JAM 6 Set of all real umbers P A..d. B, p Posso Specal Istructos / Useful Data x,, :,,, x x Probablty of a evet A Idepedetly ad detcally dstrbuted Bomal dstrbuto wth parameters ad p Posso dstrbuto wth mea N, Normal dstrbuto wth mea ad varace Exp The expoetal dstrbuto wth probablty desty fucto x e, x, f x,, otherwse t Studet s t dstrbuto wth degrees of freedom Ch-square dstrbuto wth degrees of freedom, A costat such that PW,, x Cumulatve dstrbuto fucto of N, x Probablty desty fucto of N, where W has dstrbuto C A E Var Complemet of a evet A Expectato of a radom varable Varace of a radom varable m Bm, x f x x dx, m, The greatest teger less tha or equal to real umber x Dervatve of fucto f.5.5987,.5.695,.65.7,.7.76,.8,.5.8697,.977 MS /8
JAM 6 SECTION A MULTIPLE CHOICE QUESTIONS (MCQ) Q. Q. carry oe mark each. Q. Let P. 7 The rak of P equals Q. Let,, be real umbers such that ad. Suppose P ad P P. The ad ad ad ad, Q. Let m. The volume of the sold geerated by revolvg the rego betwee the y-axs ad the curve xy, y m, about the y-axs s 5. The value of m s 5 6 7 Q. Cosder the rego S eclosed by the surface ad y. The volume of S s z y ad the plaes z, x, x, y MS /8
JAM 6 Q.5 Let be a dscrete radom varable wth the momet geeratg fucto The P equals.5 M t e, t. e t e e e e e Q.6 Let E ad F be two depedet evets wth P E F P F E, P E The PE equals F ad PF PE. 9 Q.7 Let be a cotuous radom varable wth the probablty desty fucto The E ( ) f( x), x. / ( x ) equals equals equals does ot exst Q.8 The probablty desty fucto of a radom varable s gve by x, x f( x),., otherwse The the dstrbuto of the radom varable Y log e s Q.9 Let,, be a sequece of..d. N (,) radom varables. The, as, coverges probablty to.5 MS /8
JAM 6 Q. Cosder the smple lear regresso model wth radom observatos Y x,,,,. ad are ukow parameters, x,, x are observed values of the regressor varable ad, are error radom varables wth E,,,, ad for,, f j,, j,,, Cov, j., f j ubased estmator of, the For real costats a,, a, f ay s a a ad a x a ad a x a ad a x a ad a x Q. Q. carry two marks each. Q. Let (, Y ) have the jot probablty desty fucto x ye, f y x, f( x, y), otherwse. The PY ( ) equals 8 7 9 Q. Let,, be a sequece of..d. radom varables havg the probablty desty fucto 5 x x, x, f( x) B(6,), otherwse. Let Y ad U Y., the a possble value of s If the dstrbuto of U coverges to N, as 7 5 MS 5/8
JAM 6 Q. Let,, be a radom sample from a populato wth the probablty desty fucto f e, x, x, otherwse x,. If T m,,, the T s ubased ad cosstet estmator of T s based ad cosstet estmator of T s ubased but NOT cosstet estmator of T s NEITHER ubased NOR cosstet estmator of Q. Let,, be..d. radom varables wth the probablty desty fucto If ( ) f x x e, x,, otherwse. max,,, the lm P ( ) loge equals e e e Q.5 Let ad Y be two depedet, e e.5 e e N radom varables. The P Y equals e e e e Q.6 Let be a radom varable wth the cumulatve dstrbuto fucto The E equals, x, x, x, 8 x, x, 6, x. F x MS 6/8
JAM 6 Q.7 Let,, be a radom sample from a populato wth the probablty desty fucto x f x e, x,. For a sutable costat K, the crtcal rego of the most powerful test for testg H : agast H : s of the form K K K K Q.8 Let,,,,,,, m m be a radom sample from N, ; m,. If ad, the the dstrbuto of the radom m varable s T m m m t m t m m t m m t m Q.9 Let,, be a radom sample from a Posso populato,, ad T. The the uformly mmum varace ubased estmator of s T T T T T T T MS 7/8
JAM 6 Q. Let be a radom varable whose probablty mass fuctos H ) ad hypothess f x H (uder the ull f x H (uder the alteratve hypothess H ) are gve by x f x H.... f x H.... For testg the ull hypothess : ~ H f x H agast the alteratve hypothess H: ~ f x H, cosder the test gve by: Reject H f If sze of the test ad power of the test, the. ad.. ad.7.7 ad..7 ad.7. Q. Let,, be a radom sample from a, estmator for s N populato,. A cosstet 5 5 5 Q. A sttute purchases laptops from ether vedor V or vedor V wth equal probablty. The lfetmes ( years) of laptops from vedor V have a U, dstrbuto, ad the lfetmes ( years) of laptops from vedor V have a Exp dstrbuto. If a radomly selected laptop the sttute has lfetme more tha two years, the the probablty that t was suppled by vedor V s e e e e MS 8/8
JAM 6 Q. Let y ( x ) be the soluto to the dfferetal equato The y s dy s ;,. dx x x y x y x 6 Q. Let s a e b e s for. The ad a coverges but b b coverges but a both NEITHER a ad b coverge a NOR b does NOT coverge does NOT coverge coverges Q.5 Let f x x x x, x, s,, ad g x x s x s x, x,, x. The f s dfferetable at but g s NOT dfferetable at g s dfferetable at but f s NOT dfferetable at f ad g are both dfferetable at NEITHER f NOR g s dfferetable at MS 9/8
JAM 6 Q.6 Let f :, be a twce dfferetable fucto. Further, let f f f. The there does NOT exst ay x, such that fx there exst x, ad x, such that f x f x f x for all x, f x for all x,, ad Q.7 Let f xy, x xy for all local mmum at local mmum at local mmum both at local mmum NEITHER at xy,. The f attas ts, but NOT at,, but NOT at,, ad,, NOR at, Q.8 Let y x be the soluto to the dfferetal equato The y equals d y d y 9 y, y(), y(). dx dx e 5 e e 7 e Q.9 Let g :, be defed by The area betwee the curve x g x x t e dt. t ad the x-axs over the terval y g x e e e 8e, s MS /8
JAM 6 Q. Let P be a sgular matrx such that Pv v for a ozero vector v ad The 7 5 P 7 P P P 7 P P P 7 P P P P P 5 P. 5 MS /8
JAM 6 Q. Q. carry two marks each. SECTION - B MULTIPLE SELECT QUESTIONS (MSQ) Q. For two ozero real umbers a ad b, cosder the system of lear equatos a b x b. b a y a Whch of the followg statemets s (are) TRUE? If a b, the solutos of the system le o the le x y If a b, the solutos of the system le o the le y x If a b, the system has o soluto If a b, the system has a uque soluto Q. For, let a, f sodd,, f seve. Whch of the followg statemets s (are) TRUE? The sequece a coverges The sequece a coverges The seres The seres a coverges a coverges Q. Let f :, be defed by f x x e. x x Whch of the followg statemets s (are) TRUE? lm f x x lm x f x x exsts lm x f x x exsts exsts There exsts m such that lm x m f x x does NOT exst. MS /8
JAM 6 Q. For x, defe f x cos x x statemets s (are) TRUE? f x s cotuous at x g x s cotuous at x f x gx s cotuous at x f x g x s cotuous at x ad g x x s. Whch of the followg Q.5 Let E ad F be two evets wth PE, PF ad PE F PE of the followg statemets s (are) TRUE? PF E PF C P E F PE C PF E PF E ad F are depedet. Whch Q.6 Let,, be a radom sample from a U, m,,, Y max,,. Y maxmum lkelhood estmator (s) of? Y Y Y Y 6 Y Y 8 populato,, ad Whch of the followg statstcs s (are) Every statstc T satsfyg T,, Q.7 Let,,,, be a radom sample from a, Y Y N populato,. Whch of the x,, x : x, as the most followg testg problems has (have) the rego powerful crtcal rego of level? H H H H : agast H : : agast H : : agast H : : agast H :.5 MS /8
JAM 6 Q.8 Let,, N, populato,. Whch of the followg statemets s (are) TRUE? s suffcet ad complete s suffcet but NOT complete be a radom sample from a s suffcet ad complete s the uformly mmum varace ubased estmator for Q.9 Let,, be a radom sample from a populato wth the probablty desty fucto Whch of the followg s (are) f x e, x, x,., otherwse % cofdece terval(s) for?,,,,,,,,,,, Q. The cumulatve dstrbuto fucto of a radom varable s gve by, x, 7 Fx x, x,, x. Whch of the followg statemets s (are) TRUE? Fx s cotuous everywhere Fx creases oly by jumps P 6 5 P MS /8
JAM 6 SECTION C NUMERICAL ANSWER TYPE (NAT) Q. Q. 5 carry oe mark each. Q. be a radom sample from a,,, Let Y Y Y. If 7 N populato. Suppose Y 6 6 ad has a dstrbuto, the the value of s Q. Let be a cotuous radom varable wth the probablty desty fucto f x x, x, 9, otherwse. The the upper boud of P usg Chebyshev s equalty s Q. Let ad Y be cotuous radom varables wth the jot probablty desty fucto The P Y f x, y x y e, x, y,, otherwse. Q. Let ad Y be cotuous radom varables wth the jot probablty desty fucto x y f x, y e, x, y. The P, Y Q.5 Let Y be a B 7, radom varable. Usg ormal approxmato to bomal dstrbuto, P Y 8 s a approxmate value of MS 5/8
JAM 6 Q.6 Let be a B, p radom varable ad Y be a, 5 9 P, the PY B p radom varable, p. If Q.7 Cosder the lear trasformato The rak of T s T x, y, z x y z, x z, x y z. Q.8 lm cos s The value of e s Q.9 Let f :, be defed by fucto f o x f x x e x 5 6. The mmum value of the, s Q.5 Cosder a dfferetable fucto f o, wth the dervatve f x x. legth of the curve y f x, x, s The arc Q. 5 Q. 6 carry two marks each. Q.5 Let m be a real umber such that m. If m the m x e y dy dx dz e, Q.5 Let P 5 6. The product of the ege values of P s MS 6/8
JAM 6 Q.5 The value of the real umber m the followg equato s x x y dy dx d d r r x m Q.5 Let a ad a for. The a a coverges to Q.55 Let,, be a sequece of..d. radom varables wth the probablty desty fucto f x x xe, x,, otherwse ad let S. The lm P S s Q.56 Let ad Y be cotuous radom varables wth the jot probablty desty fucto cx, x, y, f x, y y,, otherwse where c s a sutable costat. The E Q.57 Two pots are chose at radom o a le segmet of legth 9 cm. The probablty that the dstace betwee these two pots s less tha cm s MS 7/8
JAM 6 Q.58 Let be a cotuous radom varable wth the probablty desty fucto f x x, x,, otherwse. The P Q.59 If s a U, radom varable, the P m, Q.6 I a coloy all famles have at least oe chld. The probablty that a radomly chose famly from k ths coloy has exactly k chldre s.5 ; k,,. A chld s ether a male or a female wth equal probablty. The probablty that such a famly cossts of at least oe male chld ad at least oe female chld s END OF THE QUESTION PAPER MS 8/8