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