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Document 1734953
*t
Ot
t-L
.
TY 0F C::.LICUT
UNIVER.SIf
E.Sc. Revlsed SyJ.labus (1996 iidrnlssion ,nvrards)
:::,
i
1 Scheme of Exeminatirtn :
Ist year Paper - 1 : Descrlpti-ve statis'tlcs ancl Probabillty:
3 h,lufs: 80 1"1.:rks
2nci Year paper - 2 : Probability Distributicns and
Statistical inference. J hours : 12O Mark-s
l
Pap
er
r, "'u "L lii;tt#t?13;;#f;T#;i,
i*,
(r'\ brlef descripti-on of ccllecti-:-,n, cli:.sslfication and tabulation'
of statistical
data; formation cffreciuency distributions,
curnul-ative frequency distributj-cns and their graphs, Fundz;mental
characteristics of univarlate data: I{e:sures of centrcrl tendency
sh:uld bd given tc the students. )
1
Characteristlcs cf univariate d&ta :
Dispersion, Skewness and Kurtosis. ti;;rents : measures c,f
skewness and kurtosis based on mrments.
?. Characteristics of bivariate ria:ter:
Curve fitting, Principle of least squares and regression .-.':1r,
analysis. Fitting cf y=ax+b, lr=.*2b**", y=axb and y=ab;{.
correlatlon, KarI pearsonrs coefficient of correl-aticn and Rank
correla tion co efficient.
3
Basic probabjJity:
.
experirnent - Sample point- sample space - events
al-gebra of evonts - st^qtlstical regularlty. Classicar and
frequency apprcach tc probability - I,lumerj-cal prcblems.
Randr:m
-c
4. Axiomatic approach to probabillity:
P::cbability as measure - probability space -addition theorem
multiplication theorem - compcund and ccnditional prcbability hdependent events - Bayets tlieorem ernd its application.
5. Random variables:
Di-screte and ccntinucus randc,m variables - Distributirn function
Prcbability density functions - Functitns of rand,-.:m variabl-es
ehange of variabl-e. (Un1-varia:te case :-n1y)
6. . Mathematical expectaticn:
Definition, trlementary properties - Moments - Relatlr-.n between rarg
moments .rr:,J central moments - Ivloment generating f\:nctlon.
a a a a 12
cr. /z /9
\-
-i
!-
z
'7
7.
Elements cf sample surveys:
Generaf principles ,)f s.rmpling siriiple ri:ndom sampling
Esti-rnaticn c'f D-pu1ati,;n ileon ;rnd determin?ti,irl of i_ts sampling
vsri:ri
startified sampling anri Sy--"teriritic sampling (Oefiniti_c,n
and methods onJ-y).
e d is tr:ibr;t:-.,n ;f marks
(Question peper shculcl cary questi;ns f --r 120 marks ,rnd thlt the
maximum marks that can be sctred is i,O('_+
Topic-wJ-s
Seeti,:n
I
2.
3t4
E
6
7.
Topi-c
[tarks in €irch secticn
Characteristics cf univ:ri::,te data
12
Fund.:mental charecteristics :f bivi-,rirte ii- -:tl"rt.
distrlbrirtions
25
Probability
25
Randcm variables and pr-rhability ciistributions 30
Mathematical- Expectati -;n
20
Elements of seimple surveys
B
Total-
cr. /T /o
l-
120
-:s
:7 ( /*"i
'/
I
:--" t--.i
B.Sc- Degree ExamlnaLiirn Pnrt iII
t
(SulsiCiary)
SITITISTICS l,:lner ff
P,'rPER
.i
.
if
i,r-';'"'lbil i by D,';'J1.il-ru-[,i1)ris anc] Stitistici:I
Bivi,:rii-,te distribu-bions
Inference
:
Bivariate dlstribution funct:i.,rn iin,l riens_1-C1.. fuinctiln, i,Ii:rginal and
crnditicnal dls-ur-i-bul;j-cns, incie.oediie.nce :f .Lv,',:;1;:y1d:;fi variables,
expected vahie if functi,:ns r:f t_v;-, i:.ind.ri[ vi,,ril;irIes. blvar-iate
mcments, ac,diticn :rnd rnul tiplica, ti:n thr=:,re;l ..)tf expectati:_,n,
bivariate m "g-. f . , eonditj-tni,-r me3n :irrd 1r, :pi.rnce I Meon, variance
and m"E"f . of linear ctmlrini:tiln tf r::,nrj,nl vilri:bles.
2,. Standard distributi-cnl
point and tt,rr, p':int distributi;ns, iJnif rnt, Point bin:mia1,
Binc,mial, Poisscn, Gelme,r-ric r Ilectangulaf , Exponential, li,:rmal.
One
:f lar:gc' nu^nbers:
Tcheb-rrsheff rs inequa.l-ity - ccnver[ience ill prcbability - vreak ]-aw
,,-f large rtr-rIrbers - ,Xeernoullirs 1.rw ::-f l-arge nunibers - Geneaal
iclea of centrar rriir:-t therrem anrl its ,':pplrcaticns - Lindeberg
Levy i -r, , r' ilLT vri-ilr pro,sf .
1.
4
"
Law
Sampling di-.rtrrbutj-:ns
,-
.
'1. -r:;
Sampling distribution
- standard ey'T',T - san'rpling Clstributir-,n of
the n:ean and variance of r-: sample t:lke:a r'r.,m i,r norrl.l populati,:n
ehi-iiqunre, Studcittts tand F' d j-stributir,.nsr ll-"e o f -vr.ibles,
5^ Thcory of estimation:
Point estjrnatio:i desiri:b]e pr:,pertie-" :f e gl,,:ci estimator Fisher lleymann factr-,rie&ti,;n tiieorenn (v,,itli;ut prc?f ) Uetfrcds cf.
estimation - maxim'um rikelih.,r>d meth,cl - methodli cf moments, ;
Interv.':l esti-matlr;n interdal estirn-rt,1e. ;r. meali anrl vi:ri:-:nco rf,.
'"\ " ncrmal population ^ prcpcrtitn cf succes's''p in biriomial- distribution.
5,
Testr-ng
ef statisticaf hypothesis:
Idea cf testing statistici:i hypltheses tvro types ;f errors critical re'Iicn significance'level_ : i,ower of :_:ttest - Neymann
Pearscn appr::ach : Large sarnpl-e t.=s'tc : resting cf mean elualitrv
of means in twc popula:tions testing cf pr,tpcrticn of success
equality )-" t)rcporti-ons Small sample tests ; strmple tests baseci
r;ir Ch1-squi: e , t anrl F distriuut-i -,ns.
cr. /7 /9
1\-D
-\
,
O
!
(+
.|
a
J
/l
Topic-wise distributi:n'
i-;f iui'rLts '
questi'ns fi,.r 130 mlrks and thot
'
'i '";-:
marhs that c::n be scrrei is 1?-O) I
Questir:n 'Poper: sh';uld ceruY
(
the
.-
mi,--xirnum
...,,'-,- ^ \
) ,l
::: r-"..: i
Secticrn
i''li,rks in each sectlln
TcPic
Biva
2"
Standard Distributicns
3.
4.
Law crf Lange :'^r4urnbers
Sampling Dis trii:uticn
ThecrY of Esti-mati';n
tr
5.
10
45
ria te Distri-butirns
"l
Testing':if
20
ZU
30
i5
HYPrrtheses
'l ,tai
-\
1;.====
j:,lT
D c rn
r -r-lLra
Pr\RT
P+iPER
Tlme ; J
VL' rU
rl a^
-\T,. aij,.E
-rr- it .'i.irC
,i--j
III I
'
1'
ST;iTISTrcs ( suB-StL_r] .i1y )
- DESCIIfFTI,C S,Lrr-L:I,CS E, plrOB..iirILITy
l4ax.MarkS: go
-1. a,;;-:.,rr.l
..
.: .. -....i1. , i , . ;
4
10
1s
25
tl:at the varil-lnce ::f the ; bs erve
tions
a, [l+d ,
O+2dr .
a+ 2nd is n(n+1
16
T
Show
d2
(5)
Explaln the fitting of a _"tl:::ipht
line tc n pe1rs,,ff
lbserv:rtiun usi.ng the principle
c.[
4.
,
Distin6ulsh between abs.;rute enrr rcla-Li-ve
dispersicn. calculate ,.re cc-ef,icient meo-qur.s cf
of vi:riation
tor(7)
"
&'.€"lrowins;:;-;- 6-12 ia _18 1;_r;-r
z4_3o 3o_to
Frequency :
2.
--';rj,r-:ljCJ.j
hour_c
, r. l-r;
---1
'1,r,
r-,ber_"a
]ast [email protected]
(.5)
the different types ;-r. c.:i.rel_itions.
Sholv that
rorrelaticn c.;efj,j_ci-ent 1s n,;t ;,ffe.cted
by
v chengG
----"5 ot, :,rii1in
Enurr:erate
.na Sc:ile ,:f me.:surement,
5'
5.
Distinguish between rl*er-:rti
are !*q regression Llnes.
(6)
,n-" and reg.ression.
rrtrhy.
car-culate the correr-atl -n c
-;eff:-cie.t anc recressi.n
frr the f:I-l_.;wipg dati:.
x t 46 54 35 82 90 rjo
5c T1 TB. 63.
Y: , 86 To 41 79 E2 62 4sr
93 75 ;;-
there
(5)
equa tir)rn
(g)
7'
Define the terms (a) Rancr,;in €-XC,eri.!;ent
(b)
\ " '/ sample
(u)
'c:tlr sp.rce
frcl epend ent events
rc)
ur.- state end prcve adcriti..n the;rern
on prcbi:biii.ty f )r cny twrr
events, Deduce the sente f.._,r events.
3
(T)
!'
-
There are J uryl-q cr-:ntrirrrrr,= balr-s
uf different col,lui,s as
-etrted bel:w.
(6)
Urn f z 4 red, Z bllck,
4^rrcen
1,.,_
' ''
Urn ff :
a red,
2 red,
4 bfack
5 green
UTnIII:
4 b1ack, 2::reefr
rifl
1l ch,-.sen at ran61i1 irnc-i t_vv) r-:arr_s ere
1o' Eibaturyl
drawr: frlm it.
i-" tire frc,-u"iriritv
tr-,.-t--ir,iii 'u." r.,f didferent
c.,l_;ur-e?
cr. /7 /9
1 C.
St:'te 3:-.ye r -o theorem, Ihi.ce iii-:clri-.:.,c_s
::,, i:2 C nr_-,iuc..60
,3r r,O
':
"'
a'*
"
).
- '
i 2:
;"PrC produce 60'3O2
10. St.,rte Briye!s thelreir' Thre e :Il2cflines
lOpercentrespectiveJ-yofthet'':'1pr:,'clucti;n'rf:factc'ry"\
B prcouce6 3%
It is estimlted that ,\ pr,lrJuces Zoi/a defectives,
andCprr..duces4%cefectivesintiieirt':;talpr:ducti.')ns.
t:ti:I p-rroductii;n is fi'-lund
An i-terrr Ch.sen randcmly fr.:;m thd
tr:beclefective.ilhatistheprlblrbi]-itythiltithasCJme
(A)
from machine C
I
t ---
'
----.--.-
'|-''
pr",ilarbility function' (5)
11. Define ': rand -'m vari[al:le lnd t'ne
,.
' l; r'rv' :.ncl write'd ,,wn il=
(')
12. Define tlistrlbuti-r:n functions f.
prcperties.
(6)
be tire tiensitY rf x,
k
13.
di-stribution function of X
(
1
,..' 1 =0, elsewhere find
ha
14.
1
(4)
v
15, i', r.v. X hils the Pdf
f(x) = 2'"'--Zxt x
(6)
C ctherwise
n=
P(x
/t
16. Let the pr-;bability m'rss fLrnctil-n "f
E x = 1.2,3.
f(x) = -b.
i
Find tne-iistributi'-''n functirn ''f
(4)
6)
;,
(i)
':C
'"
.t
st:.te lts prcperties. (6)
Eryertietioa,.;ind
Mathem'ltical
17. Define
1
i-ie '
E. Define D .fi ' f ' lf a r:;ncllm vi::rla
' vari::nce of :-': rrv' wh'lse m';;'f is
l-ind an" *"ur, t"U
(6)
I
o x 1
*
19. If the Pdf cf X is f(x) = z(t*'x1'
.r
)
E(-x2X+1)rletermine
Fience
Shcw thet E(Xr)
):
( r+'l ,i ( r-i,l
ir,
rr
4,.rf
.:
i)
t
i-li''
20, St:te the lclvrntr:3es ' f si;r;li
'i(' t i:
:cl rver census sLlra/ey (4)
F.
I
21.Exp1ai-nsimplef.and-,mSani]Jlj-Ll:!teChr]i..]uefr.lmfinite
populaticn- without replaceiielrt'
cr,/T/9
I
(4)
.Z
\
_)
r
li_?--
YilAL
IrI -
QIJ-!?.rO]{ ?iP R
ll.Sc D..nr.r;i.: ;iAtIOlT
StMrsrrcs (st*,ofiI^liY)
FII"CO.L'TILITY
D
I:,?NE1]TTOI,TS
SIIITISTICAt I}IF.F
r{ors
Ti-nv': J
- 1.
Two ret:clon
OTC
I"
1'1"4.
(
."i
vcrlatcles X
ar:c1
Fir:d.
2.
the nr:,r'git:aI
Lrt X
'120
jointl;r clistributecl with r:lousity
Y n::e
otltcrw'ise
clistributior:s of X ar:cl Y.
(Z)
= cr
i
J,r-o
bt two i'.v.s. enclt talcing values -1,
l:.avin6 tI*- joint 1:r,obebility clistribution.
ru:c1
ifarks:
,'-.i':', f
functior:
f (*ry) = 2(x+v - ,ryt), ol.-xz-1, o.-' :r (.-1
'
iSI-.D
Y
X
ar:c1
Y
O ancl
Ir:clepcncler:t?
1 uc1
(f O)
'
0
*'1
n
0 ';
10
,.
a,1
o.2
0.1
.i i
S) Shorv tltat X r.ncl Y have d.iff;rerjt e;ql'ctntiot:s
b) Peoqve that X aucl. Y oxr rincc;r.1.-.-1-at:cl
t
c) Cal-culate V(x) er:cl V(y)
it) Giv"r: tltet Y={), what is tlre cor:rlitior:el probability clistributior: Jf
e) niuct V(y/X = -1)
Ii:t (fry) hnve thir Joint iristributior: 6iver: by
(*,y): (r,r) (rr2) (t,l) (zrr) (z,z) (z,t)
,,\
rf/-.
\xrYl,
t'T
ri
15- T
Fincl nearrs aucl variuces
4.
n
ii
1,
15-
I
1
iE
of X ruril
(G)
y?
I,ot f(xry) =2r.0 rlx ,<y .i1 be il:e clensi!! of (Xrf).
i(Y/x)=1r{,
X.
Or-l-'xl-1 mrcl-:(X/y)=I,o,l_t{.
Show
ilmt,
1
(7)
!*2(i'2
5.
Defir:e binonir:a1 clist:ibutir,-r. obtain its n.'g.f . IDr:ce fir:cl 'Llic
5.
neal: et:cl vru-iru:ca
show thct fol a poissa: clistritvrrticr:
r ;
-1 ,
t
.1
,i-i. r+1
=
r-,' +
,.A,,(r-,,t"
'
: I
(g)
.l with ',pru:a:reter-
-.
(6)
)-
7. l,st tlE two inc1e1;-nc1:r:t r:.v.'s .Xr or1 q 1:.cva th,:"sete'6,eer_.trie
clistribution, Shor,r that tlre ccnclitiounl clistributior.r rf
(xr/x., + x, = n) is ur:iforn,
8. For a t:ornal clistributiol I.T(
'-- lo
).o/l/'1
=
/a
),
..-'
prove that
1.1.5
';'i'2t =
u^^
\\
..
.. (zr-t
2r
) ,...=*
" .,..,2
\_g
I
I
lto
a
10.
ftt a notrral- rlistrir;ritior: 7'/" cr! the itens ar-e uncl!-r f;
iur,, over 5J. ir4rat is tlte neen ettcl S.D. ?
Sl:or^r
c-nd, 11/,
(s)
t"'t:at llir:anj-r:a1 clis'ir.ibutior: tr:r:c1s to t:orr:al r]-istributi-or: unclel
certair: coUlitior:s tci be str"tc.c-t,
11. nef ir:e cor)v.)r.aj,inco iu,gu.obebility state ar:r1 pr.ove Bernoulli ?e
kw of lelge r:lrn'bers.
12, Stato e.uc1 prove Linc1brry6-*r^vy fca.n of centrnl lilit T}rcoren.
11. Tire nunber of air plcnas ar.rrivinE: at ru: eirport in er:y 20
nitntes pu'riocl fo]rJ-ows a poissor: 1r.* *ith nean 10C.,. TJsir\B
Chcbyclrevts inequality futer=rius .e.lot*rer bourrl fo:: tlre probability tlnt tlre r:unber of airplru:es"iu.rivir:6 ir:'a given 20 nirute
ir:tervd- will 1j-c betr,m.:r: BO r"ncl 120"
1/r. Der:ivu. the clist::ibu-tiolr of sanple ncru: j.n tlu case of sir:p1ir:6
fron r:oma1 populctio, with neiur .'''/- c.i:cl viu.iru:ce ,,-3
15. Def ir:e chisques+ clist^ibution. Obtcin its n,,,al:, vc.]-i-1,;rc-. lutcl n.g.f
15. D*fine F statistic. r,^i'liat is its pclf . Drplair-r two j-npo-r:trurt
uses of it ir: sto.tistrcal ru:aIysis,
17. lrrhat ar-e tlic, clesirable properties of e 6oo11 estinr.tor:. Give or:e
exenple er.ch of estinat:s pcssessinl.l aecir of ti:e clasirablo
properties.
'
'!8. ,kiuir:e tlrc consistency of tlu sani:I.: neciiilr for estinating
.k ir: _r( ,/.1 , <;-?).
,,
',.,
19. Jl tire r,v. Xltas pclf f(X) = (i:' +1 )xr'.,
o 5 x-i-,l:i)O=a 9fl1--p.1is-,
J.;l/"-
(s)
(z)
(a)
(5)
(6)
.
(A)
(5)
(g)
(z)
''1
a
'
2A.
Obteiu tlr.. n.I,e, ,l li
Obteir: (1- i{) 'IOG.Z cor:fir1ilnce ir:t-'rval
for the vru.iauco of
.'
(7)
e
clistribut j on
(Z)
2i, t,Ihat c1o you nerur by e otatisticalL h;rpotiresis? lrrirat ruru the tuo
types of errors? Outlinc tite l;leyrerr-PurJ:son nethocls of testiug e
ste.tistic aI l\ypot lr*s is.
(z)
'-'''
22. ff f (x) =:t1 ;.
XVlt *>" oc.r:cl i-I^ zl-i=1,
,
r(,, _ 2, f iud tii: powexi"f *t* it""t b""..l?o,: J
o
.
rbs.,rvation
't
":-ngr=
w[ich lejects iIO w)ie-o -;t
(5)
';].='l='r",,
21, hcplir,in tlu lru'g+ sc'npl-,
t+stii:3 ltri ,qr.tity or tiro
no:r-ra1
21i.
?5.
popula.ti ou llioiro::ti ol:s.
A srulple of 10 obscrvaiior:s gives ir, ni.roJl eEr:--L io 18
Crui ue cot:clucle thai th': populatiolt E--iul i= 40.
,15)
rutct
S.D.4,
(5)
-a
sanples of B rurc].7 itans:--esTr-.ctivc1y
followii:6: vdues c:f -il,,,r viu'iabi.:.
Tr+o il:clcpet:c1et:t
serr1rl; r: 9
11 G
11 15
g
tZ
hadr
tl:e
1tt
Siu:rp1e IIs 10 12 10 14
9 B 10
Do r:o the csti"Erators of ti,e popr-rlatior: vsrirutca 61ivcr: by
sirr:u1es cliffer signific'rurt1;u,
th:
above
(to)
r:: .t).q t/'t /,1/
I / .!/
,
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