Expectation and Moments
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- Ambrose Harrington
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1 Her Sr d Joh W. Woods robbl Sscs d Rdom Vrbles or geers 4h ed. erso duco Ic.. ISB: Cher 4 eco d omes Secos 4. eced Vlue o Rdom Vrble 5 O he Vld o quo Codol ecos Codol eco s Rdom Vrble 9 4. omes o Rdom Vrbles 4 Jo omes 46 roeres o Ucorreled Rdom Vrbles 48 Jol Guss Rdom Vrbles Chebshev d Schr Iequles 55 rov Iequl 57 The Schr Iequl ome-geerg Fucos Chero Boud Chrcersc Fucos 66 Jo Chrcersc Fucos 7 The Cerl Lm Theorem Addol mles 8 Summr 8 roblems 84 Reereces 9 Addol Redg 94 oes d gures re bsed o or e rom merls he course eboo: robbl Sscs d Rdom rocesses or geers 4h ed. Her Sr d Joh W. Woods erso duco Ic.. B.J. Bu Fll 6 o 46 C 8
2 4. eced Vlue o Rdom Vrble 5 e Vlue: he eeced me vlue o mesuremes o rocess volvg rdom vrble. Ths s commol clled he eeco oeror or eeced vlue o d s mhemcll descrbed s: For lboror eermes he eeced vlue o volge mesureme c be hough o s he DC volge. d For dscree rdom vrbles he egro becomes summo d r Geerl coce o eeced vlue I geerl he eeced vlue o uco s: smg rmeer: g g d g g g r g g I e o he eeced vlue ou hve smle esme o uure eeced oucomes. d Or or g ˆ ˆ g oes d gures re bsed o or e rom merls he course eboo: robbl Sscs d Rdom rocesses or geers 4h ed. Her Sr d Joh W. Woods erso duco Ic.. B.J. Bu Fll 6 o 46 C 8
3 mle 4.- eced vlue o Guss d e or e d Leg h d d e d e d e d e d e oes d gures re bsed o or e rom merls he course eboo: robbl Sscs d Rdom rocesses or geers 4h ed. Her Sr d Joh W. Woods erso duco Ic.. B.J. Bu Fll 6 o 46 C 8
4 omes The momes o rdom vrble re deed s he eeced vlue o he oers o he mesured ouu or d r Thereore he me or verge s somemes clled he rs mome. eced e Squred Vlue or Secod ome The me squre vlue or secod mome s d r The secod mome s reled o he verge eerg or oer sgl here he eerg d oer re deed s lm T T T d d lm T T T T d oes d gures re bsed o or e rom merls he course eboo: robbl Sscs d Rdom rocesses or geers 4h ed. Her Sr d Joh W. Woods erso duco Ic.. B.J. Bu Fll 6 4 o 46 C 8
5 oes d gures re bsed o or e rom merls he course eboo: robbl Sscs d Rdom rocesses or geers 4h ed. Her Sr d Joh W. Woods erso duco Ic.. B.J. Bu Fll 6 5 o 46 C 8 Cerl omes d Vrce The cerl momes re he momes o he derece beee rdom vrble d s me. d oce h he rs cerl mome s d d d The secod cerl mome s reerred o s he vrce o he rdom vrble d The squre roo o he vrce s deed s he sdrd devo σ oe h he vrce m lso be comued s: The vrce s equl o he d mome mus he squre o he rs mome.. Aoher esme o uure oucomes s he vlue h mmes he me squred error. ˆ m m error
6 oer d erg Cosderos DC oer/erg s reled o he me squre AC oer/erg s reled o he vrce oce h he d mome hs boh AC d DC oer erms. es d Vrces o Deed des ucos. oes d gures re bsed o or e rom merls he course eboo: robbl Sscs d Rdom rocesses or geers 4h ed. Her Sr d Joh W. Woods erso duco Ic.. B.J. Bu Fll 6 6 o 46 C 8
7 Ler o he eeced vlues ler oeror The ler llog ler oere chrcerscs c be erreed s llog he eeced vlue o uco o be he eeced vlue o he erl elemes o uco. For emle Bu hs c be eded or rere s g g g g The order o egro d summo c lso be chged So h e hve As oher emle g g g g d d d d d d d d d d d oes d gures re bsed o or e rom merls he course eboo: robbl Sscs d Rdom rocesses or geers 4h ed. Her Sr d Joh W. Woods erso duco Ic.. B.J. Bu Fll 6 7 o 46 C 8
8 oes d gures re bsed o or e rom merls he course eboo: robbl Sscs d Rdom rocesses or geers 4h ed. Her Sr d Joh W. Woods erso duco Ic.. B.J. Bu Fll 6 8 o 46 C 8 mles Frs ome o eoel or u e d e Iegrl Tble Formul e e e e e Secod ome o eoel or u e d e Iegrl Tble e e e e e e e e e
9 oes d gures re bsed o or e rom merls he course eboo: robbl Sscs d Rdom rocesses or geers 4h ed. Her Sr d Joh W. Woods erso duco Ic.. B.J. Bu Fll 6 9 o 46 C 8 mle 4.-7 Vrce o Guss d or e d e d e Leg h d d d e d e Iegrg b rs: dv u u v du v e e d e e
10 oes d gures re bsed o or e rom merls he course eboo: robbl Sscs d Rdom rocesses or geers 4h ed. Her Sr d Joh W. Woods erso duco Ic.. B.J. Bu Fll 6 o 46 C 8 mle 4.-5 Geomerc Dsrbuo m Deerme he eeced vlue oe: or d d d d d Ths llos he me vlue o be qucl oud oce s o. Deerme he d mome oe: d d d d
11 oes d gures re bsed o or e rom merls he course eboo: robbl Sscs d Rdom rocesses or geers 4h ed. Her Sr d Joh W. Woods erso duco Ic.. B.J. Bu Fll 6 o 46 C 8 Deerme he vrce
12 4. Codol ecos eced Vlues or Codol robbl The codol eeco s deed s B B B d For dscree R.V. B r B m B B B As oeror he deo should be eeced. mle 4. Codol eeco o uorm R.V F The codo B Thereore: F F F F B F B Usg some umbers d erormg eecos. Assume h he RV s uorm rom o. Wh s he e eeced vlue o. F d d 5 oes d gures re bsed o or e rom merls he course eboo: robbl Sscs d Rdom rocesses or geers 4h ed. Her Sr d Joh W. Woods erso duco Ic.. B.J. Bu Fll 6 o 46 C 8
13 The codol s bove 65 h s he e eeced vlue o. The codo B 65 B F d 5 B B B d 5 65 B eced Vlues or Jo Des ucos d Codol robbl The eeced vlues o jo des ucos here mulle rdom vrbles re volved d m or m o hve codos occurs oe. ublc Helh Cosderos ecs o oe vrble o oher or sscl or robblsc eerme. Deo 4. The eeced vlue o codol robbl. For he jo des uco gve s: We o o he eeced vlue We o rom beore Useuless or lco s he eeced vlue o d d d oes d gures re bsed o or e rom merls he course eboo: robbl Sscs d Rdom rocesses or geers 4h ed. Her Sr d Joh W. Woods erso duco Ic.. B.J. Bu Fll 6 o 46 C 8
14 Alco d d Bu d d d Ths m or m o seem le logcl resul. d d oes d gures re bsed o or e rom merls he course eboo: robbl Sscs d Rdom rocesses or geers 4h ed. Her Sr d Joh W. Woods erso duco Ic.. B.J. Bu Fll 6 4 o 46 C 8
15 roeres o Codol ecos: roer I: eced vlues o codol eeced vlues: Dervo d d d d d d d d roer II: I d re deede ( should o mers) Dervo d d oes d gures re bsed o or e rom merls he course eboo: robbl Sscs d Rdom rocesses or geers 4h ed. Her Sr d Joh W. Woods erso duco Ic.. B.J. Bu Fll 6 5 o 46 C 8
16 oes d gures re bsed o or e rom merls he course eboo: robbl Sscs d Rdom rocesses or geers 4h ed. Her Sr d Joh W. Woods erso duco Ic.. B.J. Bu Fll 6 6 o 46 C 8 roer III: A codol ch rule I he he eboo roo d d Tg he eeced vlue ol rem d d d Bu d d d d Bsed o hs equo ll vlues o hve bee cosdered. Thereore d
17 oes d gures re bsed o or e rom merls he course eboo: robbl Sscs d Rdom rocesses or geers 4h ed. Her Sr d Joh W. Woods erso duco Ic.. B.J. Bu Fll 6 7 o 46 C 8 4. omes o Rdom Vrbles 4 mle 4. Boml R.V. m Deerme he eeced vlue roo bsed o Wed hs://e.ed.org//boml_dsrbuo Bu m m m m Deerme he d mome
18 oes d gures re bsed o or e rom merls he course eboo: robbl Sscs d Rdom rocesses or geers 4h ed. Her Sr d Joh W. Woods erso duco Ic.. B.J. Bu Fll 6 8 o 46 C 8 The secod erm s revousl comued Deerme he vrce q Fgure 4.- Vrce o boml RV versus.
19 Jo omes 46 Whe e hve mulle rdom vrbles ddol momes c be deed. g g d d All eeced vlues m be comued usg he Jo d. There re some e reloshs. Correlo d Covrce beee Rdom Vrbles The deo o correlo s gve s d d Bu mos o he me e re o eresed roducs o me vlues (observed he d re deede) bu h resuls he he re removed ror o he comuo. Develog vlues here he rdom vrble mes hve bee erced s deed s comug he covrce COV d d Ths gves rse o oher cor he he rdom vrble mes d vrces re used o ormle he cors or correlo/covrce comuo. For emle he ollog deo correlo coece bsed o he ormled covrce COV d d Also oes d gures re bsed o or e rom merls he course eboo: robbl Sscs d Rdom rocesses or geers 4h ed. Her Sr d Joh W. Woods erso duco Ic.. B.J. Bu Fll 6 9 o 46 C 8
20 oes d gures re bsed o or e rom merls he course eboo: robbl Sscs d Rdom rocesses or geers 4h ed. Her Sr d Joh W. Woods erso duco Ic.. B.J. Bu Fll 6 o 46 C 8 The shor dervo The eeced vlue s ler oeror coss rem coss d sums re sums roeres o Ucorreled Rdom Vrbles 48 For sdrded rdom vrbles he correlo coece c be solved or s he correlo vlue. For eher or ero me vrble For deede rdom vrbles
21 oes d gures re bsed o or e rom merls he course eboo: robbl Sscs d Rdom rocesses or geers 4h ed. Her Sr d Joh W. Woods erso duco Ic.. B.J. Bu Fll 6 o 46 C 8 e d Vrce o he Sum o o R.V. For d ucorreled leg =+ VAR oe Ler O: VAR VAR VAR COV VAR VAR VAR For ucorreled d VAR
22 oes d gures re bsed o or e rom merls he course eboo: robbl Sscs d Rdom rocesses or geers 4h ed. Her Sr d Joh W. Woods erso duco Ic.. B.J. Bu Fll 6 o 46 C 8 A secl oe Ideede rdom vrbles re ucorreled. Hoever ucorreled rdom vrbles re o ecessrl deede. mle 4.-5: Gve j j j j j j oe o deede j j j j j Thereore COV
23 oes d gures re bsed o or e rom merls he course eboo: robbl Sscs d Rdom rocesses or geers 4h ed. Her Sr d Joh W. Woods erso duco Ic.. B.J. Bu Fll 6 o 46 C 8 mle 4.-4 Ler redco me squre error Fgure 4.- rse observos o ( ) cosue scer dgrm. The relosh beee d s romed h srgh le. Uder ler ssumo o he relosh beee d b ˆ A error c be deed s b ˆ We sh o orm he vrce o he error ˆ b b b b b b b b b b b mo ss o e he dervve d se he dervve o ero d he dervve b d se s dervve o ero.
24 oes d gures re bsed o or e rom merls he course eboo: robbl Sscs d Rdom rocesses or geers 4h ed. Her Sr d Joh W. Woods erso duco Ic.. B.J. Bu Fll 6 4 o 46 C 8 Tg rl dervves b b quo #: b b b b quo #: b Fdg (subsue b # o #) Fdg b (usg d #) b The ler redcor becomes b ˆ ˆ Or ˆ
25 oes d gures re bsed o or e rom merls he course eboo: robbl Sscs d Rdom rocesses or geers 4h ed. Her Sr d Joh W. Woods erso duco Ic.. B.J. Bu Fll 6 5 o 46 C 8 d he error s ˆ Fll deermg he mmum me squre error b b b Deleg here ossble leves Thgs o oce. see he ollog ge.
26 oes d gures re bsed o or e rom merls he course eboo: robbl Sscs d Rdom rocesses or geers 4h ed. Her Sr d Joh W. Woods erso duco Ic.. B.J. Bu Fll 6 6 o 46 C 8 Thgs o oce. he ler redcve s bsed o he robbl vlues comued. Fgure 4.- rse observos o ( ) cosue scer dgrm. The relosh beee d s romed h srgh le. ˆ eg o he correlo coeces or ler correlo o some e s eeced I : ˆ I : ˆ ˆ s o oe s o here or he esmo error mus be ero
27 oes d gures re bsed o or e rom merls he course eboo: robbl Sscs d Rdom rocesses or geers 4h ed. Her Sr d Joh W. Woods erso duco Ic.. B.J. Bu Fll 6 7 o 46 C 8 Jol Guss Rdom Vrbles 5 I o R.V. re jol Guss e I : e e e Vsulg Jo Gusss Fgure 4.-4 Coours o cos des or he jo orml ( = = ): () σ = σ ρ = ; (b)σ >σ ρ=; (c)σ <σ ρ=; (d)σ =σ ;ρ>.
28 oes d gures re bsed o or e rom merls he course eboo: robbl Sscs d Rdom rocesses or geers 4h ed. Her Sr d Joh W. Woods erso duco Ic.. B.J. Bu Fll 6 8 o 46 C Chebshev d Schr Iequles 55 There re umber o robbl reloshs h boud secs o egeerg roblems. The re cll bsed o momes rculrl he me d vrce. Ths s he rs. The Chebshev equl urshes boud o he robbl o ho much R.V. c deve rom s me vlue. Chebshev equl Theorem 4.4 Le be rbrr R.V. h o me d vrce. The or Dervo d The d d d d d Resuls #: I e lso cosder he comleme o he robbl descrbed d usg he comleme Thereore Resuls #:
29 oes d gures re bsed o or e rom merls he course eboo: robbl Sscs d Rdom rocesses or geers 4h ed. Her Sr d Joh W. Woods erso duco Ic.. B.J. Bu Fll 6 9 o 46 C 8 I m be covee o dee he del uco erms o mulle o he sdrd devo. The he Chebshev equl becomes mle 4.4 Devo rom he me or orml R.V. The Guss orml CDF s er dv v v e Thereore er dv v v e er d er The ollog ble comres he Chebshev equl o he bove uco.
30 rov Iequl 57 The rov equl ocuses o he me vlues d ses. For o-egve R.V. or The Dervo d d d Thereore mle 4.4 Bd Ressors Ressors hve me vlue o ohms. I ll ressors re o be mesured d hose bove 5 ohms dscrded ho m mgh ou esme ould be dscrded? 5 5 oe: hs s boud o he ec vlue he ec vlue could be eeced o be smller h 67% s he equl suggess.. oes d gures re bsed o or e rom merls he course eboo: robbl Sscs d Rdom rocesses or geers 4h ed. Her Sr d Joh W. Woods erso duco Ic.. B.J. Bu Fll 6 o 46 C 8
31 The Schr Iequl 58 Schr Iequl cosders he mgude o he covrce o o R.C. COV equl ll hold d ol s ler uco o ( correlo coece o +/-). The equl c be re s or Cov Cov Dervo: The covrce deo leds o srgh orrd recogo o hs equl Thereore Cov COV h ou hve lred eereced he Schr Iequl oher seg For he er roduc ou m lso see he covoluo orm s h h g h g h g h g * d * * g d h h d g g * d oes d gures re bsed o or e rom merls he course eboo: robbl Sscs d Rdom rocesses or geers 4h ed. Her Sr d Joh W. Woods erso duco Ic.. B.J. Bu Fll 6 o 46 C 8
32 L o Lrge umbers o h e hve dscussed he Chebshev equl e c rovde roo o he L o Lrge umbers. The dscussos here rovdes he codo h he smle me coverges o he esemble me h s he sscl me equls he R.V. esemble me. mle The smle me equls he eeced vlue me For lrge eough umber o smles e s h ˆ I e e he eeced vlue ˆ ˆ ˆ d Vr ˆ Vr Vr ˆ Vr ˆ Thereore usg he Chebshev equl e se h The or ed vlue o del lm ˆ ˆ lm d lm ˆ oes d gures re bsed o or e rom merls he course eboo: robbl Sscs d Rdom rocesses or geers 4h ed. Her Sr d Joh W. Woods erso duco Ic.. B.J. Bu Fll 6 o 46 C 8
33 4.5 ome-geerg Fucos 6 The e s o movg o some dvced coces h suor mhemcl dervo o hgher order momes. I hve bee eosed o roblems here he 4 h mome o R.V. requred s r o soluo. I ou rell le d re comorble h Llce d Fourer Trsorms hese roch rovde soluos ser d more esl h more brue orce egrl roches. The mome geero uco (GF) s he o sded Llce rsorm o he robbl des uco (d). I he GF ess here s orrd d verse relosh beee he GF d he d. The GF s deed bses o he eeced vlue s Thereore e e I ou le s beer h our Llce rsorms s s e For dscree R.V. e erorm dscree Llce rsorm d d s m e s es Wh do e do hs?. I ebles covee comuo o he hgher order momes. I c be used o esme () rom eermel mesuremes o he momes. I c be used o solve roblems volvg he comuo o he sums o R.V. 4. I s mor lcl srume h c be used o demosre resuls d esblsh ddol bouds (he Chero Boud d he Cerl Lm Theorem). oes d gures re bsed o or e rom merls he course eboo: robbl Sscs d Rdom rocesses or geers 4h ed. Her Sr d Joh W. Woods erso duco Ic.. B.J. Bu Fll 6 o 46 C 8
34 oes d gures re bsed o or e rom merls he course eboo: robbl Sscs d Rdom rocesses or geers 4h ed. Her Sr d Joh W. Woods erso duco Ic.. B.J. Bu Fll 6 4 o 46 C 8 I ebles covee comuo o he hgher order momes Bsed o he deo e erorm he Tlor seres eso o he eoel e e or e m m m The m re he h momes o he des uco So ho ould e solve or he momes? B g dervves d seg = m m m b seg = m Soluo doe b erormg -sded Llce Trsorm d dereo
35 oes d gures re bsed o or e rom merls he course eboo: robbl Sscs d Rdom rocesses or geers 4h ed. Her Sr d Joh W. Woods erso duco Ic.. B.J. Bu Fll 6 5 o 46 C 8 mle 4.5-: The GF o Guss or e GF: d e d e e Whe egrg Gusss orm egrl o correcl ormed Guss uco d eque o.. d e d e d e e d e e The egrl s o equl o.. Ad e hve 4 4 e e e o e c geere he momes o Guss uco.
36 oes d gures re bsed o or e rom merls he course eboo: robbl Sscs d Rdom rocesses or geers 4h ed. Her Sr d Joh W. Woods erso duco Ic.. B.J. Bu Fll 6 6 o 46 C 8 The s ome e e e The d ome e e e The rd ome e e e
37 oes d gures re bsed o or e rom merls he course eboo: robbl Sscs d Rdom rocesses or geers 4h ed. Her Sr d Joh W. Woods erso duco Ic.. B.J. Bu Fll 6 7 o 46 C 8 The 4 h ome e 4 4 e e See: hs://e.ed.org//orml_dsrbuo Addol useul emles: mle 4.5- GF o Boml q m GF: d e q e q e gcl mh se o rell bu I hve doe he dervo msel q e
38 oes d gures re bsed o or e rom merls he course eboo: robbl Sscs d Rdom rocesses or geers 4h ed. Her Sr d Joh W. Woods erso duco Ic.. B.J. Bu Fll 6 8 o 46 C 8 The s ome e e e q q q e e The d ome e e q e e e e e q q q q q
39 oes d gures re bsed o or e rom merls he course eboo: robbl Sscs d Rdom rocesses or geers 4h ed. Her Sr d Joh W. Woods erso duco Ic.. B.J. Bu Fll 6 9 o 46 C 8 mle 4.5- GF o Geomerc Dsrbuo or m GF: d e e e The s ome e e e e e The d ome e e e e e e e e e e e Checg he resuls rom Tble 4.- All hese resuls mch s eeced.
40 4.6 Chero Boud 64 The Chero boud s bsed o coces reled o he GF. Loog he l robbl or here s deed cos. As l e eec he des uco o hve grel decresed hs rego le eoel or Guss. The e c se d u I he l s such h u e d The Bu he u d e d e e d e To mme he boud he vlue o h rovdes he smlles vlue should be used. Thereore deree h resec o d he vlue o or he mmum d use he derved vlue or he bouds. mle 4.6 Chero Boud o Guss Le be Guss d cosder he bouds here > [] For he Guss e or e oes d gures re bsed o or e rom merls he course eboo: robbl Sscs d Rdom rocesses or geers 4h ed. Her Sr d Joh W. Woods erso duco Ic.. B.J. Bu Fll 6 4 o 46 C 8
41 oes d gures re bsed o or e rom merls he course eboo: robbl Sscs d Rdom rocesses or geers 4h ed. Her Sr d Joh W. Woods erso duco Ic.. B.J. Bu Fll 6 4 o 46 C 8 The Boud becomes e e e e To d he mmum vlue o e d d e Thereore e e or e
42 4.7 Chrcersc Fucos 66 The chrcersc uco s o he GF s he Fourer rsorm s o he Llce rsorm. Ised o beg comle vrble (smlr o s) e se = j. Thereore he chrcersc uco s For dscree RV j e j e d m e j ej I geerl he CF hs smlr roeres o he GF. I ddo o hose revousl meoed he Fourer rsorm s ver useul he erormg me dom covoluo. For sums o R.V. =+ s revousl sho he des uco s he covoluo o he o des uco d. d Bu hs c lso be erormed he requec dom b mullco As resul mes much eser o coemle d Whch ould become oes d gures re bsed o or e rom merls he course eboo: robbl Sscs d Rdom rocesses or geers 4h ed. Her Sr d Joh W. Woods erso duco Ic.. B.J. Bu Fll 6 4 o 46 C 8
43 oes d gures re bsed o or e rom merls he course eboo: robbl Sscs d Rdom rocesses or geers 4h ed. Her Sr d Joh W. Woods erso duco Ic.. B.J. Bu Fll 6 4 o 46 C 8 mle 4.7 For here he re deede d decll dsrbued (IID) solve or he e dsrbuo he s Guss orml R.V. ( me u vrce). or e e G e e j j j G The or ero me u vrce e d or IID R.V. e e e e The verse rsorm (bsed o he orrd rsorm h vrce) resuls or e
44 oes d gures re bsed o or e rom merls he course eboo: robbl Sscs d Rdom rocesses or geers 4h ed. Her Sr d Joh W. Woods erso duco Ic.. B.J. Bu Fll 6 44 o 46 C 8 ome Geero h he Chrcersc Fuco AS h he GF he CF c geere momes b dereo. m m j or j m
45 oes d gures re bsed o or e rom merls he course eboo: robbl Sscs d Rdom rocesses or geers 4h ed. Her Sr d Joh W. Woods erso duco Ic.. B.J. Bu Fll 6 45 o 46 C 8 Jo GF d CF Jo GF d CRF Fuco c d re deed. As m be eeced he c be used o comue cross-roducs o rdom vrbles jus s he geered momes. The Jo GF s deed s e The Jo CF s deed s j j j e All he momes become or he jo d mrgl vrbles c he be comued bsed o l l l l m ) ( or ) ( l l l d l l l m j ) ( The Cerl Lm Theorem The cerl lm heorem ses h he ormled sum o lrge umber o muull deede R.V. h ero me d e vrce eds o he Guss orml CDF rovded h he dvdul vrces re smll comred o he sum o he vrces. Covoluo me dom s mullco he requec or Llce doms
46 Geerl coce o eeced vlue I geerl he eeced vlue o uco s: Rec o eced Vlues g g d g g g r es d Vrces o Deed des ucos. rcce clcule he mes d momes d vrces or he ollog: oes d gures re bsed o or e rom merls he course eboo: robbl Sscs d Rdom rocesses or geers 4h ed. Her Sr d Joh W. Woods erso duco Ic.. B.J. Bu Fll 6 46 o 46 C 8
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dom Sgl Processg Cher4 dom Processes Cher 4 dom Processes Coes 4 dom Processes... 4. Deo o dom Process... 4. Chrcerzo o dom Process...4 4.. ol Chrcerzo o dom Process...4 4.. Frs-Order Deses o dom Process...5
Unscented Transformation Unscented Kalman Filter
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