, where θ [0, ] is the ukow parameter. The the maximum likelihood estimate of θ is (A) 5 (B) 5 (C) 5 7 (D) 5 9 MS 4/7

5 Q.9 Cosider four cois labelled as,, ad 4. Suppose that the probability of obtaiig a head i a sigle toss of the i th coi is i, i =,,, 4. A coi is chose uiformly at radom ad flipped. 4 Give that the flip resulted i a head, the coditioal probability that the coi was labelled either or equals (A) 0 (B) 0 (C) 0 (D) 4 0 Q.0 Cosider the liear regressio model y i = β 0 + β x i + ε i ; i =,,,, where ε i s are i.i.d. stadard ormal radom variables. Give that x i =., i= y i = 4., i= (x j x i) j= i= (x j x i) (y j y i) =.7, j= the maximum likelihood estimates of β 0 ad β, respectively, are i= i= =.5 ad (A) 7 ad 5 75 (C) 7 4 ad 5 75 (B) 7 ad 75 5 (D) 4 7 ad 75 5 MS 5/7

6 Q. Q. 0 carry two marks each. Q. Let f: [, ] R be defied by f(x) = x + [si πx], where [y] deotes the greatest iteger less + x tha or equal to y. The (A) f is cotiuous at,0, (B) f is discotiuous at, 0, (C) f is discotiuous at,, 0, (D) f is cotiuous everywhere except at 0 Q. Let f, g R R be defied by f(x) = x (A) f(x) = g(x) for more tha two values of x (B) f(x) g(x), for all x i R (C) f(x) = g(x) for exactly oe value of x (D) f(x) = g(x) for exactly two values of x cos x ad g(x) = x si x. The Q. Cosider the domai D = { (x, y) R : x y } ad the fuctio h: D R defied by The the miimum value of h o D equals h((x, y)) = (x ) 4 + (y ) 4, (x, y) D. (A) (B) 4 (C) 8 (D) 6 Q.4 Let M = [X Y Z] be a orthogoal matrix with X, Y, Z R as its colum vectors. The Q = X X T + Y Y T (A) is a skew-symmetric matrix (B) is the idetity matrix (C) satisfies Q = Q (D) satisfies QZ = Z MS 6/7

7 Q.5 Let f: [0, ] R be defied by Now, defie F: [0, ] R by The (A) F is differetiable at x = ad F () = 0 (B) F is differetiable at x = ad F () = 0 (C) F is ot differetiable at x = (D) F is differetiable at x = ad F () = 0, 0 x <, f(x) = { e x e, x < e x +, x. x F(0) = 0 ad F(x) = f(t)dt, for 0 < x. 0 Q.6 If x, y ad z are real umbers such that 4 x + y + z = ad x + 4 y z = 9, the the value of 9 x + 7 y + z (A) caot be computed from the give iformatio (B) equals 8 (C) equals 8 (D) equals 8 Q.7 Let M = [ ]. If x V = {(x, y, 0) R : M [ y] = [ 0 x 0 0 ]} ad W = {(x, y, z) R : M [ y] = [ 0 z 0 ]}, the (A) the dimesio of V equals (B) the dimesio of W equals (C) the dimesio of V equals (D) V W = {(0,0,0)} MS 7/7

8 Q.8 Let M be a o-zero, skew-symmetric real matrix. If I is the idetity matrix, the (A) M is ivertible (B) the matrix I + M is ivertible (C) there exists a o-zero real umber α such that αi + M is ot ivertible (D) all the eigevalues of M are real Q.9 Let X be a radom variable with the momet geeratig fuctio M X (t) = 6 π et /, t R. The P(X Q), where Q is the set of ratioal umbers, equals (A) 0 (B) 4 (C) (D) 4 Q.0 Let X be a discrete radom variable with the momet geeratig fuctio The M X (t) = ( + et ) ( + e t ), t R. 04 (A) E(X) = 9 4 (C) P(X ) = 7 04 (B) Var(X) = 5 (D) P(X = 5) = 04 Q. Let {X } be a sequece of idepedet radom variables with X havig the probability desity fuctio as x f (x) = { / Γ( ) e x ( ), x > 0, 0, otherwise. The equals lim [P (X > 4 ) + P( X > + )] (A) + Φ() (B) Φ() (C) Φ() (D) Φ() MS 8/7

9 Q. Let X be a Poisso radom variable with mea. The E((X + )!) equals (A) e (B) 4 e (C) 4 e (D) e Q. Let X be a stadard ormal radom variable. The P(X X X + > 0) equals (A) Φ() (C) Φ() Φ() (B) Φ() (D) Φ() Φ() Q.4 Let X ad Y have the joit probability desity fuctio, 0 x y, f(x, y) = { 0, otherwise. Let a = E(Y X = ) ad b = Var(Y X = ). The (a, b) is (A) ( 4, 7 ) (B) ( 4, 48 ) (C) ( 4, 7 ) (D) ( 4, 48 ) Q.5 Let X ad Y have the joit probability mass fuctio m +, m =,,; =,, P(X = m, Y = ) = { 0, otherwise. The P(X = Y = ) equals (A) (B) (C) (D) 4 Q.6 Let X ad Y be two idepedet stadard ormal radom variables. The the probability desity fuctio of Z = X Y is / (A) f(z) = e { z, π z > 0, 0, otherwise (B) f(z) = { π e z /, z > 0, 0, otherwise (C) f(z) = { e z, z > 0,, z > 0, 0, otherwise (D) f(z) = { π (+z ) 0, otherwise MS 9/7

10 Q.7 Let X ad Y have the joit probability desity fuctio f(x, y) = { e y, 0 < x < y <, 0, otherwise. The the correlatio coefficiet betwee X ad Y equals (A) (B) (C) (D) Q.8 Let x =, x = ad x = be the observed values of a radom sample of size three from a discrete distributio with the probability mass fuctio f(x; θ) = P(X = x) = {, θ + x { θ, θ +,,0,, θ}, 0, otherwise, where θ Θ = {,, } is the ukow parameter. The the method of momet estimate of θ is (A) (B) (C) (D) 4 Q.9 Let X be a radom sample from a discrete distributio with the probability mass fuctio f(x; θ) = P(X = x) = { θ, x =,,, θ, 0, otherwise, where θ Θ = {0, 40} is the ukow parameter. Cosider testig H 0 : θ = 40 agaist H : θ = 0 at a level of sigificace α = 0.. The the uiformly most powerful test rejects H 0 if ad oly if (A) X 4 (B) X > 4 (C) X (D) X < Q.0 Let X ad X be a radom sample of size from a discrete distributio with the probability mass fuctio θ, x = 0, f(x; θ) = P(X = x) = { θ, x =, where θ Θ = {0., 0.4} is the ukow parameter. For testig H 0 : θ = 0. agaist H : θ = 0.4, cosider a test with the critical regio C = {(x, x ) {0,} {0,} x + x < }. Let α ad β deote the probability of Type I error ad power of the test, respectively. The (α, β) is (A) (0.6, 0.74) (B) (0.64, 0.6) (C) (0.05, 0.64) (D) (0.6, 0.64) MS 0/7

11 Q. Q. 40 carry two marks each. SECTION - B MULTIPLE SELECT QUESTIONS (MSQ) Q. Let {a } be a sequece of real umbers such that a =,. k k=+ The which of the followig statemet(s) is (are) true? (A) {a } is a icreasig sequece (B) {a } is bouded below (C) {a } is bouded above (D) {a } is a coverget sequece Q. Let a be a coverget series of positive real umbers. The which of the followig statemet(s) is (are) true? (A) (a ) is always coverget (B) a is always coverget a (C) is always coverget a /4 (D) is always coverget Q. Let {a } be a sequece of real umbers such that a = ad, for, a + = a a + 4. The which of the followig statemet(s) is (are) true? (A) {a } is a mootoe sequece (B) {a } is a bouded sequece (C) {a } does ot have fiite limit, as (D) lim a = MS /7

12 Q.4 Let f: R R be defied by f(x) = { x4 ( + si ), x 0, x 0, x = 0. The which of the followig statemet(s) is (are) true? (A) f attais its miimum at 0 (B) f is mootoe (C) f is differetiable at 0 (D) f(x) > x 4 + x, for all x > 0 Q.5 Let P be a probability fuctio that assigs the same weight to each of the poits of the sample space Ω = {,,,4}. Cosider the evets E = {,}, F = {,} ad G = {,4}. The which of the followig statemet(s) is (are) true? (A) E ad F are idepedet (B) E ad G are idepedet (C) F ad G are idepedet (D) E, F ad G are idepedet Q.6 Let X, X,, X, 5, be a radom sample from a distributio with the probability desity fuctio f(x; θ) = { e (x θ), x θ, 0, otherwise, where θ R is the ukow parameter. The which of the followig statemet(s) is (are) true? (A) A 95% cofidece iterval of θ has to be of fiite legth (B) (mi{x, X,, X } + l(0.05), mi{x, X,, X }) is a 95% cofidece iterval of θ (C) A 95% cofidece iterval of θ ca be of legth (D) A 95% cofidece iterval of θ ca be of legth Q.7 Let X, X,, X be a radom sample from U(0, θ), where θ > 0 is the ukow parameter. Let X () = max{x, X,, X }. The which of the followig is (are) cosistet estimator(s) of θ? (A) 8 X (B) X () (C) ( X i=5 i) (D) X () + + MS /7

13 Q.8 Let X, X,, X be a radom sample from a distributio with the probability desity fuctio f(x; θ) = { c(θ) e (x θ), x θ, 0, otherwise, where θ R is the ukow parameter. The which of the followig statemet(s) is (are) true? (A) The maximum likelihood estimator of θ is mi{x,x,,x } (B) c(θ) =, for all θ R (C) The maximum likelihood estimator of θ is mi{x, X,, X } (D) The maximum likelihood estimator of θ does ot exist Q.9 Let X, X,, X be a radom sample from a distributio with the probability desity fuctio f(x; θ) = { θ x e θx, x > 0, 0, otherwise, where θ > 0 is the ukow parameter. If Y = i= X i, the which of the followig statemet(s) is (are) true? (A) Y is a complete sufficiet statistic for θ (B) Y is the uiformly miimum variace ubiased estimator of θ (C) Y (D) + Y is the uiformly miimum variace ubiased estimator of θ is the uiformly miimum variace ubiased estimator of θ Q.40 Let X, X,, X be a radom sample from U(θ, θ + ), where θ R is the ukow parameter. Let U = max{x, X,, X } ad V = mi{x, X,, X }. The which of the followig statemet(s) is (are) true? (A) U is a cosistet estimator of θ (B) V is a cosistet estimator of θ (C) U V is a cosistet estimator of θ (D) V U + is a cosistet estimator of θ MS /7

14 Q. 4 Q. 50 carry oe mark each. SECTION C NUMERICAL ANSWER TYPE (NAT) Q.4 Let {a } be a sequece of real umbers such that The a a = coverges to ( ),.! Q.4 Let S = {(x, y) R : x, y 0, 4 (x ) y 9 (x ) }. The the area of S equals Q.4 Let S = {(x, y) R : x + y }. The the area of S equals Q.44 Let The the value of J equals J = π t ( t) dt. 0 Q.45 A fair die is rolled three times idepedetly. Give that 6 appeared at least oce, the coditioal probability that 6 appeared exactly twice equals Q.46 Let X ad Y be two positive iteger valued radom variables with the joit probability mass fuctio g(m) h(), m,, P(X = m, Y = ) = { 0, otherwise, where g(m) = ( )m, m ad h() = ( ),. The E(X Y) equals MS 4/7

15 Q.47 Let E, F ad G be three evets such that P(E F G) = 0., P(G F) = 0. ad P(E F G) = P(E F). The P(G E F) equals Q.48 Let A, A ad A be three evets such that P(A i ) =, i =,, ; P(A i A j ) = 6, i j ad P(A A A ) = 6. The the probability that oe of the evets A, A, A occur equals Q.49 Let X, X,, X be a radom sample from the distributio with the probability desity fuctio f(x) = 4 e x e x 6, x R. The X i= i coverges i probability to Q.50 Let x =., x =. ad x =. be the observed values of a radom sample of size three from a distributio with the probability desity fuctio f(x; θ) = { θ e x/θ, x > 0, 0, otherwise, where θ Θ = {,, } is the ukow parameter. The the maximum likelihood estimate of θ equals MS 5/7

16 Q. 5 Q. 60 carry two marks each. Q.5 Let f: R R be a differetiable fuctio such that f is cotiuous o R with f () = 8. Defie The lim g () equals g (x) = (f (x + 5 ) f (x )). Q.5 4 Let M = i= X i X T i, where X T = [ 0], X T = [ 0 ], X T = [ 0] ad X 4 T = [ 0]. The the rak of M equals Q.5 Let f: R R be a differetiable fuctio with lim x f(x) = ad lim x f (x) =. The equals lim x x f(x) ( + x ) Q.54 The value of equals π x ( e si y si x dy) 0 0 dx Q.55 Let X be a radom variable with the probability desity fuctio 4 x k, 0 < x <, f(x) = { x x, x <, 0, otherwise, where k is a positive iteger. The P ( < X < ) equals MS 6/7

17 Q.56 Let X ad Y be two discrete radom variables with the joit momet geeratig fuctio M X,Y (t, t ) = ( et + ) The P(X + Y > ) equals ( et + ), t, t R. Q.57 Let X, X, X ad X 4 be i.i.d. discrete radom variables with the probability mass fuctio P(X = ) = {, =,,, 4 0, otherwise. The P(X + X + X + X 4 = 6) equals Q.58 Let X be a radom variable with the probability mass fuctio P(X = ) = {, =,,, 0, 0 0, otherwise. The E(max{X, 5}) equals Q.59 Let X be a sample observatio from U(θ, θ ) distributio, where θ Θ = {,} is the ukow parameter. For testig H 0 : θ = agaist H : θ =, let α ad β be the size ad power, respectively, of the test that rejects H 0 if ad oly if X.5. The α + β equals Q.60 A fair die is rolled four times idepedetly. For i =,,, 4, defie Y i = {, if 6 appears i the ith throw, 0, otherwise. The P(max{Y, Y, Y, Y 4 } = ) equals END OF THE QUESTION PAPER MS 7/7

IIT JAM Mathematical Statistics (MS) 2006 SECTION A

IIT JAM Mathematical Statistics (MS) 6 SECTION A. If a > for ad lim a / L >, the which of the followig series is ot coverget? (a) (b) (c) (d) (d) = = a = a = a a + / a lim a a / + = lim a / a / + = lim