Page I
M,h ,6 Fdr2M O&,er 6st
Math 166- Exam 3 ReYiew
NOTE:For rcriewsof theothersectionson E\arn 3, referto thefrst pageof WIR#7 and#8.
SectionM.1 - Markov Processes
. Markov Process(or Markoy Chain) - a sp€cialclassof stochastic
processes
in whichtheprobabilitiesassociated
with theoutcomesat anystageof theexperimentdependonl) on the outcomesof theprecedingstage.
. Theoutcomeat anystageof theexperimentin a Markovprocessis calledthesttte of theexperiment.
. TransitionMatrix - A transition
matrixassociated
with a Markovchainwiihrl statesis annxn matrixf with
:
j)
P(rnoving
to
state
il
currcntly
in
state
suchthat
entriesat
L ai; 2 0 for all i andj.
2. Thesumof theentriesin eachcolumnof I is 1.
. Any matrixsatisfying
thetwoproperties
aboveis calleda rtochaiticmatrix.
. If 7 is thetransition
matrixassociated
wilh a Markovprocess,
thentheprobability
distribution
of thesystemafter
(or steps)is givenby
n obsewations
Xn: T"Xo
. Powcrsof the Tlansition Matrix - The entry of Ir in row i andcolumnj givesthe probabilitythat the system
(i.e.,n stepsin theMarkovchain)giventhatit wasinitiallyin stotej.
movesto statei in n observations
SectionM.2 - Resular Markov Processcs
. Thegoalof thissectio[is to investigate
long-term
trendsofcertainMarkovprocesses.
. RcgufarMarkov Proc$s- A stochastic
matrixI is a r€gularMarkov chainif thesequence
T,Tz,TI,.. . approaches
a steadystatematrixin whichall entriesareporiti)e(i.e.,strictlygreater
than0).
. lt canbeshownthata stochastic
powerof 7 hasentries
matrixf is regularif andonlyif .rorr?e
thatare6ll posilive.
. FindiDgthe Steady.State
DlstributionVector- Let f be a regularstochastic
matdx, Thenthe scady-siate
vectorX maybe foundby solvidgthematdxequation
distribuiion
7X = X together
with theconditionthatthe
sumof theelements
of thevectorX mustequal1.
SectionM.3 - Absorbin! Markov Proc€sscs
. Absorbing Markov Process- A Markovprocessis calledabsorbingif thefollowing two conditionsaresatisfied.
state.
l. Thereis at leastoneabsorbing
stateto oneof theabsorbing
2. It is possible
to movefromanynonabsorbing
statesin a finitenumberof stages
(i.e.,stepsin theMarkovchain).
. The transitionmatrix for an absorbingMa*ov processis said to be an absorbing stochasticmatrix. When
studyingthe long-termbehaviorof an absorbingMarkov process,we reorganizethe transitionmatrix so thal
absorbingltatesarelistedfirsl, followedby nonabsoftingstates.
. Theorem1 - In an absorbingMarkovprocess,the long{ermprobabilityof goingftom anynombsorbingstateto
someabsorbins
stateis l.
Page2
Ms16rt!eofudq&|@
Theorem2 CfomastikandEp6tein,pg. 350)- Id f bethetransitionmatrixofm absorting Mad@vproc€sswith
a alrsorting statesand6 nonabsoftingstates.Reorderthe statesso thatthefirst a of themareabsorbingandthe
.emaining, ofthem ar€nonabforbing.Panidonthematrixasfollows:
Thenth€followingaretrue:
l. As , becomeslargewithoulboun4 thematrixfo, whichis th. transitionmatsixftom theinitial stageto lhe
stage,hesdsfor thelimiting matrix
',d
Ir
o*"
Ag R) t1
--Obn
l
(/- 8)-' is thesamedimensionas8, thatis, Dx ,.
wherctheidentityrnatix l in lhe expression
Th€entryin theid row and/d columnofthe sub"mafix,{(/ - B)_l girestheprobabilitythatthesysternwill
endup in therd sbsorbingslatewheninitially in th€/d nonabsobing3late.
All columnsurnsof,1(1-- 8)-l arel, andthuseverythingis ery)ectd tobe abso ed.
L True/False
TRUEa tALsD
a\ I^A= AIr: A rora]lmatrices
,{ fyug
rnur 6G\
u
(l*G')
TRTJE
,nrs
nXn,
A\f
Whenusingrrcf h thecalculator,s rowofall 0's at thebottomofthe resultirig
mabixguarante€s
thatth€ s)stemofequationshasinfinitclymsnysolutions.
c) To be ableto computetie mafix product,{8, the numberof colurmsof,{
musteoualth€dtlrnberofrows of8.
@
d)
@
e) Iff is al x 3 transitionmaDrix.then13 raprcsenBthe probabilityofgoi!8 io_.
,/ TRUE \
FALSE
(TU")
FALSE
6;)
;{
If8 isa 2 x 2 InaEix,
tllenB+12=B.
9!r=$
in stlrel. trb, p\r1fi6.fttWt 3t -+e t,,?tr''-tlatLl
srate
3 nextifcurrEntfy
ct'r.v'o* t' StLk t 5 3lrrf -3
If itr solvirya syst€itlof equltionsin thcvaliableg
r and/ usingtlc CaDss| | o lll
Jordancliminationme6odyouobrain| ; ; I : l. tbenrbesycremofequaL U rl6 j
tionshasonly onesolution,
FAISE
lf thepor.mctricsolutionto s rFtem ofequationsis (3t + 2, -t - 3,r), dren
(-7,0, -3) is a particular
solution.
f - -3
h)
If,4 is a3 x 4 matrixandB is a2 x 4 [t6trix, thedBl! is a 2 x 3 matrix.
l./
nt(t^B
1t
'l \4
t
w
At- 4\--___J
^
|
L-,
-..,--
t-x5
,t
4 -."a
L)
- ---
I
, .
lf -
Pagel
M16tumol]leF|lq
2. Forthercxt two wordprobl€msdo thefolloq,rng:
I) Defirc ttrc€dables tlnr areusedin settingupthesystemof equations.
n) Setup thesys&mofequatiorsthatRp.esentsthisproblem.
m) Solvefol thesolution.
I\) Ifthe solutiotris param€tsic,
thentell whatrestr'rctionr
shcnd be placedon lhe peramet<s).Also givethree
soecicsolutions.
(a) Fre4 Bob,andG€orgeareavid collectorsof tuseballcards.Amonglhe lhlee of tlero, theyhave924cards.
Bobhrs thrEetimesasmanycnrdsssF&4 andCoorgebat 100ftot€ cardsthanFrcdad Bob docombined.How
({t,thI^A'lD1(q,lsrBobLv't b9 ut/5, 2,J
H'^6t Ltt
'
Ca/\A 5 ,
=l)w6nt3to providea rabbitwith o(ecdy1000unitsof gihmi! A, cxactly
O) ln.lsbor.tory €xperimelt a researcher
1600unih of vitunin C andoxastly2400utritsof vitaminE. The nbbit is fed a mixtrrc of rhi€r foode. Bach
grsrnof food I ooDisins2 utritsofvitami! A,3 unitsofvitaEin C, aDd5 unitsofvitaEtinE. E3chgrab of food2
conbins4 lmit! of vitaminA, 7 unitr of vitaEin C, a|rd9 unitsof vitlmin E, Bschgralr of food3 coDfriff 6 units
of vitdnin A, l0 unih ofvitarnh C, ad 14utrib of vibmin E, Howmlny grdnsof€achfoodshoulddrcnbbit be
tan /1 | 1= t&t d,qM!r{^- 4 q!4e
q ktod I
d* q ={" .t-.|',^trtr{{u.-aq hil>
11*i --\kw."t*^ {,fra*
u t qt^il3
u
foodl
t0 (rlnt
Tota.l
lU)0'-t,'itl
lU,rr.irS lu fD.*4i'5
vr'tt'is
t{,r,r"ii s }4 0u
&N+ 4 q tb t- - p o o
'rcu)
3y+14Hnt
5r$4vt9z--)4o()
r X *1 t n; t > * t' -ro o
l$if if r ooI 'q * e --,*
lo trDlo
!*
z--t t wl,.w t satlru!
\ -- -? +zoo
x -" t r3tD
-L+5oo>/D
t.LhsD
I
O,
*,u'wJUlr'7q t t = l tr D
"
l-'-t+rva
J --cttou
_t|007O
tL-tu)
Ir u"lHi c
-tr+luDrL)
{r,X,e)= (-c t-aoo)
v!.r.1
AL't atoo
P4rg3s-.{4!5lr44!Is1
z (60orl0o,D)to
,rtfil-- o, 1-1,517)
Ir tv1}$lobd 2-)05!tr lool1
loetlr.a'esbt@d
. tOl.,r t: loo,(X9t?)= (lDo, OrrDo)rr)
^ rl toudI r Ottb{rola, rooqqfood
2^'0o\
f.
= r) ('r'A,A)--
(fq4, qq, lY 9i)
. W"o,^t
qqld[tsora)
,q.5 dLfoD4
\,
lSr\lDcd3
Page4
e16lr16(o|&&bfy
r. solvethesistsn
alYte\
\
='-rusingthecauss-Jorderelirliruli@rncfiod".(s€.mR#?forrdcxmplcwith
*;1:,
lnrrc
ud
tlnkrow''.)
rhrcc.su.dons
Rl-.r.R_,
\
p-a 1r'ln,-a,n^-c,
[,--l=l:J
tr t-'J
-'))_.
L*
_o
-t)
[r
1 =Lr-o fl-elzo
-,, i1
Lt,:"|
,',
t:
"*. ol 'Jt -rs
-j*i_+
?1
Ra-JR,.-R*'
[r -flr-l i"K^-R'-,f, -,trlr-l'-'-'=--<--.-*-ltp-t
'
>/_h-^
Lo' rr'rtffiil
"x 'ri
----x
n
/ g,tl
5Raaa, .- - " 1 * * l
g,tlsgaaAr.
ilt*l
f
x' fu
x=
-*
[,
['
ffi
l)
I
l''l-al l,-'r:;t
It+,-")\
|i
po$ible,exphinwhy.
4. Solvrfrr thcvsriiblorr,"
X'ts1=q
y=-gu
y=-L
u
x-:rn;=
,.,"=j
(',u)=lRr
<--:_ 6ji{f4'ot
0ll
l2
41.17
I
0l =-[^l 1- r
5 |
_l
j*1"
,
-tJ[v-1
-iJ
['
'J
tl
-,tl
f:ara*
rt-zl I L, (J - L v
lx-rr5",>
"l
l+
lz
€l
[r
z+1 f
t--
st"zf
Ltt-'
9: -Jtdu
ll 'z 34
n+ Z - - - t o
f-t+'n
|
-,OJ
L v
s\ -t =3
ox--b+ z
SX: - lt t ]
tY: -15
lv ---31
-^
1)
--t,
X-5-r
-5 =-?-x
5 -- x--1
tu
Page5
16rdfl@@ni'ry
5. A smalltownhasor y two dry cleane(s,AcrneDry Clean,andEmcaDry Clearcrs.Acme3 mrnrgerhopesto
ircrEasetbc finn's marketshareby conductingan extensiveadv€nisingcampaign.After tbecampoign,a martet
of Acme'scwtomerswill retumto AcmeDry Cl€€nwith their next load of dry
researchfim finds thal 65010
cleaning,and25%of Emca'sc'trstomefs
ry!11-$dtc$qAcmefor theirn€xtloadof dry cle&ing. Swposethateach
cleani(&ler
w!9Lg,N thal beforethe adcampaign.35%of aI cuslome$us€d
custorierbringsonelosd of &y
AcmeDry Cleanard 65%of all customenusedEmcaDry Cl€anets.
(a) Whatis theaansitionmafix for thir Mar*ovprocess?
Ifthis is anabsorbing
Markovprocess,
ordertbesbtes
Lurre.^t
ar€listedfirst.
sothalall abEorbing
srates
A- A.'//u ?r\(lr.an"
E- Ewc li3Clr"^rY
,4
(- -t
A l,{6 .25'l
r--
ruu',lr
"'s1
O) what will thc perconllgcsofthe marketsharelook Iike 4 weelcssfterthe
rounded!o 2 decimalplaces.
answgraspercedages
h'lT4 xr-{4x"= 1.414.v1
toaqj
L
Alt<v
4wlc.,4l.so9o I I
apto /ltnt WU{.Aq^. ",i
a+ol
tE.so?"wit\ {D faEw''4^,
(o) Whatis dtepoblbility thst aniEdividualwho itritially took his or h€f dry clc.ningto AcmcDry Cl€anwill
besr[s?
thentrke his or herdry cleaningto EmcaDry Cleonen(|nereek afierdred campalSn
(e) Is thiss regularMa*ov $oces6or anabsorbing
Markovprocess?
If&.gulsr,find thesteady-sbte
disfibdioo
process.
Ifabso.bin& ddtermircthe long-termbehaviorofthis Markovproc€ss.If
vectorfor thir Markov
neilhcr.explsinwhy.
(,l1tlll.f ti tU
6!tre'^lytQsarzgoVb;li1;<s
fi .V^"vroo's. L+X'[-i] \LIIA
TX-X
l.u<
L.t5
t:lttl'lil
1el0zr,4rr5.t n .4, I , t ^nl
rthr^+!iq-ehkl.;'t..it
vrt--f "n'-1
y',1^)
\.\
-;l;;j
=i " ;:?,:
?::li'G
=;fu+fl*&\,4
Pag66
6. Us€ the gii€n matricesio computcesch oflhe following. lfan op€rationis not possibl€,explain why.
r 2l
s
1=ILir ;r'1l .8 : f| 5 8 l, ._l
"6
t
5
[3
(e)R+C
-3
1l
harAV anfttl bt-adllA 'trxcnuP'
, .Uft az-cnDt It-P A-n"-e'+W
c
_9-
]
4 1f
o) 8c
bc-,f,. +4
P'
-a
L't5 -v
(c)ABr - 5c
a$I$ rtut_d"
am
1r,
l-
1 O6r_5s--_\-atl -v
-r4
\.
5e_
tW*^*1
q
jtz"
-u) i
5lr
l i
l_-r
-)
(d)cA
aj-W
,'nd.t|,,k
tN4r a - +4,
0r")-
l---.)--
7.
(b+r.d t
x !e.o d.e/h!e
4\,r'At
Nobpo::iL\e
frC'/D()rtot
rd rlrreb,
,tbrltut"
"rbll'
,t d ,ry s@ r b)
^rrffii
p--ritemarcsunrma.ized
in thefiIst
A company
du.luficturesthreeproducts:A, B, andC. Its productioncxp€nses
trble below.Thesec.ndbble givesthenunberofeaohtypeofptoductproducedin ea.hqutrterof 2007.
Pmjlg&I!gq&tle!g_949!L"'r)
qurndty ProdN.d Erch Qurrt r
prcduct
-T-----B-----C
Expenser
Materials 20
Assembly 12
Psckaging 2
26
15
|
P ro d u c t l2 3 4
---T---T30--3t5-
11
l0
2
B
c
320 2n n5 250
485 410 415 3q)
Writetwo matricesC rnd I $mmarizingtheaboveinfofi$tiod a.Ddshowhowmrtrix multiplicotiotrcanbeus€d
to poducea matix tbal givesthetot6lcostsfor €achtypeofexpensefor eachquarter.
r#Ft?"il
u (-
Erbrtle:
f\+t t{\artiia.otoSt fr"
,10,L:
n7i*',h<
b11291)
)atD 215
I
c f Qt{ .1ro ar< "w
Uqo)
&,o't+ttl
xg* rffr +?v*bo + r'1e+(5
"
t*t,l absuS 2toto
ttqgo
Mrrn.lt!t<o
(wL L lqqo
lst'o
3lb05
tl115
t'6+t
4
2o,{ooI
11,01e I
,1"0l
Page?
Mdr&rjrm@ll*n'ry
a. 1de6or*n.4
4to.tuMbd,!4rlll:'6,h.4brd.4sBtu,dsi4r)
wortsoutin thegymeveryday.Thewortoutsiary ftom stscnuous
A groupofphysicalfitr€ssdevotees
to moderate
to hght. Whedtheir exercis€routinewasrecorde4thefollowingobsenationl,a3 made:Ofthe peoplewhowork
out skenuously
otra panicularday,40olowill workoutstrenuoullythenextday,45%will wod<out moderately
the
people
next
p€dicula.
nextdat endtherestwill do a lighl workoutth€
day.Oftho
whoworkout moderatelyon a
moderatel,and liehtly (resp€ctively)
d^y, 5V/", 30% arfr 2(Y" will work out strenuously,
the no( day. Of the
peoplcworkiry out lightly on a pqnicularday,45yo,3U/oLad25o/owill \xork out strenuously,
modemtely,and
lighdy (r€specli!€Iy)thenextday.
(a) Writethe transitionmarrixfor this Markovproc€ss.Ifthis is anabsorbingMarkovprocess,orderthe states
q,lYt{.^+
sothat all at8orbingstalasaralistedfrst.
Jj*a^,c*5P5r':*t
iA
"f? T'J '1r|
--I2
(as- .-Uf€u*
fl4-Ynodn
Jl 'a
r,ls+cnl,a5 ,j ,101
t-hal* *nlr o,rr
:u* s[rd ,r ,rdl
(b) wlrat is the probability6!t 6 per6onwho did a sifmuousworkouton Thesdaywill do ! light,\rorkouton
Friday,(same
week)?
.\--J'
. .-.
tA4t
G7
1
ifiu
\!_,
(od j"
5 tl4..4
r rw VrmL*z*nlt n-i-r, 1
L o * - a * + 0 .a
L',,
<rr1
'r
s[{453b-'l
o.l Xte l f
X"--i$ia-d,/,r=r'" x,'fEftl17
L_
^T t '.wortout,
lat Uandro
l do
(c)-Suppose
$ar ona panicularMonday80''/odo a sEmuousworkout.IOZ do a mo&rate
10/o
percentages
a light workout.Whatwill th€
?
'rsgl_ v -:[.g -l
n. -tl.\
for eachtypeof workoutlmk like ooGnel@f
Oe-.1!!!gggg_
0n,wrli05da14 ',tr t '
^o'lt
I
= [,rrlr,ol
,1.,,,y_,,X0
l."i"rrl
ttr t1t*"hi,*{i;"!fr.2
y ,T1v
[.
ffif,#r;rt.#'J''"
(d) Is lhis o rc8ularMorkovprocessor anabsorbirgMarkovproocss?
lf r€gular,find th€steady--staG
distriturion
proce$.
voclorfor this Markov
lf ahortin& determinethe long-termbehaviorof this Markovproc€ss.lf
ncithcr,cxpleinwhy.
av posit.,Q-'
(rg^tar einno a!'a-a'^"kt'ost6l
\'[i)
*
dr;trivutior' ./octar
IX=X
[.+ .s .+<lf'.] fxl
. 4x+ .5 J+ .* s L=x
4 , + s d r,4 r,4 t -- 5
Lli7 kll2J',St1
S yt 5uoa"-s.k
-.{aX +,55 r,zl5.t --o
-', .45i{-;t1 +,5t =o
, 15*+ ,.r5 t ,*st'- L
,r< * + ' \-,ts t
=o
-I
h:i+fl^i ii\n:q
Pag€8
h6rdMol]i&R'ry
9. Sotvedrc followitrgsyst€nsof €Wari63. If lhqr lre irfuitely h6It.Jsoirtions,stde so andgive6€ ldr'dneEic
solutio[ Ifther€ is no solution,stateso.
= 4y+z
= -r+52
= 14-y
-3'
(a) y-1
2t+z
-1X- 4q-L= D
l
f-r -4r -5
y r i4-st'1
'
|
.
-
aXft1r€-''r
\- z
X
Li\
ttl
---\
,
/'
-
1"^ \ r r ok lloo
11 . l- : - : - n l^
, ,
I
IU \ D
r llrl I
loo
( €r.r
-.
lot
9t +2y - 62
3x-v-22
-3r-iy+z'
6x-2y-42
x 9?
h ' t.e - b
l,r -1
xt
\-aol-Lo
x - - i\ - Lu
2"
O=l Fo-!e!
\ 0:D +"n"s
t-rt
3 r-tr ..,+rb -! 3 t-tf
,r+J
4
LL .L
7
)
8Jt.1-i&'-)
-7
-20
kis&tr3
l"Y 5=t utt^r/o
\ --
\'1/t
\l
'w _ D.l \
2x-Y =
=
"'6x- 3 y+42
(u -t
L.- Lu '
Y"r.I
L
o' lh
^ f\ r a
Ito
r,1
t
g'/21
= l)
-* t7 l _t I *o l o
l1
ll
o
I
o
l b -q ,, ll5
,[2
[
|
1., t
-t -+ lr.1
x
\\7
.L
t'- +
o g)\
rw!. ttY'"lW'
t=l
L--
-2 ,
Nofd.,,A
Page9
Md16rdldolj!&klry
10. Dercrminewh€tler eachof the following is the tralsition matdx for a r€gularMarkov process,an abdorbing
Mar&ovprocess,or neith€r.If tio trdsitiotr malrix is for anabsoibiryMad@vFoc€ss,Eordertbe ststesso that
,ff slnsnrbinesraresaJ€first. sndthenfind de llnitins narrix.
(.1
. 11
^
.2-s1
,,8
a+ T7 t ,'= l'-tr.n
I
L-oot,
r
.08
.31J
L.
trt fl :1:
l r'%ltI 3?3ll. -
a.^;_yaslfiy
A,rttz aWoW;rs ll$T2
0
Ahryt
ot^ A.bis<bi^^
ttL+e-,
(,
n 0.,
ef it i i r
olol o 0 1l =\
at porr5
Lo"rc
LLI 0 0l
ilJd-**
p-.\
l|
o,"J19
LO*l
l0
T:
r^ "fi
ll,',-l i 3 l ,
lo r uJ
r - loe
r l- \
-4DJ
O
LI
aWb\i5
L'[4-21l1
$
(o , ol-r
itnkl^Q
(
i5.LrXu2ar.
\J'.1
u,t
o
o'5
I
o'u
I
:l
19o 'J
Page10
M16FnreOlrleRlq
I l,
trctr6rd,sd*e.ds6r)
1r&dtu*tr,r!.1sd.ejfub4rr61r'l&/k
The 6gur€belowgiveslhe layoutof a housewith four roomscomectedby doors. Room I containsa 0ive)
mou3etmp,
and Room2 containsch€es€.A mouse,after beingplacedh eitherRmrn 3 or 4, wi[ s€archfor
chees€:ifunsuccessfirlaftertwo minutes,it will €xit to anothdroomby selectedoneofthe doon at random.A
mous€edterjngRoomI will b€ trnp?edaDdthereforeno longermoveto othermoms, Also, a mouseent€ring
Roorn2 will rqnain in thatroom.
(a) Writg lhe traruitionmatrixfor this Markovprocess.If this is anahorbing MarkovFocess,orderthe sbt€s
sothat all abso$ingsta!.sar€list€dtusL !t!ks
LiStl
bq fte{u ,/]'/r-'..J.fli-,
tL
-t{- -
34
r f r olo +l
alo rli
tl
rlo o l0 fi l
4Lo o I ? o J
th) Tfr mouscbesiN in Room l. whar is the ombabiliw that i! will find the c[ees€.aft€r8 minur€s?
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