MODEL QUESTION Statstcs (Thory) (Nw Syllabus) GROUP A d θ. ) Wrt dow th rsult of ( ) ) d OR, If M s th mod of a dscrt robablty dstrbuto wth mass fucto f th f().. at M. d d ( θ ) θ θ OR, f() mamum valu at M. (Fll th blak) If two varabls ar roortoal th thy do ot hav ay corrlato. (Wrt tru or fals) Fals. ) I cas of rakg som dvduals wth rsct to a charactrstc td raks may occur. (Wrt tru or fals) OR, Two raks of a dvdual must b qual wh som dvduals ar rakd wth rsct to two attrbuts. (Wrt tru or fals) Tru OR, Fals v) To study th trd of a tm srs o ds to tak (a) oly o obsrvato (c) oly daly obsrvatos (Wrt th corrct aswr) (d) obsrvatos for a suffctly log tm. (b) two obsrvatos (d) obsrvatos for a suffctly log tm v) A radom varabl X s such that P(X ) P(X ). What s th valu of P(X > )? P(X > ) P(X )
v) If a dscrt varabl assums valus, 6, 7, 8, 9 ad wth qual robablts th ts robablty mass fucto s gv by..., f,,6,7,8,9, f () (Fll th blak), othrws OR, I cas of a cotuous radom varabl X P(X.). (Fll th blak) f() 6 f,,6,7,8,9,, othrws OR, P(X.) v) For a bomal varabl X E(X) < Var (X). (Wrt tru or fals) Fals (E(X) > V(X) for B.) v) If th aramtr of a Posso dstrbuto s 7.8 th ts mod s (Fll th blak) Mod [7.8] 7 ) If a dgt s tak from a radom umbr srs, th robablty that t s 9 ad th robablty that t s ot 9 ar ot qual. (Wrt tru or fals) OR, Ca ay dgt rat a radom umbr srs? Tru. OR, ys, ay dgt ca rat. ) If th obsrvd valus of a saml ar gv, th valu of a statstc bcoms fd. (Wrt tru or fals) Tru. GROUP B. ) Wh oly two attrbuts A ad B havg two forms ach ar cosdrd, how ar th margal frqucs ad th total frqucy rlatd? Sum of th two margal frqucs for ach of two attrbuts A ad B Total frqucy. ) Df bvarat data wth a sutabl aml. OR, Wh ar two varabls sad to b ostvly corrlatd? S, Gr or Roychowdhury.
OR, Wh th valus of o of th varabls cras (or dcras) wth th cras (or dcras) th valus of th othr varabl, th th two varabls ar ostvly corrlatd. ) Show th cas wh th valu of Sarma s rak corrlato coffct wll b +. OR, What do you ma by rak corrlato? Sarma rak corrlato coffct s gv by th formula r s ( d ) for dvduals btw two sts of raks. Ad hc r s + wh d for all ().. Q R.. th two raks ar qual for ach dvdual. OR, S Gr or Roychowdhury. v) If for a bomal dstrbuto m ad th what s th valu of th coffct of varato? OR, If for a bomal dstrbuto th ctd valu s 4 ad varac s, what ar th valus of ts aramtrs? 6 mq C.V of B(,, % % µ m % 4% OR, Hr m 4 & mq q 4 8 % 4 4 m 6 4 4 4 Paramtrs ar m 6 & 4 v) Show that a ormal dstrbuto s symmtrc about ts ma. Th.d.f. of Normal dstrbuto wth arators µ & σ s gv by
µ σ f() ; < < πσ, othrws. Now, f(µ + ) & f(µ ) σ πσ σ πσ f(µ + ) f(µ ) Hc th dstrbuto s symmtrc about µ, whr µ s th ma of th dstrbuto. v) Stat th mag of a Poulato Statstcs. S Gr or Roychowdhury GROUP C ( µ ) / σ. ) Dtrm th valu of d σ π + [Gv that r π] OR, Fd th frst ordr raw momt of th robablty dstrbuto havg dsty fucto, gv by f() / θ, f >, θ> θ, othrws πσ µ σ d z µ dz uttg z π σ z dz ; sc π z s a v fucto. t π dt t z [ uttg t
zdz dt; dz dt t. t t dt π π.. π π OR, Frst ordr raw momt s gv by / µ E(X) f() d. θ / θ d - t θ θt.. θdt /θ t; d θ dt θ. t. t - dt θ. θ. θ. ) I th as of two attrbuts A ad B dscuss th cass of assocato. OR, Draw scattr dagrams cass of rfct corrlato btw two varabls. S Gr or Roychowdhury OR, Prfct ostv corrlato Prfct gatv assocato ) If th bass ad scals of two corrlatd varabls ar chagd, dtrm th ffct o corrlato coffct. OR, If two varabls ad hav a commo varac ad corrlato coffct
r, rss r trms of θ so that + ad + θ ( θ ) bcom ucorrlatd. Cosdr th ar of valus (, y ), (, y ),.. (, y ) o th varabls (, y), ad u a y c & v ; () ; u & v b w varabls shftg bas & b d scal. Th th corrlato coffct btw & y s gv by y Cov(, y) s s γ, whr Cov(, y) ( ) ( y y) y ( s s.d() ) & s y ( s.d(y) y y) Now, Cov(u, v) ( u u)( v v) a a y c y c b b d Σ d ( )( y y) b.d Cov(, y) b.d su Σ(u u) Σ( ) b d ad s Σ(v v) Σ( y y) v s b s y d Hc th corrlato coffct btw u & v s gv by γ uv Cov(u,v) Cov(, y) b. d s.s b.d s.s u v b. d + γ y b.d Hc umrcal valu of th corrlato coffct rmas ualtrd wth th chag of bas & scal. OR, Lt v( ) v( ) σ Cov(, ) r or Cov(, ) r.σ σ.σ Now, + ad + θ ar ucorrlatd. y
Cov( +, + θ ) or, Cov(, ) + Cov(, θ ) + Cov(, ) + Cov(, θ ) or, ( ) σ + θ Cov(, ) + Cov(, ) + θ v( ) or, σ + θ.r σ +.r σ + θ σ or, + θr + r + θ or, r(θ + ) ( + θ) + θ r θ + v) Drv th two ormal quatos cas of obtag a lar rgrsso l o th bass of bvarat data. OR, If th ormal quatos ar gv for obtag a lar rgrsso l th dtrm th rgrsso l. S Gr or Roychowdhury. OR, S Gr or Roychowdhury. v) Dscrb th mthod of trd dtrmato by th movg avrag mthod wh th rod lgth s a v tgr. OR, Stat th dsadvatags of dtrmg trd by th mthod of movg avrags. S Gr or Roychowdhury. OR, S Gr or Roychowdhury. v) Gv th dfto of a robablty mass fucto of a radom varabl X ad rss P(X X < ) by usg such fucto. OR, If a ubasd co s tossd thrc, fd th robablty dstrbuto of th umbr of tals. For a dscrt radom varabl X, f thr sts a fucto f() such that P(X ) f() satsfyg th codtos () f() > ad () f(), th f() s calld th robablty mass fucto (.m.f.) of X. P(X X < ) P(X, X ) P(X ) f < P(X ) P(X ) f() f() &, othrws.
OR, Th saml sac s S { HHH, THH, HTH, HHT, TTT, HTT, THT, TTH } Lt X b th r.v dotg th o. of tals. Th th ossbl valus of X ar,,, ad th robablty dstrbuto of X s gv th Tabl. Valus of X Prob. P(X ) 8 8 8 8 Total v) Dtrm th mda of a uform (, β) dstrbuto. OR, Dtrm th varac of a uform (, β) dstrbuto. Th.d.f. of th dstrbuto s f(), < < β, othws. If µ b th mda. Th P(X < µ ) P(X > µ ) P(X < µ ) or, µ f()d or, µ d β. β or, ( µ ) β + β Mda, µ +. OR, E(X) f() d d β
. + E(X ) β Hc varac of X s V(X) E(X ) E (X) β β. d β ( β ) + β + + β + β + β 4 + 4β + 4β + β β β ( ) β. 6β v)if a Posso varabl X s such that P (X ) P(X 6) th what s th valu of P (X > )? OR, If a bomal (, ) varabl X s such that P(X ) P(X 6) th what s th valu of? W assum that X ~ P(λ).m.f.; f() P(X ) P(X 6) λ λ λ λ. λ.! 6! P(X > ) P(X ) OR, 6 λ.. λ λ! ; () X ~ B (, ).m.f. s f() P(X ) P(X 6) q ; () or, q q!! 6! 6! 6 6 6 q 4 or, 6 6 6
) I cas of fttg a bomal dstrbuto to a frqucy dstrbuto of a varabl X, how wll you stmat ad us t to fd out a stmat of P(X )? Usg th obsrvd frqucy dstrbuto of r.v. X w ca fd th valu of A.M. as momts as. Th w stmat th ukow by th mthod of f f E(X) ˆ whr X ~ B (, ).. Hc momt stmat of s gv by ˆ, whr A.M. Now,.m.f. of X s f() P(X ) f () +. ; f( ) q. So usg th stmat of as ˆ w ca fd. + ˆ P(X ). f( ) ;. ˆ Usg P(X ) f() ( ˆ ). q ; () ) Dscrb th rocss of slctg boys from a grou of boys by usg a radom umbr srs. OR, I a saml survy mor accurat rsult s obtad ad samlg rror may b gaugd. Ela. S Gr or Roychowdhury. ) Gv th dfto of mmum varac ubasd stmator. Show that f f b th umbr of succsss out of Broull trals wth succss robablty th f s ubasd for. M V U E : A ubasd stmator T of a aramtr θ s sad to b MVUE f T has th mmum varac amog all othr ubasd stmators of θ. Hr f ~ B(, ). Hc E(f). f f E s a u. of.
GROUP D 4. ) For a Posso (λ) dstrbuto th d, rd ad 4 th ctral momts ar µ λ, µ λ ad µ 4 λ + λ rsctvly. Us ths to masur skwss ad kurtoss of th dstrbuto. Hc commt o th skwss ad kurtoss of th dstrbuto. OR, Gv that for a ormal (µ, σ ) dstrbuto th ma s 6. chs ad P(Z > 6.).9. Fd th trval (µ σ, µ + σ) [Gv that for th stadard ormal varabl Z, P (A <.8).9] Th masur of skwss s gv by [ s m ] g g µ & that of kurtoss s gv by µ µ / 4 µ. λ Hr g > sc λ > for ay Posso dstrbuto. / λ λ Hc Posso dstrbuto s ostvly skwd. λ + λ Now, g >. λ >. λ λ Hc Posso dstrbuto s ltokurtc. OR, Assum that X ~ N(µ, σ ), whr µ 6.. Hr P(X > 6.).9 X 6. 6. 6. or P >. 9 σ σ or P(Z < /σ).9 or Φ ( /σ).9 or Φ (/σ).9 Φ (.8) σ.8 σ.8 σ.7.8 (µ σ, µ + σ) (.78, 77.)
) If thr ddt stmators T, T ad T b ubasd for a aramtr ad thr varacs ar th rato : : th whch o of th followg stmators would you rfr most for θ ad why? T + T + T T + T + T T + T,,. 4 OR, O th bass of a radom saml (,,., ) draw wthout rlacmt from a ft oulato wth varac σ fd th ctato of th saml varac ad hc suggst a ubasd stmator for σ. Lt V(T ) σ, V(T ) σ, V(T ) σ T + T + T E 4 E(T ) + E(T ) + E(T ) 4 θ + θ + θ 4 θ T + T + T E E(T ) + E(T ) + E(T ) θ + θ + θ θ T + T E E(T ) + E(T ) θ + θ θ T + T + T V 4 V(T ) + V(T ) + 4V(T ) 6 σ + σ + 4 σ 6 6 σ T,T,T ar ddt. T + T + T V V(T ) + V(T 9 T + T 4V(T ) + V 9 ) + V(T ) σ + σ 9 + σ ; T,T 6 σ σ 9 V(T ) ; T, T, T ar ddt.,t ar ddt. 4 + σ σ 9 9 σ 9 σ. Hr T + T + T V OR, S Gr or Roychowdhury. s mmum ad hc w rfr T + T + T most.