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trtrCs //-3?of -ra?o W kM c hapltr Q H./"* 3 tt C *ln, a. 3 , ?, iss o rt prruo ) v,k;. JTt A,a rz-r3

EX3 Poisson Process

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Page 1: EX3 Poisson Process

trtrCs //-3?of -ra?o W kM

c hapltr Q H./"* 3tt

C *ln, a. 3 , ?, iss o rt prruo )

v,k;.JTt A,a rz-r3

Page 2: EX3 Poisson Process

/' (',"i .)'.'

Tr^ d*l* Gtlt,) ) *l1a: f*ckat'i*A::tXi;Fyrcqss /s avt, * 4 e!*; e*J & F*i:e-r*rr'Foo.*{ { r{on* #=yur':!E u'suJ pwcess

itl &ulNlli"? fr' €bY7)

fr s+nCosfi. P"{€srf*ror lr>"}

't?or)e O!W@FT^,T &eTntevuc& a?e

:,5

,- ,rl.J\

l

Page 3: EX3 Poisson Process

:j,.iil

e>-1 ,'/ t

.' /-.. ., ':=i ] ..:r, ,'::, / I i- -('i" *,,' iit, I

t 'ft

7. ^'Pa -,1t t r, V{7) VY-J J \/ \-q-E

i 'i-'t-

't'

/\ty't 'i.if. --1.I tl l"9tN .' i'.VJ' ,\'1Y-

Bc*l s

/',tl .1 )

lK tt-l ,.', ()'\ /'l'/!\

, r?rJ)" (,

, l-+ )'t'-"',1 : \ , )'-,'

"'ilJr, ,,u{',"r,,,: +,3:t' t),1{1//'

t'f ' ft"'l \/r t \t \

?nL ."t' 'v1

Fr*uvvJ i

J\" i .{ } )" n t? i -4"t

C^yy irJ rrH i:n t{Ytl€ l /'

)-$,kt}= ?11

aM / r/]c. */ lt> >"

1, ?wyezties L$$qxi t' C{',,n 6 P*^sliv'

?';cass ,

;;; o r'oes ; {" 5f 4+

=?*l At+*++

n!

{Hotl

0ru,r) t,!u'#=f,prab ,{ ffi'ia,^ tuteYuJ aL

,.b- pn?rt+l-o$4 To'*4Y'r& r'tftl* '7'i

Page 4: EX3 Poisson Process

\.J>, i

< B? Arrir&

'r).t-,# --L ->)

,.'\" <';t :'f' l) i, ':;' -'

/,ah- l. )

A,re ' lvl€ Dt +r I l€fS//^ t/

1,;d /

Pu

{ Sam* #-4 h Jef,^,fior

h nr** , ltp way wribe

{at} ' Pot frd*c+} * frt+}= oJ

| ;A*€ + e( ^+)PT'i'

R\?, te+1 = P" f A tt+*+) -fi{tJ = c:

= Aat * ltar}

h 1) = W(#r++o4 *A&J) ?J

iL/aX,

!,Q'&*a o

# Deyit'&iu^ o$ '?tWn

G,Yl ivt* inel

-"t

= C { e4 } + fr rsqnp t i pa a&ts{Ln-

f)r t

(nLfior€ *han 0,r, ewi ,.,r! /nC"g:

-n C <ae)(u*o, *

nrcb [n e

€j7n&w*8

Page 5: EX3 Poisson Process

, ..+'

\J ]Lrt I *-- .+l'-)\t'

.:c) E /n( ?) Arirtr).0,,' IJ t,-I ,fuf'L t &ra t?rit -{2 "*i4 l

f"t ffi *t *rr,o4 tr *lF^{ g #.rvi u& *ie l

,#"4.H

{tri

6\!F U !-+t*' | 7""'-

I ii'..*2n42/- :,' ')?( 'i/ {.'. 4rval\

'.:,':' :i ti. .-.'.i. '.

,D-, rei g O

. .-bs"-q-p

$-{fu #e e{?'rA *Ad fihe ?nr*z {a-* y

WrrA'U ra*,cL #' E

P"t q} -t* aYri'to!' bf-3 &?*t &' dvryo! -n,-a+J$ gt f,ssu,wPtiou b S

{n*L *hJ,t f*r',r"4 teYYo

Crn, y,pT g,...'^,4yi$.{+€ € 4"n #,J WS..,i**

I S,J4r?)e

P* tt+ ^& )

?*{4 + At} = Potn ayyiool b {'J ' {,0 o ziuJ

/'* a+J + |*Cv+ Ew|uJ in€ J"

?oC 4 a";sJ l" *4 I ,

).f l- }'A*+#rst)]+-)tA A,t,+ Ct**lJ

f* fi*^ b2 l, 2t *zd ) *J

Page 6: EX3 Poisson Process

t' )>D

+[i + eHYl?y,qtfuL-frl

l-._i:f'/ii \.,rj I

i

'1r/, '' t' :,1 ..7; -.'-:;\-:,

l+1,vJ

nIt

the /uL"*io(ev b1

/\n' B sP-{sJ:

@ bE #"nd Lrl {. *ro,.{.vY:r\r

fi,o"ttuneous C'ff fr )

Ag r.l' 3 {'i(s} =

hro;= -; ?r# +A&d{$)

P, tol + iPo-r (s;

s+J\?{ {^}t

+h#tsulwQ w*?J Gd*l,sr.c

+lA4 &"Yiud f>eeeg

Page 7: EX3 Poisson Process

i -t- 1, I,ir7' ;itfE .j ,i 'l'-

\)l ) '',,' in)= Ii f, \t*./lv tlrrtfI ?n u L. ,t 0 nYrlo*-Y J

| | " * ot'

Ir-' ?" f,*"e €*t *J

.) _

cun4 P" e n) =' I .''J

{ P rn$ ' "/ n e'?'/ '' '-l lL;dlne + hZ) J

ti,r* t^ P' es] =

Pn* t'i u'a :{1

L5+A

AP--t {s}aA-,

s+A

TottJ=(a*)*<-G

n{

Tytcfl,'a*/^i

He &^ou?f,hitss1n

bia{ri *,,r*}**'

{,Qrdlrv

fc*t'

-A+ tlo€+a, ?l,nts'g

P***)=

G) Avercpe ,#.:{ r*'ye-ieA # "l *" ' 4

Ei N{"+:JE E *7^t+} s{* t . _=/ eneIo'f.,rtd{, ;--: ^h-!.r* r^ : iryM)f{,}_{

n=? &4 !

1r*41 \1

^, & ta rf,

-4fl T * {J14J? t ,. -L, 4.i y/

B=g '/ '

E ^-if. i,-,f Qilo= e- e,a4fuw

Page 8: EX3 Poisson Process

tJ.!ll ,\,-.,: /,>/i .r t;

-l/J*rtl € { u€{**

:.'!._j 1

liI,::a

-/ :--:j;t--'_- - i --

- -.>=_

{,,){:tI

-d- q a .::':!)aa /'t')

..+aafl; filru =: e FtN l( hr6 us?ng t f'1/{ N4}3 }

= F f/v {N:il3 + Ef fl3 - €Zrv]

sltwz EtN{N*fiJ= f n{n+P,{+tn= o

= u4,efl" u(n+)Q+)'

= Z-** ;A*):,ZW?: €,-Hei#*

fuil"

ffiNrc; 3 A= iL*i Y Exveftl ( eutev*ft+ )

# # "i &,/iaJu "cureJ

at$ah

Page 9: EX3 Poisson Process

-: - ) ,). )--ia' i ;',t r i b e;( r' ar.t

( 7n ,Da i ss, zt iDva ee;a )

,,

y. V denutiry dJe Tn av- ryivoli *ffvt'z

( 7, e - Trne fufueco /4'4jee'ss?ue arviva{

of & fe?srt"n frrri u'a! futeess )

Met": Tn po?-sson A"irr7 pracesr , ^# ':f g*dfs Ti?scT gts{'ib'PJ ' 8"f

( 'fr Disv'ede r' v l'

Tnt'erwvi{^(' t)r*e C [*rhw"

6wiu*{s ) 7s &ntin uoL{s ?' v-+ ttt"i is

I

t t -r--,l atP ,7L.gU | )

7,e

C I -v *le kh'vv'J LE*;u, 6rr tu

7n'hrvJAeJrne4"

fr;F. Tr'fur-4n; '-! €me 7r-

?risson fiwiuJ fuc<ss '

Page 10: EX3 Poisson Process

iDn b. ri,sl = "/ -/- Lt, . :'i':"'_: - _, ) '-t'?)

t /'ta =i'=t,--'|"fti 4 A. I T3+ I = '"f :(si evriry=t* -E"'. 'r '- l"u g : e :

' 4= u

/ *' fu { w, &*iv*0 r^ ,( as *;5 " .,

= I * Po*] = I - e-Aa ("={^',,' :;t';',,i,,F7t4) * l- P^ { T> +} \

f* & aYri u,! >n { o,ffil /

t):ff) 3 i)

*n-FI/ 'ef,/

'' il1'1't

i ) ,.

..\ -DY-+ / 11) 'Y

@ Wsfr; l''frJ

_LA huor," u{ A ,

x

)

t Variorr,* tf TcIT'J = lF"{*r, Ju',[t"an] {dt

*ry'= sIT'J -e7TJ= *

/ b€caeEs e(

Tis dens;'fi {" A.n *

", - -J t i;t4lsi ^ u,! )fr(*] = Frc{; = }e (,,Af-r?uf," ,,

#*rul &J€ @, @c&J. -drt// P.io., frrru"/ ?n^ss lo,

| --

Me* ( frveraee ) .f T? C,alinous ' Y' V

€til"- [^*frc+tJt = [*uJe-]+/t,Jo Lrt Jo

Page 11: EX3 Poisson Process

14a r -':.,/a e:F,f ofl€,ri t'n-f 4) s-lyi A rr-/) j / -., 7t /t) t.,' t :,"'1

/l ,)rl / ' //^l ) -] rflesnay E/e ss Ck etze afe r At =th e.T

b +

a ^ o

T,€ . TAer€ Aa_r &ero &n grrivd & t* & ,a*J gane rn pt'er y# tor*) , ffi+"

tc, t+ Til?/

-Shre fuL"t // PrJ. "f a/n

F#€=*

arriy& rn ht"vvn{,--(*t #o}

{,J{€f is 4" WrA t an 6a,rtv& D,atz i#

oer+7,'ft,

I,i€L#i.-- iz-| -tl

1 J'

Mse no-@alqd:=...;t''.=::-

7n tL)s eaA-e- )_;. _-4. _ __ ) da* .f *JlSTi a,ie dn,s;7ood- T o,o #77r"/ ooira! ,P r T< t+z iTzt]{/L &

= E,f T< t+TJ t) P*t T>tJf^ t TZt 3

n1I'n I f I IIt, \ v

-f" tt+z) - Frt*)

P^ LTzt J / - F7t*)1,

c l - aAtt+t'J -[, t .-,"-^. ]Te-^t J

-\2.?I -

lC-/'.-ll.n

Page 12: EX3 Poisson Process

':/i/,

Jl,>, i,lrtil,/j 2 ( :[ i + ,Fz f r7t, :1 --- /,A f -T]:1 =id {A C ld / -> i

Fn* #;s yps^H, 'te **a s?y €$a'*;!/ n ! r A rt n J.* : r oln

t' n6 [ /]n fryrivo-{ in Cey{ain Tntentr*u*tI

deyer4" _un/y elt 4e durnfian 'r' &EthM ie @ (n,r deynlsoa)

"F furvi ous ars,id, tlt- t

7e. r*pwcrx ,{' tl*ses

,&

po issory f,ry;,r.!' 't{s. fuferaryrvr!dre 7nJ'rf*"Ja"C a-,{

zfi. lnem"yy/ess Pf$ t*€nt)omrJ o!/,dr4n u; -{' ',tfuoi&ey-

s: * Gtar;{,

ATsh&rrtilJ/ dist;J,^t / !

/1 t4 o-Ls {!Lu*s fro

lnTOt'

rt!% t;i+L nJ'fn^*o*"^frwiuJ *in€,

d"^J P"isssn rt*iY*( ra-Z*

Page 13: EX3 Poisson Process

pnmss u6*re flas*cn tit fir\rn ,{ z*

ryealfeJ.

:1

:;' -i j:1. i

,,: ?'.1t -''7'7.,)):/s zu .Jffi ,e/s as//.o

'fr o. ,n,4€ias &o D4fusat #,.J

fffinrc ,7s

I.++*Y,t * r I t t>t

Le* {Xr*, f +>o} L* a. f4.P#T t*or

7{ tn_/ 1*, , dea,l

Tit { Xtrrs < x" I frr*s, t{ in-,j= fr { Kttnt<r, lxftr-,)}

N(+n-r), --'"^lq

xceny, ) f-Ket,'t?j

fr, 0a, &ce/n, hle tJt// d€"/ "n&7 a fuf-u;

/. EitrrrA *r?re,arrJ p, &+4uoo, *rwe

/* as,+L gitrr& s{e.t4o,kr Cfia,\ ) fr l4,f *stA