JAM 05 JAM 05: Geeral Istructos durg Examato. Total durato of the JAM 05 examato s 80 mutes.. The clock wll be set at the server. The coutdow tmer at the top rght corer of scree wll dsplay the remag tme avalable for you to complete the examato. Whe the tmer reaches zero, the examato wll ed by tself. You eed ot termate the examato or submt your paper. 3. Ay useful data requred for your paper ca be vewed by clckg o the Useful Data butto that appears o the scree. 4. Use the scrbble pad provded to you for ay rough work. Submt the scrbble pad at the ed of the examato. 5. You are allowed to use oly your ow o-programmable calculator. 6. The Questo Palette dsplayed o the rght sde of scree wll show the status of each questo usg oe of the followg symbols: 7. The Marked for Revew status for a questo smply dcates that you would lke to look at that questo aga. If a questo s 'aswered, but marked for revew', the the aswer wll be cosdered for evaluato uless the status s modfed by the caddate. Navgatg to a Questo : 8. To aswer a questo, do the followg: a. Clck o the questo umber the Questo Palette to go to that questo drectly. b. Select the aswer for a multple choce type questo ad for the multple select type questo. Use the vrtual umerc keypad to eter the aswer for a umercal type questo. c. Clck o Save & Next to save your aswer for the curret questo ad the go to the ext questo. d. Clck o Mark for Revew & Next to save ad to mark for revew your aswer for the curret questo, ad the go to the ext questo. Cauto: Note that your aswer for the curret questo wll ot be saved, f you avgate to aother questo drectly by clckg o a questo umber wthout savg the aswer to the prevous questo. 9. You ca vew all the questos by clckg o the Questo Paper butto. Ths feature s provded, so that f you wat you ca just see the etre questo paper at a glace. Aswerg a Questo : 0. Procedure for aswerg a multple choce questo (MCQ): a. Choose the aswer by selectg oly oe out of the 4 choces (A,B,C,D) gve below the questo ad clck o the bubble placed before the selected choce. MS /8
JAM 05 b. To deselect your chose aswer, clck o the bubble of the selected choce aga or clck o the Clear Respose butto. c. To chage your chose aswer, clck o the bubble of aother choce. d. To save your aswer, you MUST clck o the Save & Next butto.. Procedure for aswerg a multple select questo (MSQ): a. Choose the aswer by selectg oe or more tha oe out of the 4 choces (A,B,C,D) gve below the questo ad clck o the checkbox(es) placed before each of the selected choce (s). b. To deselect oe or more of your selected choce(s), clck o the checkbox(es) of the choce(s) aga. To deselect all the selected choces, clck o the Clear Respose butto. c. To chage a partcular selected choce, deselect ths choce that you wat to chage ad clck o the checkbox of aother choce. d. To save your aswer, you MUST clck o the Save & Next butto.. Procedure for aswerg a umercal aswer type (NAT) questo: a. To eter a umber as your aswer, use the vrtual umercal keypad. b. A fracto (e.g. -0.3 or -.3) ca be etered as a aswer wth or wthout '0' before the decmal pot. As may as four decmal pots, e.g..5435 or 0.003 or -93.67 or.8 ca be etered. c. To clear your aswer, clck o the Clear Respose butto. d. To save your aswer, you MUST clck o the Save & Next butto. 3. To mark a questo for revew, clck o the Mark for Revew & Next butto. If a aswer s selected (for MCQ ad MSQ types) or etered (for NAT) for a questo that s Marked for Revew, that aswer wll be cosdered the evaluato uless the status s modfed by the caddate. 4. To chage your aswer to a questo that has already bee aswered, frst select that questo ad the follow the procedure for aswerg that type of questo as descrbed above. 5. Note that ONLY those questos for whch aswers are saved or marked for revew after aswerg wll be cosdered for evaluato. Choosg a Secto : 6. Sectos ths questo paper are dsplayed o the top bar of the scree. All sectos are compulsory. 7. Questos a secto ca be vewed by clckg o the ame of that secto. The secto you are curretly vewg wll be hghlghted. 8. To select aother secto, smply clck the ame of the secto o the top bar. You ca shuffle betwee dfferet sectos ay umber of tmes. 9. Whe you select a secto, you wll oly be able to see questos ths Secto, ad you ca aswer questos the Secto. 0. After clckg the Save & Next butto for the last questo a secto, you wll automatcally be take to the frst questo of the ext secto sequece.. You ca move the mouse cursor over the ame of a secto to vew the aswerg status for that secto. MS /8
JAM 05 JAM 05 Examato MS: Mathematcal Statstcs Durato: 80 mutes Maxmum Marks: 00 Read the followg structos carefully.. To log, eter your Regstrato Number ad Password provded to you. Kdly go through the varous coloured symbols used the test ad uderstad ther meag before you start the examato.. Oce you log ad after the start of the examato, you ca vew all the questos the questo paper, by clckg o the Questo Paper butto the scree. 3. Ths test paper has a total of 60 questos carryg 00 marks. The etre questo paper s dvded to three sectos, A, B ad C. All sectos are compulsory. Questos each secto are of dfferet types. 4. Secto A cotas Multple Choce Questos (MCQ). Each MCQ type questo has four choces out of whch oly oe choce s the correct aswer. Ths secto has 30 Questos ad carry a total of 50 marks. Q. Q.0 carry mark each ad Questos Q. Q.30 carry marks each. 5. Secto B cotas Multple Select Questos (MSQ). Each MSQ type questo s smlar to MCQ but wth a dfferece that there may be oe or more tha oe choce(s) that are correct out of the four gve choces. The caddate gets full credt f he/she selects all the correct choces oly ad o wrog choces. Ths secto has 0 Questos ad carry marks each wth a total of 0 marks. 6. Secto C cotas Numercal Aswer Type (NAT) questos. For these NAT type questos, the aswer s a real umber whch eeds to be etered usg the vrtual umercal keypad o the motor. No choces wll be show for these type of questos. Ths secto has 0 Questos ad carry a total of 30 marks. Q. Q.0 carry mark each ad Questos Q. Q.0 carry marks each. 7. Depedg upo the JAM test paper, there may be useful commo data that may be requred for aswerg the questos. If the paper has such useful data, the same ca be vewed by clckg o the Useful Data butto that appears at the top, rght had sde of the scree. 8. The computer allotted to you at the examato cetre rus specalzed software that permts oly oe choce to be selected as aswer for multple choce questos usg a mouse, oe or more tha oe choces to be selected as aswer for multple select questos usg a mouse ad to eter a sutable umber for the umercal aswer type questos usg the vrtual umerc keypad ad mouse. 9. Your aswers shall be updated ad saved o a server perodcally ad also at the ed of the examato. The examato wll stop automatcally at the ed of 80 mutes. 0. Multple choce questos (Secto-A) wll have four choces agast A, B, C, D, out of whch oly ONE choce s the correct aswer. The caddate has to choose the correct aswer by clckg o the bubble ( ) placed before the choce.. Multple select questos (Secto-B) wll also have four choces agast A, B, C, D, out of whch ONE OR MORE THAN ONE choce(s) s /are the correct aswer. The caddate has to choose the correct aswer by clckg o the checkbox ( ) placed before the choces for each of the selected choce(s).. For umercal aswer type questos (Secto-C), each questo wll have a umercal aswer ad there wll ot be ay choces. For these questos, the aswer should be etered by usg the mouse ad the vrtual umercal keypad that appears o the motor. 3. I all questos, questos ot attempted wll result zero mark. I Secto A (MCQ), wrog aswer wll result NEGATIVE marks. For all mark questos, /3 marks wll be deducted for each wrog aswer. For all marks questos, /3 marks wll be deducted for each wrog aswer. I Secto B (MSQ), there s NO NEGATIVE ad NO PARTIAL markg provsos. There s NO NEGATIVE markg Secto C (NAT) as well. MS 3/8
JAM 05 4. No-programmable calculators are allowed but sharg of calculators s ot allowed. 5. Moble phoes, electroc gadgets other tha calculators, charts, graph sheets, ad mathematcal tables are NOT allowed the examato hall. 6. You ca use the scrbble pad provded to you at the examato cetre for all your rough work. The scrbble pad has to be retured at the ed of the examato. Declarato by the caddate: I have read ad uderstood all the above structos. I have also read ad uderstood clearly the structos gve o the admt card ad shall follow the same. I also uderstad that case I am foud to volate ay of these structos, my caddature s lable to be cacelled. I also cofrm that at the start of the examato all the computer hardware allotted to me are proper workg codto. MS 4/8
JAM 05 E P A Specal Istructos / Useful Data Set of all real umbers x,, :,,,, x x Expectato of the radom varable Probablty of the evet A Ua, b Cotuous uform dstrbuto o, ab, ab N, Normal dstrbuto wth mea ad varace B, p Posso Bomal dstrbuto wth trals ad success probablty p Posso dstrbuto wth parameter Geom p Geometrc dstrbuto wth parameter p, whose probablty mass x P x p p, x,, fucto s gve by Gamma, Gamma dstrbuto wth parameters ad, whose probablty x x e, x 0, desty fucto s gve by f( x) ( ) 0, otherwse. P The sequece of radom varables coverges probablty to the radom varable d The sequece of radom varables coverges dstrbuto to the radom varable! Bomal coeffcet, equal to x x!( x)! log x Natural logarthm of x I Idetty matrx Cumulatve dstrbuto fucto of N (0,) x Specal values.85 0.900,.645 0.950, e.78, 3.4 log 0.693.96 0.975 0 MS 5/8
JAM 05 Q. Q.0 carry oe mark each. SECTION A MULTIPLE CHOICE QUESTIONS (MCQ) Q. Let,, be a radom sample from a populato wth probablty desty fucto x e, x 0, f ( x) 0, otherwse, where 0 s a ukow parameter. The, the uformly mmum varace ubased estmator for s Q. Let,, 00 be depedet ad detcally dstrbuted 0, 98 The correlato betwee ad s equal to 00 3 N radom varables. 0 96 98 98 00 Q.3 Cosder the problem of testg H : 0 0 agast H : based o a sgle observato from U, populato. The power of the test "Reject H0 f " s 3 6 56 3 3 MS 6/8
JAM 05 Q.4 The probablty mass fucto of a radom varable s gve by P x k, x 0,,,, x where k s a costat. The momet geeratg fucto M (t) s t e t e e t t e Q.5 Suppose A ad B are evets wth P A PB c P B A B s equal to c 0.5, 0.4 ad P AB 0.. The 3 4 0 Q.6 Let,, be a radom sample from a, Gamma populato, where 0 s a kow costat. The rejecto rego of the most powerful test for H : 0 agast H : form s of the K K K K Q.7 Whch of the followg s NOT a lear trasformato? T : T : T : T : 3 defed by T( x, y, z) ( x, z) 3 3 defed by T( x, y, z) ( x, y -, z) defed by T( x, y) ( x, y - x) defed by T( x, y) ( y, x) Q.8 If a sequece x s mootoe ad bouded, the there exsts a subsequece of x that dverges there may exst a subsequece of x that s ot mootoe all subsequeces of x coverge to the same lmt there exst at least two subsequeces of x whch coverge to dstct lmts MS 7/8
JAM 05 Q.9 Let f : be defed by f( x) x( x)( x ). The f s oe-oe ad oto f s ether oe-oe or oto f s oe-oe but ot oto f s ot oe-oe but oto Q.0 Whch of the followg statemets s true for all real umbers x? x e x x e x x e x x e x Q. Q. 30 carry two marks each. Q. Let,, be a radom sample from a Posso populato, where 0 s ukow. The Cramer-Rao lower boud for the varace of ay ubased estmator of g( ) e equals ( ) e ( ) e ( ) e ( ) e Q. Let ad Y be two depedet radom varables such that U0, ad Y U, 3 The P Y equals 34 78. Q.3 There are two boxes, each cotag two compoets. Each compoet s defectve wth probablty /4, depedet of all other compoets. The probablty that exactly oe box cotas exactly oe defectve compoet equals 38 58 5 3 7 3 Q.4 Cosder a ormal populato wth ukow mea ad varace 9. To test H : 0 0 agast H : 0, a radom sample of sze 00 s take. Based o ths sample, the test of the form K rejects the ull hypothess at 5% level of sgfcace. The, whch of the followg s a possble 95% cofdece terval for? 0.488, 0.688.96,.96 0.4,.598 0.588,.96 MS 8/8
JAM 05 Q.5 Let,, be a radom sample from a populato wth probablty desty fucto ( ) x, 0 x, f ( x) 0, otherwse, where 0 s ukow. The maxmum lkelhood estmator of s log log log log log Q.6 Let,, be a radom sample from a populato wth probablty desty fucto 4 4 3 x x e, x 0, f ( x) 0, otherwse, where 0 s ukow. The, a cosstet estmator for s 4 3 4 Q.7 Let the probablty desty fucto of a radom varable be gve by x f( x) e, x. If E( ), the 4 e ad e 4 ad e 4 ad e 4 ad x MS 9/8
JAM 05 Q.8 Let be a sgle observato from a populato havg a expoetal dstrbuto wth mea. Cosder the problem of testg H : 0 agast H : 4. For the test wth rejecto let PType Ierror ad P rego 3, TypeIIerror. The 6 e ad e ad e 6 e e ad 6 e ad e e 6 Q.9 Let Y be a expoetal radom varable wth mea, where 0. The codtoal dstrbuto of gve Y has Posso dstrbuto wth mea Y. The, the varace of s Q.0 000 cashew uts are mxed thoroughly flour. The etre mxture s dvded to 000 equal parts ad each part s used to make oe bscut. Assume that o cashews are broke the process. A bscut s pcked at radom. The probablty that t cotas o cashew uts s betwee 0 ad 0. betwee 0. ad 0. betwee 0. ad 0.3 betwee 0.3 ad 0.4 Q. Suppose,, are depedet radom varables ad k s ukow. The maxmum lkelhood estmator for s N 0, k, k,,, where k k ( k ) k k k k k k k k Q. Let,, 0 be depedet ad detcally dstrbuted 5,5 dstrbuto of the radom varable Y log 5 0 s U radom varables. The, the 0 0 0 0 MS 0/8
JAM 05 Q.3 Let f : be a dfferetable fucto so that f( x) f( x) 0 for all x. The, whch of the followg s ecessarly true? f s a creasg fucto f s a decreasg fucto f s a creasg fucto f s a decreasg fucto Q.4 Let M be the matrx 3. Whch of the followg matrx equatos does M satsfy? M M I 3 5 0 3 M M M 5 0 3 M M M 3 0 M M I 5 0 Q.5 If the determat of a matrx A s zero, the rak( A) the trace of A s zero zero s a egevalue of A x 0 s the oly soluto of Ax 0 Q.6 Let f :(0, ) be gve by The, the umber of roots of f s f( x) logx x. 0 3 Q.7 The umber of dstct real values of x for whch the matrx x x x s sgular s 3 fte MS /8
JAM 05 Q.8 Suppose f : The h () s equal to s a cotuous fucto. Defe h : x x hx ( ) f( uvdudv, ). 0 0 by f (,) f(, 0) f(0,) f (,) tt dt f (, t) f( t,) dt 0 0 Q.9 Let A be a 5 3 real matrx of rak. Let space of. A Let S x Ax b 3 x0 as the set x0 V x0 v: v V 5 b be a o-zero vector that s the colum 3 :. Defe the traslato of a subspace V of. The S s the empty set S has oly oe elemet S s a traslato of a dmesoal subspace S s a traslato of a dmesoal subspace 3 by Q.30 Let f : be a dfferetable fucto whose dervatve s cotuous. The lm ( ) x f( x) dx 0 s equal to 0 s equal to 0 f x dx s fte s equal to f () MS /8
JAM 05 Q. Q. 0 carry two marks each. Q. Suppose, SECTION - B MULTIPLE SELECT QUESTIONS (MSQ) a b are sequeces such that a 0, b 0 for all. Gve that a coverges ad b dverges, whch of the followg statemets s (are) ecessarly FALSE? ( a b) coverges a coverges b b coverges a ab coverges Q. Cosder the ordary dfferetal equato dy x y x for 0. dx x Whch of the followg s (are) soluto(s) to the above? x x yx ( ) yx ( ) x x yx ( ) 0 yx ( ) x Q.3 Let f :[0,] be a cotuous fucto such that The f attas the value 0 at least twce [0,] f attas the value 0 exactly twce [0,] f attas the value 0 exactly oce [0,] the rage of f s [,] f(0), f, f(). Q.4 Let f :. Defe g : by The g s eve for all f g s odd for all f g s eve f f s eve g s eve f f s odd g( x) f( x) f( x) f( x). MS 3/8
JAM 05 Q.5 Whch of the followg matrces ca be the varace-covarace matrx of a radom vector? Q.6 Let,, be a radom sample from a N (,) populato, where s ukow. Whch of the followg statstcs s (are) suffcet for?,, 3, 4,, 3, 4 Q.7 Let,, be a radom sample from a T N(, ) dstrbuto, where 0 s ukow. Let ad T Whch of the followg statemets s (are) correct? T s ubased for T s ubased for T s cosstet for T s cosstet for. Q.8 Suppose ad Y are depedet ad detcally dstrbuted radom varables wth fte varace Whch of the followg expressos s (are) equal to. E EY E Y? Y E E Y m E a a MS 4/8
JAM 05 Q.9 Let,, be a sequece of depedet ad detcally dstrbuted radom varables wth mea ad varace 4. Whch of the followg statemets s (are) true? P P 4 d N(0,) E 4 for all Q.0 Let,, (assume ) be a radom sample from a, N populato where ad 0 are ukow. Whch of the followg statemets s (are) true? The maxmum lkelhood estmator of attas the Cramer-Rao lower boud The uformly mmum varace ubased estmator of attas the Cramer-Rao lower boud The maxmum lkelhood estmator of s a ubased estmator of The relatve effcecy of the maxmum lkelhood estmator of wth respect to the uformly mmum varace ubased estmator of s strctly less tha MS 5/8
JAM 05 SECTION C NUMERICAL ANSWER TYPE (NAT) Q. Q. 0 carry oe mark each. Q. Let ad Y be depedet expoetally dstrbuted radom varables wth meas 4 ad 6 respectvely. Let Z m, Y. The EZ. Q. Let be a (0.4) Geom radom varable. The P 5. Q.3 s a sgle observato from a B, p populato, where 5,45 p s ukow. If the observed value of s 0, the the maxmum lkelhood estmator of p s. Q.4 s a radom varable wth desty E. x f( x) e, x. The 4 Q.5 A system comprsg of detcal compoets works f at least oe of the compoets works. Each of the compoets works wth probablty 0.8, depedet of all other compoets. The mmum value of for whch the system works wth probablty at least 0.97 s. Q.6 Let be a ormal radom varable wth mea ad varace 4, ad ga Pa a The value of a that maxmzes g( a) s. ( ). Q.7 The volume of the sold formed by revolvg the curve y x betwee x 0 ad x about the x -axs s equal to. Q.8 Let [ x ] be the greatest teger less tha or equal to x. The [ xdx ]. Q.9 The umber of real solutos of the equato 3 3 7 0 s. 3 x x x MS 6/8
JAM 05 Q.0 Let f : be a o-costat, three tmes dfferetable fucto. If tegers, the f (). f for all Q. Q. 0 carry two marks each. Q. Let Y ~ U 0,. The codtoal probablty desty fucto of gve Y s The E. f Y x y, f 0 x y, y 0, otherwse. Q. The probablty desty fucto of a radom varable s gve by, f x, 4 f x, otherwse. 4x The P. Q.3 Let,, dstrbuto. The 34 lm P. be depedet ad detcally dstrbuted radom varables wth U 0, Q.4 Based o 0 observatos x y x y,,,,, the followg values are obtaed. 0 0 0 0 0 0 x x y xy 6, 4, 9 ad 0. For, the predcted value of Y based o a least squares ft of a lear regresso model of Y o s. MS 7/8
JAM 05 Q.5 The cumulatve dstrbuto fucto of a radom varable s gve by The P 0, f x 0, Fx 4 xx, f 0 x, 4 6, f x. 0 0. Q.6 The probablty desty fucto f ( x ) of a radom varable s symmetrc about 0. The x f ( u) du dx. Q.7 The legth of the curve y 4 x from x to x s equal to. Q.8 The system of equatos x yz x 3yz 5 4x 7ycz6 does NOT have a soluto. The, the value of c must be equal to. Q.9 Let y( x ) be a soluto to the dfferetal equato y'' y' y 0, y(0) ad y'(0). The lm yx ( ). x Q.0 The area of the rego the frst quadrat eclosed by the curves y 0, y x ad equal to. y s x END OF THE QUESTION PAPER MS 8/8
JAM 05 Aswer Keys for the Test Paper: Mathematcal Statstcs (MS) Secto A - MCQ Multple Choce Questos Secto B - MSQ Multple Select Questos Secto C - NAT Numercal Aswer Type Questos Q. No. Key Marks Q. No. Key Marks Q. No. Rage Marks C A;C 0. to 0. B A;B;C 0.086 to 0.087 3 B 3 A 3 0. to 0. 4 A 4 C;D 4 to 5 B 5 A 5 3 to 3 6 A 6 A;B;C;D 6 to 7 B 7 B;C;D 7.04 to.05 8 C 8 A;C;D 8 0 to 0 9 D 9 A;C;D 9 to 0 B 0 A;B 0 0 to 0 A 0.5 to 0.5 C 0.5 to 0.5 3 C 3 to 4 C 4.40 to.45 5 C 5 0.30 to 0.34 6 A 6 to 7 A 7 3.40 to 3.43 8 A 8-7 to -7 9 D 9 0 to 0 0 B 0 0.8 to 0.9 C C 3 D 4 B 5 C 6 C 7 B 8 D 9 C 30 D