tochoskic paocwsas defination
TRANSCRIPT
ONIT-1
tochoskic PAocwsAS Defination
o Stochastie phoCEssS a foubluy
Of hanolorm VaniaublsXCt)s teT , wneet
wSualy donekis time not is cut ewey time at
t n the St T. o hancom hundber xtt) as
o bseveo.
Dpincbion:x(t), teT} a oliscacke ime
Pho Css of t the Acdr inte on countable.
Pcvhacdine Hus neall mion =In
To,1,2,3. . -
Thus a olisCARAA time pAociss is
x Co), XC1), x(2), - a Aanolom ousso clotiol wtHh eNe time to, I,2. -
=Xte)teT} o continuo us tinml
PsoCsiTAS hot finitt on countobe.
= In Phouctce this qenehaly meons
T Lo,oJ ov T-Lo, kJ jok 3omt h.
Thws Countinuo ws Ai mi PAO CS
t)teT hos Aondom hilben xl t)
assoctaliel h eve fnstoant An tume,
Clossiications of Stochastic psocSeS.
A Stochassic PAoceA O
PAnbab &tid moolul dUseribing o Collock on
of me ohdonol Aanoloh yaiaubles that
hap hsent the poebo Sounpll pouths, Stocbasbe
Phocss e on Le chcsbepiol on Hhe bass ef
th& natunl of ther pakamttii s poucl ocnX 3 tate oLnel S tate
S Pacl.
CAoskirtcation of Stolls Ounol choin
Iiuoucib lt onel cuuicto
Choun which ail not nlolucible
s Coullel heuiccb.le oi noh -ineduicibl the
t.P. m AReluiciLLL.
I has tuwo clossipicotton (i paêmitive f oplnlcolic) tp.m PAerniik
Cii) Imphimbtive lperioolic) it t.m.p inpemitie In Mecucible choin al St abs
belongto he Saunn& clovsS.
CAossitcation @ Statis Mn a Markov choin To wdnslamol the n-StipP thansite To
dtail we netol to Stuely bow mol
mathlmabict ons classity he Stolis ef o moukoy
Choin,
The fo lousing transhion meubin
usknatus hiDSE of t joMowing ufinitnous.
haphical Axprsentotion S houwn m Hh tooK
CStat&-Tlansition diaak
-b
-5 O
3
2
EOumplDs Of Stochasic PhocoSS
Eoumpls 1 LeA wwr ,ws. nel and
Hhe iml incle hbe finits oLnLN. A
Stochasic PAo S thssetlng C two
dimtrbiCnal ameuy oh m otru uch at.
X (e) Xa(w)
Xa( ) X2 (ua)
LXNtW)
Eucn Aow Aphsemts a hcnclom vatiabAl
anel eouch Colum s a Sonnple Poth a
lalization of Hhe Stochastic Phocssx i the
time inole ts wnmbouneieol ecch Sanmpli Pabh
iven initt sec,ucl. OUn
E Xanmples of 6uch 3 tochasMc Piocesss
ncuele the stene PAocUss Of bAouwnion motiC
PAocOsseS use b Aoces oh bo howyion mo tion
ocis Bacheil o Stuoy Psic emonqes on the
Paws Bowrse aned the Posson PAo C5 useol
A.K.erlamg to Stuely t he nunmber of Phoht Ceulls
m
UNIT-IL Monkon chotns
Ifx(e) ster} o 3tochosiic S
Phocoss Such hak oiven 'Veue of xs) Hue Velut
tt) t7s olo not ollo eMl ovne he Voulut c
X(u) LLS hl Stochcsttc Pho ess n s calllcl cus
malken chaine
HLahl rd thansitiou PhoboblEt In 9pnehal a hiopher hdti thansilis
PAObacbilty Coun e olefintol ous o pAobacbilEty
PAmeydY==P () which means that the Po CWSs have a
S taiteol fhom the
Thorstion Paobability
Consiole atsms amallculs that ae
Aoulh Cnta St outem getcngexcitiel oma
to a hiqher tng state 'n' by te absarptlon ob
Let Nnn be he
humb eh o hounsitions PL
Seconol PAom m bo n. Them m
NmN Popartohal to the numo er of SPecits v State
m anol to thl riumle er of Photons, folllnc, on the
S ournp pli econod,
N mn Nm(Vmn) => Nmn Bmn Wm(Vmn Dmn -Emslerih hansition Proboub ity of cub sarption.
Chapmon - kolmeeyrov E9,uaiions
Thack btots evoluution boslol en
Aransit" on PAobabiALs betwrn all stovtesRn)
the Probabihns of beine n 6tall ouft
uansiu en /S teps when &tariene n s tote i
Py (a)he Pej P Ca)-he Peg On-1) Pim) Pe tn-bn) oL mbn
K K
Ths Pho vicles o 3implt wston to
COmpuuke higher OAoles bransite'on Probe-lLtes.
CAosificalion of stoutes
1. A ccs.cbte
a. COmmunicodle
(a) Rohtnt vity Cb)Sq mm
Cc) TAansitivity
3. Absorbinq
ROcuunt oY) PÜsAAint
Ciy TAanint stoll Cii) PAouooie 8tate
Civ) PAlsAAnt Stote Cif) E Aesdic Alete
R The miting behoytous o stp bhanmition Phobabillty
katemlk:1
weduclable oncl Fo o nite
O- peuioolic makov chodn fo all n zN he nh
Stap Po babiuty moutst Consists M known os
o eltrments.
The chouin finitk onel imloluciable
h Th skole ommunicoule each Others,
Then by olehinAon therk e bSts on b
indusea hltdy . sucn Hok Py(n) So Probaub ulity
thoot the State Alaches n nhsbep pati
The choin 0-perioolic anel oubove 4
tue fo oll Lnij) z NCidaltnibults of Steps now
Then @ anynz m > Peljy (m) 7o .
Statemunt a In a nete maukov hoin allths
St outl ccLhot be thanstton. Th statemlnt St ale
oMod s fiom the Soct that arny thansiti on 3tate
Ltm P (m) = o h
ancl n a thonsistent PADbocblEt maniv
Pn)=1 rd
Enourmple1 ot
Thansistnt Paobouh uty of Makov chouin wtn 3tatus Co, l.- .. ) at
P 1/2
The 9iven mankov choln heoluass teuble
becausl every Statt Coun be ana othh btotus
aut aun no of tiansistnt.
boulalloh
thot P iven
-PxP =P V2 22 /2 0 2
1/2 p t/2 /2
/2 /2 1/2 P3-Pxp= 1/2
2 2 D
1/21/2
p P 2 o/2
OLO V2 o/2z
1/2 o1/2
LV2 o 2
PCn) Th hunthal p)- p(2) , pn-pCn)
that iplos Pj (an) 7 far each i) Pr2n- fat each ?
thenR staks at perloolic toith paramekr to Pez
m
Stochaste PhoMs UNIT ()
r'oiss on Phowss
T Nt), tzo} C
O Contina Conting Counine PAoOAS wsi th nlo) -0 , Nlt)
Suid t o be Potson Phoass. OL
oisson Procoss - 91Asunmpioh With ALspecA to thl Pouson Phoss5
the 3ollo uwina assump tlon aL macde.
1 Th nCnRnveut rCoull inclepoandent.cfe) For t' iml Peint t,ta. .tn Such hok ti t2 .-.. tn tn
XCEi), X(t)- X(t,),. . X¬n) -X (bn-l) arl nolepevdliuit
Thawsiton Phobabdlb 5olisbles in 3takiera PAobab ly. tie) thi distbibuttons of t)- x(S), SLt
H ag time intewU buk hot on (S') dup enels Chly
. Probabillty that atleask one eveut happln in tthus
Smaull nkivoul of time nn7o aiven by
PCh)-Ch) +o (h) hese o (h)/h 0 shsb
Pobobity hot Hae mar then cne eNen
happen n Small inbtiva oCh h' ch lnaplEsh thhe
4PL1 Occwmune dn lt, t +Dt) Dt 1 o(DE)
5.POocuvne in (t,t +0t)1-)D6 +a (4t) 6.PL2 lor) Mone occwvutt fn lt,t t14t)= o A (E)
7. Xt) depenolnt ot
Othe event any intovoul pAah lov) aftir he
fntenva to, t)
8. The Phobabic theut Hhe evenis oCC ui e Spiciel
u mbeh o mls in (to tott) depencs during eny t
but hot Cn to.
Drivation of dith eAunce ot disteontiabe Pobssen
Pho cs Let be the holo O CCwlMcs loy)
Ne of o cc wwwncs POR umt me aiol PnCty b e the
Phob aub Mty of OCC nCus of eyent in the
intllval to t ) a Peiss ba dlistrcbuton oith Peicmetes
-At CAt) e PLxt)-n n 0,', 2
-At Prlt) e CAty
Proof PCt)-P xtt)=n]
Plt bE). PX(tiDt ) -h r Ch-1) dccwvuncus (o, t) ,
1OCwsentts s tt, t10tJ n Occwirtnces aco, t3 Y
ho OccwnLnCAs Ctt Fot) J. Ch-) occUvencis 4s Co, t ) cuel Sn noccWiin
occuvencts t, ttot) J o OCC nt
Co,t
tttDE)J|
-Ph Lt) A2Pnlt) C1- ) t)
Pnlt) )DtPn(1) -Pa lt
PhLt+0t -Bal)a Pa-,(t)-Pt
Aim PrltDt) -Pa lt) im n-,t) -Pr (t) 1-0
ol Prlt)à Pn.Ct3 -Pa (t
A SSume he doltion o 0 be
Pnlt) (at)f(t)
(ty [PaCt
P(4)Aht t) a thft tt).
Publeng nol in 0
CAtyh- Ln
t"f'tt) = Ah f(t) In-)
'tt )=- -hAt)tt
olf Lt) àf lt) ol
dflt = -h olt
olfLt) -A Jolt fCe)
Aoftej- ft) kera
Co) o lo) -?l (o) roT Pluo event o ccwes ih to.0 fco) -
Sub siitutung n -o
flo) ke kà )
Put K-l fn At -flt) = e
Substltulinq din Phlt)- P)x lt) = h
Pn lt) h 0 , 2
Mean o he Po-Aon PAo uss =? Elx (t) zAt
Vanionc of th PolMon Phows=> Vair te)ye At
Conelatieh of me Poisen PAocoss
Rxy (t1, ta) - At, ta. ) min (ti, ta)
Stochostic Pho (OASS UNIT - V
Statlonoy Phos886
hounclom Pho ess s soiol s to be
Stottonauy ts Mean , varionde, Moment, ctc., al Cons tant Othi Phoess oe c alltel on- 3tctuovay
Tst oneln stationahy Phocus Aehintbion:1
hondom Pro ass Calllel Stabienaty to ohel oul @ finst @rl Stalioney if ts
36t olsn staionauy denbta funtion does not
Clhonae wth o shit in thime arieyn.
M ust be trul for au ti aunel t, ans helel numeep
ctn/t?10 ber a t orelr statoveq Pprow5.
hinitton Tf Hhe Phoctss JAt aholis 3tationary
Mecun- E(XCt)) = Cos tourt
StcOol -0oles statiooy Pho (6
haholom PAo Cesss socol to loe
o secenel olus tottovefug PAOCCss. Tf s seconl O
Aeler &talionay odansity nciioh,
XXati.ta) (x1, X2, t, tC1, ta tCa) Ys2 So hot chan witn H°me wht X =A lti) ; X2 =Xlta)
S hong stationany Pho cass5 hanclom Proes calliel as o
Sthony Bkaliorcu phcaS n shiot 3ense Stoioncry
PhrocRss (sss Piocwss ) is he cll ils inik dlemensiGual
stibuitons wnelen thanilaton af ohk Varfaut
4ime
(x X2s t, ta) = K (1,2 tp9Ci starc)
Fx (1,2,M3 , ti, t2 , ta) = (1, K2 , M3 , b1tCi p t2t C2 s t3tCs)
F (1,a ..-n st, t2 - tn) = FxLxI 5Xa 6tC tn
Jewntly Stoutionoa in he sbruct 3 ense
xlt) cunelYlt)f or celcl to le
ont S talioua Sbaict 3ensl. IF TS
oint disthi bulton exlt onel, y L6), inolepenoluut
Amom VarualeA.
Meon o O Aondom Phouss
HRte)E Lxtt - t Lo
A xE)J s also callel mom dunctioh (ar) ens Cmbil
av eraat honclm pho WS
Auto Comelaltou oo hanelom PAo SS
Let xlti) anol X (t2) be ttne Tuto ven
numbes cof the cunelom PAoCAS Xlt)y tme cuto
Commlaton s iven b
Rxx (tt2 )- E xt 1) , A lta)
Mecn S9ua Valuk F ot Aonelon Pho ceS
Putting lti,t2) - t in eL hove
Rxt,t) Elxlt), x(t)
Rxx (t,t) Et) te me cn 69pues Value
o honolom PAo cLss
Auto Co Vatioucl a ahcunelo m Aocess
KreLtDtes) -Et-Ealto] Ixlts) EL (t]} Rxr (tiit2) - Efxlt]-Ex(t3)
Rxx It) (t+) e Ex (tn-EK (t)
Comelallen coecLent
The comelatton coetciet o the
Sonolom PAocosslt1 s ujintel os b
Exx (ti, ta) -A (XE) lta)]
Van lti)XVcnx Lé2)
hehe Xx lED lt2)) lenotes he auto covaionil,
CADO'ss conelatuon
The CAOSs Coreloion af the tio
|hanclom Pho oss fxte0 onel ytt)}s dgieel by
Rxylt,t) - E )xti).y lt2)
Wiole-Sns tattovory PhocoM
haelom Potsxtt)} Coulleel
w Rekly Skaionauy PAoctsS Cor) covarioticn
Staticne Phocess Cov) cloROEncr st ablonahy
Pho ws
ibExlt )} = constent
f E Xt)r (tt0]-Rxx tt depents
ow onT
herR T - t1-t
M
Stochastc PhoAS - UNIT (7)
Bhownian Movemlnt
The Aonolom motion ot Small colloiclel
Pattcles Suspenelil n a uuiol ah os mkoli um,
Cousel by me Colsion of he mlelium's molecules with Hut Porticles. Aso Calllel BAoush ian movemlnk.
Wienei PhoasK
nt (MY) as a nomad olishibatio with ml oun onel Vcuncunde V. A Vaiable z follouss
winh PAo CSif
Ci) The chounee n z n a bmallinteaVal oTime Ab z
(1T) A = Z vAt herR s lo,)
hn) The valuls o Dz }6n an olitennt (hOn -
overlappang ) perisds ot time ae 1nolep emolent.
E Vauttouoy Phocss
hcnolo m PiocoS hot not
Statiouou in n SncR as Collil eNaluiuove
Pho coss.
Probm for evautuionosuy Phoccss
Let as Conelilh a Aanolo m PAoOss XCt)- A CoS (8t +G) niiR A k w ahL Cons tant ane
unitornmly Asib utiel Acnelo m vorlablk T noaval (0,2_T)
Since nlyorrmlg olisbcllil n lo.27) colhave L7 f(8)-
EXLEJ xle) f (0) dr x(t) fo)dx 27
)A cos (wt} f0). %n ole 27
AJcos CuOt tD) olu 2 27 Alp n3in fwt i8]
A/2n Sin(an; wt)-Sin Cot))
AlnSin twt -Sfnwt O
theik o k huuren tbtion s Statlotag
ecun o he PeVSh , _) Ext) -= ht Precss
Vaience Hhe Poisson =>Van x ti = t
Pro CRs
Cololton o the Rxy (ti, ta) =t,ta tA tmin lbitj
Pe-bs on Pho coss
CO-VouNance oTE -Cxy (ti,t2) - A minlt), ta) Polbson Phoccss J