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IIT JAM 2013 : Mathematical Statistics

35 pages, 30 questions, 11 questions with responses, 11 total responses,

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~ Special Instructions I Usefnl Data lR: Set of all real numbers Q: Set of all rational numbers peA): Probability of an event A i.i.d.: independent and identically distributed Exp(,1): The exponential distribution with density !(X;,1)={,1e-Ax, 0,. if X>.O, otherwise, ,1>0 > N(f.l,(y2): Normal distribution with mean f.lE lR and variance t,,: ,t.: Central Student's t-distribution with 11 degrees of freedom Central Chi-square distribution with 11 degrees (Y' N(O,I) K = Complement of the event A <1>(2.33)= 0.99 E(X): Expectation of a random variable X Var(X): Variance of a random variable X Corr(X ,Y): Correlation coefficient between random variables X and Y r(a)= -e xdx , J o -x a-I a>O ~ IMPORTANT NOTE FOR CANDIDATES Questions 1-10 (objective questions) carry two marks each, questions 11-20 (fill in thl blank questions) carry three marl.s each and questions 21-30 (descriptive questions) carry.fu.!!. marks each. The marking scheme for the objective type question, is as follows: (a) (b) (c) (d) (e) For each correct answer, you will be awarded 2 (Two) marks. For each wrong answer, you will be awarded -0.5 (Negative 0.5) mark. Multiple answers to a question will be treated as a wrong answer. For each un-attempted question, you will be awarded 0 (Zero) mark. Negative marks for objective part will be carried over to total marks. There is no negative marking for fill in the blank questions. Write the answers to the objective questions in the Answer Table for Objective Ouestioll provided on page 4 only. Objective Questions Q.l Let E and F be two events with P(E)=0.7,P(F)=OAandP(EnF')=OA. Then p(FIEuF') is equal to (A) 1 2 (B) 1 (C) 3 1 (D) 4 1 5 02 . Q.2 Let {an} >1 be a sequence of positive real numbers such that lim an +1 =.!-. Then lim e ' + an n_ n---+- an 2 n---+- 4 tS equal to (A) Q.3 (B) = Let f: [O,=)~[O,=) be Suppose that, for any x=O and x=t is (A) 11 3 eY. 1 4 8 -+- eY. 4 (D) 1 4 a twice differentiable and increasing function with f(O) =O. t~O,the ~[(I+t}%-ll (B) (C) length of the arc of the curve y=f(x), x~O Then f(4) is equal to 13 (C) 3 MS-l/32 14 3 (D) 16 3 between ~ QA Let f :JR:. 2 ~ JR:. be defmed by f(x,y)= j Sin(2(X' , +, y' x-+y- /' ""(;,), if (x,y) *- (0,0), a, if (x,y)=(O,O), where a is a real constant. If is continuous at (0,0), then a is equal to (A) Q.5 f 2 (C) 1 (B) Let A be a 3x3 real matrix with eigenvalues 1,2,3 and let B = K ' +A'. Then the trace of the matrix B is equal to (A) Q.6 (D) 4 3 91 6 Let XpX" ... be 95 6 (B) a (C) sequence of i.i.d. 97 (D) 6 random variables with 101 6 variance 1. Then limp((XI -X2 )+(X, -X 4 )+ .. +(X,,,_I -X,,,) <xJ is equal to .j;; 1I~t>Q (A) Q.7 Let (x) (B) XpX" ...,X,oobe (2x) a (C) random sample 1 99 1 99 X=-LXi,S= -L(Xi-X)' and W 99 (A) Q.8 i=I , %98 98 i=I (B) , from a (D) (JzJ N(2, 4) population. Let X IOO - 2 . Then the distribution of W is S (C) %99 (x12) t" (D) t99 - I" Let XpX" ...,X",X,,+I be a random sample from a N(Il,I) population. If X" =- LXi and 11 i=-I T =~(X" + X,,+I)' then for estimating il (A) T is unbiased and consistent (B) T is biased and consistent (C) T is unbiased and inconsistent (D) T is biased and inconsistent MS 2/32 ~ Q.9 Let X be an observation from a population with density f(x) = A.zX e-Ax, if x> O,A. > 0, 0, elsewhere. { For testing Ho:A. = 2 against HI:A. =1, the most powerful test of size a is given by "Reject Ho if X> e", where e is given by (A) Q.lO 1 , 4 M." (B) 1 Z -%3" 4 . 1 z (C) -%'" 4 -. (D) 1 2 4%1." A continuous random variable Xhas the density f(x) =2 rp(x) (x), XEJR. Then (A) E(X O (B) E(X)<O (C) P(X 5::0 0.5 (D) P(X :2:0)<0.25 , MS 3/32 ~ Answer Table for Objective Questions Write the Code of your chosen answer only in the' Answer' column against each Question Number. Do not write anything else on this page. Question I Answer Number Do not write in this column 01 - 02 03 - 04 05 - 06 - 07 - 08 - 09 - 10 FOR EVALUATION ONLY MS 4/32 ~ Fill in the blank questions If X has the probability density function Q.ll f(x) =_I_ xa- 1e- x ; x >O,a >2, then , , lAns: , , , , , , , , , , , , , rea) var(~) is equal to X ,-~--------------------------------------------------- ---------------------._-------._--------------------------------------------._----- '' : ' '' '' ' ''' ' '' ' ~----------------------------------------------------- --------------------------------------.-.---------------------------------------_.: Q.12 Let the joint density function of (X,Y) be if -x<y<x, O<x<l, . 0, otherwIse. Then the value of c is equal to f(x,y)= c (x+y), { ,-----------------------------------------------------------------.--------.--------.--------._-------._--------------------------------: tAns: : , , , , , , , , , , , , , , , '' ''' ' '' ''' ' '' L. Let X be an observation from a population with density fImction f(x). Then the power of the most powerful test of size a = 0.19 for testing Q.13 x -, Ho:f(x)= 2 0, ! !3X2 if 0<x<2,. - , if 0<x<2, agamst H1:f(x)= 8 otherwise, . 0, otherwise, is equal to . rAiis:-----------------------------------------------------------------------------------------------..,--------------------------------1 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , !.._--------------------------------------------------------------------------------------------------------------------------------------, Q.14 Bulbs produced by a factory Fi have lifetimes (in months) distributed as EXPUi) for i=I,2,3. A fIrm randomly procures 40% of its required bulbs from F;, 30% from F2 and 30% from F,. A randomly selected bulb from the firm is found to be working after 27 months. The probability that it was produced by the factory F is :Ans : i . I , .J 3 ,---------------------------------------------------------------------------------------------------------------------------------------, MS-5/32 ~ Q.15 Let Xl"'" X" be a random sample from a population with density j (x,fL) = { and let XCII = min { Xl' X 2 , , ep-x. if x > fL, 0, otherwise, X,,}. Then (XCI) - ~ log, 5, X(ll) is a % confidence interval for fl. 1A~~~----------------------------------------------------------------.-----------------.--------------------------------------------1 , , , , , , , , , , , , , , , lo Q.16 , , , , . , , , , , , , , , , .----------------_.-----------------_.--------------------------------------------, w Ten percent of bolts produced in a factory are defective. They are randomly packed in boxes such that each box contains 3 bolts. Four of these boxes are bought by a customer. The probability, that the boxes that this customer bought have no defective bolt in them, is equal to ,-------------------------------------------------------------------------------------._-----------------------------------------.------ ... tAns: : , , , , , , , , , , , , , , , , , ' , , ,--------------------------------------------------------------------------------------------------------------------------------------_! . Q.17 Let j : lR ~ lR be defmed by j(x) ={ 2 x, if XE :)l, ax+j3, if XE lR- :)l, where a and j3 are real constants. If j is differentiable at x =1 then the value of 3a + j3 is equal to r Ani - - - - - - - - - - !._-------------------------------------------------------------------------------------------------------------------_._----------------' Q.18 Let {an},,'1 be a sequence of real numbers such that Ia" 1<;,.,[,;, n =1,2, .... Then !~{e~; +.,[,;Sin( ~J} ,_~~_~g!:l~_~? __. . . . . . . ._. .. lAw !.._------------------------------------------------------------------------------------------------------------------------------------- MS-6/32 _ [E] Q.19 Consider the linear system x+y+2z=a x+4y+z=4 3y-z=r in the unknowns x, y and z. If the above system always has a solution then the value of a+ is equal to r ,~---------------------------------------------------- .--------------._------._------._-----._--------------------------------------._-- ---------_.-._---_.-._---_.-----_.----------._-----._----------------------._--------------._------._------.----------------------.--- The general solution of the differential equation (x' - y) dx+ (y' - x) dy= 0 is equal to Q.20 rAI{~~ ------------------------------------------------------------------------------------------------------------------------------ i , , , , , MS-7/32 ~ Descriptive questions Consider the matrix Q.21 1] 1 a1 13. [a 0 13 p= 0 If P has eigenvalues 0 and 3 then determine the values of the pair (a, 13). r---------------------------------------------------------------------------------------------------------------------...--------.-.... , , , , , , , , , , , ~ " " ~ .s ~ u " " "- r/) , , ._----.-_ ... _-----------------------------------------------_.---------------._---_.---------_.--._-----------------.----------_.------- MS 8/32 Z /6-SW ,----------------------------------------------------------------------------------------------------------- ----------------------------~ ! , ' '' '' '' , , , , , , , ! l ! I 1 ii 1 ! , , , , , , , , , , , , , , , , , .. _-------._-----------------------------------------.----------------------_.-----------------------------------------------------------,' ~ Q.22 Let a function f: [O,l]---7lR be continuous on [0,1] and differentiable in (0,1). If frO) =1 and [f (1)]3 + 2 f (1) = 5, then prove that tbere exists aCE (0,1) such tbat ~ ~ a <lJ - ,s , <lJ g 0- '" MS I0/32 J'(c) 7 -, . 2+3 [f(c)]- j.. ... .._--.--.._-.-- . -1 , , , , , , , , , , I [!J Q.23 Let {a,,},,", be a sequence of real numbers such that 2:a" converges absolutely. Prove that 11=1 ~ the series 2: log, (1 + a:) converges. n=I , , , ,~---------------------------------------------------- -----------------_.------._---_.-------._------------.---------------------------., , , , ~ " -5 ~ .B g " 0. UJ MS-12/32 Z / 1-SW ----------------------------------------------------------------------------------------------------------------------------------------1 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , en , , , , , , , , , , , , , , , , , , , , , ,. , , , , , , , L . ,'0 , , 1'; " : 8' , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ~ , ~ Q.24 Let D ={(x, Y) EJR2: O:s; y:S; x:S; I} and let j :D ---7 JR be defined by j(x, y) = x 2 -2J.y+2, (x, Y)ED. Then determine the maximum value of j in the region D. ,~~--------------------------------------------------- , , -----------------------------------------------------------------------------------: : , , . : , : , : ' , , , , , , , , ''' ''' . ! ' , " a <U ~ <U -5 oS <U ;:: 0- '" . , .------------------------- .. ----------------------------------------------_.------._------------._-------------------------------------_. MS-14/32 ,. Z /SI-SW , , , , , , , , , , , ------------~--~-------------------------------------- -----------------_.-.--------------.----------------------------------------.----- ~ Q.25 Let X,Yand Z be independent random variables with respective moment generating functions 1, t<l; M y(t) = eIX M x(t) = 2 =Mz(t), tE JR. 1-t determine the value of peW > 2). \ -_._. - _ --_. - -_. _.. _. - _ _ -.. _.. - - I . - I I Let ' +Z . ? W = 2X + y. Then _ --1 - _.. I I ; ! " o ' o '' " o " ''' . . , 4'< j" MS 16/32 'l /Ll-SW , , , , , , , , , , '' ' ' ' : : , , , , : , , ~-------------------------------------------------------------------------------------------------.-------------------------------------, ~ Q.26 Ram rolls a pair affair dice. If the sum of the numbers shown on the upperfaces is 5, 6, 10, 11 or 12 then Ram wins a gold coin. Otherwise, he rolls the pair of dice once again and wins a silver coin if the sum of the numbers shown on the upper faces in the second throw is the same as the sum of the numbers in the first throw. What is the probability that he wins a gold or a silver coin? i-----------------------------------------------------------------------------------------------------------------------.---------------, ~ ~ ::3 ..s OJ -" h . : , ' i OJ ' U) 1 : '" . "': u , , , , , , , , , , , , , , , , , , , , , , , , , , . , . ~-----------------------~----------------------------- ----------------------------------------------------------------------------------, MS 18/32 Z /6I-SW , , , , , , , , , , , , CIl :'0 , , , " n " : ~ : ~ S' : " : ~ : ~ , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , -----------------------------------_.-----------------.-----------------------------------_.--------.-----------------------------------, [E] Q.27 Let X!," .. ,X" be a random sample from a uniform distribution on the interval [e, 2e], e > O. Find the method of moments estimator and the maximum likelihood estimator jof e. Further find the bias of the maximum likelihood estimator. _ . ~ ~ c oj ~ ~ ..e g " '" 0. , , , , ---------------------_.-._------------._------------.------.-.----._-------------------------._------------.---------------------------, MS 20/32 Z IlZ-SW ~ Q.28 Let (X"Y,), (X 2'y2)' ........ be a sequence of i.i.d. bivariate normal random variables with (X,)=75, (Y,) =25, Var(X,) =36, Var(Y,)=16 and 1 " U =L (X; + 1';). Find tile minimum value of 11 so that P(U :;104) 2': Carr (X"Y,) = 0.25. Let 0.99. Jl i=l r-------------------------------------------..-----------------------------------------------------------------------..----------------. ~ " '" ~ <l) -5 ~ <B <l) u : ' '" "':' , t _ ~ MS-22/32 Z I Z-SW j---------------------------------------------------------------.----------------------------- --- -------- --------- --------------------- , , , , , , , , , , , , , , , , , , , , , , , , , ~ Q.29 The joint probability density function of (X, Y) is f(X'y)=l~ y e(l-x- ), 0, if x+y>l, x>O, y>O, otherwise. Find the probability density function of X and E(Y I X =x), x> O. ---I ~ .s ~ <8 u " 0. " '" -------_.---------------._-----_.-----._---------------------------------------._------------.-------------.-------------.-------------. MS-24/32 I . ............z /sz-sw . . , , , , , , , , , , , , , , , , , , -----------------------_._._--------------------------------------------------------------------------------.----------------_.--------- ~ Q.30 Suppose that F is a cumulative distribution function, where 0, F(x)=i 1-e -x c, if ' 1- e-X, if x<O, O:S;x<l, if 1:S; x < 2, if x~2. I. Find all possible values of c. 11. Find P(0.5:S; X:S; 2.5) and P(X =1)+ P(X = 2). ------------------------------------------------------.---------------------.--------------------------------------------------------- .. ~ " " ~ .E ~ .a ": ? p. '" , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , .-.-------------------------------------------------------------.----------------------------------------------------------------------_. MS-26/32 Z IL:l-SJ\j f--------~-------------------------------------------.------------------.-----------------.--------------------------------------------\ ! I ! I I i , , , , , , , , , , , , , , , , , , , , , , ------------------------------------------------------------------------------------------_.--------------------------------------------, Z /8Z-SW JI.lOM qil no.l .loJ a;)llds '1.10M. Z /6Z-SW ql'lno.l .IoJ a;mds Z /O -SW :l\.I0M qil no.l .IoJ a;mds )j.lO," Z III:-SW qil no.l .loJ a:mds z m:-sw :l\.I0M Ill! no.l .I0J a:l1lds 2013 - MS Objective Part Question Number 1 -10 I Total Marks

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