Mathematical Engineering - Probabilità

Probabilità

Complete course

TEORIA PROBABILITÀ LEZIONE 1: 26/02 LEZIONE 3: 28/02 DEFINIZIONI : •EVEN ro E-UNFAITO Dewi, AL TERMINE Di UN ESPERI MEND ALEATORIO, Si Poo- SRDBIL'RE u GRSDO DI VERITE Csi E- verifies ro O MENO) -• L' E VENTO MARE debris A RISPETO DD UN E VENTO Dare E- iINSEME DEGLI E Siti w cow wi si AFFERMD CUE LEVEN ro si sin VERIFICATQD spsrio CAMPION ARIO, W Esiri EEmENrbRi (BISOGND aspire BENE IL SIGNIFICAND MAREN Arico DI ELENE NRARI ) -Vi E- DUN QUE UNS CERRA Corri SPON DENA TRD L'EVEN TO REDE E DUELLO MATE NANG:# Pridi #Lotto ESET :• A⇒m ESCE ON NUMERO Primo de Corto: A = 12,3 , -..,89 ) El t,. .-, 901=52 ; • Tiro DUE bars. ED Escano sue Numero' uuunii: A = /l 1,1 ) , ( 2,2 ) ,...,(6,6) ) El 1,...,G ! A• Esrraaao UN NUMERO RDWONDE Dsu' INTERVAL he Co, I]: A = ④ A [ 0,1 ] E [ O, 1) =D .Possible ORS MEITERE IN RELDZIONE DELEE OPERAZIONI Loaicue TRA EVENT con OPERSUONIINSIEM.is riche TRA SOCTOIN Semi DI A-• ⑤ : EVEN IMPOSSIBLE ; • A: E- VENTO CERTO; • A ': EVENRO CONTRA Rio AD A; • AUB: A OB, so verified AMEND UNO DEI DUE;⑥A NB : A EB , si VERIFICDNO ENTRANTS'; • AaB = 0 : EVEN ri Incompatibility •A EB : A # B- DEFINIZIONE : si DICE IN SE ME DELE PARTI Di r E si scrive 2"LD COLEZIONE DI Tori iSOTO IN SEMI Dl D INCLUSI LlINSIENE ⑦ E D sores so- NIS : # ( I ) = 2 #A ( # CARDINAL ith ) proprietor Maro ComesaCHE Mi INTER ESSA DVERECo, I]EMANTENERECio- CUE UORREI FARE E- AITRIBOIRE UND PROBABI Lins DIREITDNENTE Ai soitoinsreaei A ER,MA It IP su 2 t.cl/7Ee,bT)=b-eV-oscecbzLCio-CteEPosslbah FARE Duerst ASSE ANDRE P s about' A E 52, NON Dori- DEFINIZIONE : A E I si Dice G- ALGEBRA SE RISPEITA LE SEGUE Nri Proprietor:s) Oe it , rect ; 2) A E A =& ACE A ( A E- Caius PER coupe NENRDZIONE). 3) Ak E A FKE IN ⇒⇐ Uk AKE A ,NK Ak E A (AE - Cteiuss PER UNION I ED INRERSEZIONI NUMERSBILI LSEW Fosse saoxurroor, ED INTER. FINITE SAREBBE sad ALCEBRD )) PROPOSIZIONE : Ct QALGEBRA Dis ⇒ Ct cieiuss PER union i E INTERSEZIONI FINITE.NHI Niro) VoLTE INFINITE NUNERSBIE Divorce.Didhsrrnzi ONE. Ai,...,An E A ⇒ Ai VAZU .. .Van E 'A. UAZU ... . Van VDV to U...... E) Ct ; A A Azn .. . Attn A , NAZ A ... . MAN ADNAN .... E) A %aa.gs ESEMPi :• A = I = 0-ALGEBRA DELE Parte;• A =/ 0 , A ) =D-ALGEBRA BANDE €2 ";• A =/ 10,52 ,A, Ac ) #G-ALGEBRA NON BANDE piu-SEMPLIE-autiENELcssoaeilNRERES.si soo L'EVEN to A-SCOMMEITO CUE Esco3Dae CANCIO Dl UN DADO- Posse impostor Ls reis G-ALGEBRA come LA =/ 0 ,D, 43 ) , 11,345,6 )) OPPURE cows, Are I =/ PdI )P→pERDo ,V → VINGE QUVNDI CONSIDER ARE UND 0-DLAEBRS Empiricism: Lt = 25=10,1 , 4%4 , 1£41 ,Definition sin I spszio csdlpionbrio CON A 0-ALGEBRA Fissbrb so I, Avers cuisueeeneo EVEN ri solo i Satoinsierei A ES TALI ate A E A ⇒=£ Lt VIENE DELTA 0-ALGEBRA DEGLI EVEN N-LA Coppi A ( SL, A ) E- DEITD SPDZIO MisuRABhE.A E dei SURABIE =⑤ A E- UN EUENRO =p A E Ct . PROPOSIZIONE : An E A th EIN ⇒ hiuginf An E A , limnsup An Ect -Dimosrrazione: bing.nfan-U.FI??AI*=n4oBnEit;liwysopAn= ET Ft ! = Bn Ect q.mgr DEFINIZIONE : Dara LA CLASSE E E 2 si DICE 0 ALGEBRA GENE Rsrb Ds & E si SCRIVE 0 ( E ) LA Pio Picco LD G-ALGEBRA CHE GNTIENE C PROPOSIZIONE : 0 ( E) E BEN Definite V-E.*Dino STRDZIONE: 1) PER ipotesi E E 2h Quinn Esisre amend UNS G-ALGEBRA A Z E -2) E sister UNA 0-DLCEBRS Dio piccolos DRUPE? PRESS It a FAMIGLIA Dl 0-see BRE E A = ⑥ Ita ( GN L Finrro oiwfinrro ) ⇒ A E- UND 0-ALGEBRA, INFDITI:Q) LN Ltd Corriere 0 E I IN QUAND QUEST DUE soitoiwseaeiAPPDRN-NGONOATuuiisiNC.eu' Ltd , Quinol ANGE sus hero INTERSEZIONE-b) Atta E'cuivss per comp EMENRAZIONE: A E Ct = an Aa ⇒ A E Aa Fa Ms slurs ACE Cta Fa ineuarro o.seaersre ⇒ ACE aNAa=ct -c) Atta E- cniuss PER union: Numero Bili: Are E A -- Acta fr , ⇒ Ak E Aa f KEIN ,fa =, YAK E A Fda YAK EP Vta 3) 0 ( E ) = ! y £ ( ESISTE LA Pio piccolo E ↳ DEANisno o (e), Pir piccolo it co NRENENRE L'insane E) EBBA Esempi : • A E o-DLAEBRS =a 0 ( A) = A ; • LA =/ 0,52 ,A,A' } = 0 ( IN ) EO (IAct ) . • E =/ Ck ) KEI ⇒ PDRTIZIONE Discern Deuospsziocsupiowbrio A, OWEN: EEFisriisui.se liueninf EnE A NEMO Solo 2 SOCUE E- VERO 1 PERFORM-I E'su Pio NUMERABLE Ck A Ce = 01 FKFE , ¥I Ck =D. 52=10,11 " I En= I w Er t.c. Wn =LI A=o( E) = 0 (lent new ) €2 ? ssppisaeo persona: (hinsupene A µ It = G (E) =L A ED A = ¥ , Ck ,JE I ) ESENP.io#:DiuosrRsvcimoEs-mPro- anti Cn 01/03 LEZIONE 4: DEFINIZIONE: si AD= IR , E = I = L A- E IRI A APERTO I E- Ls TOPOLOGID EUCLIDED Co NATURALE) Di IR - si DEFI Nisa o-ALGEBRA Di BOREL B = O ( I ) €2 "Gli EEMENN' Deus 0-ALGEBRA Di BOREL Sono DERI BORELIDNI: B EIR ,BE B⇒m . BBORELIANO - B z la,b ) , [ a ,b ] , ( a,b ) , [ a,b ) , txt , IN , ⑨ , I .TEOREMA: B = o (E) = o CE ) doveE-- ( ( e. b) I- eosacbstoo ) , E -- I I-ee, b) I be IQ } coronation. B = O (E ) Dove E =/ (-oo, b) Ib EIR)-*Dime STRDZIONE: E EEE ⇒ B = G (E ) E 0 (Eh ) - MD ETE B , IN Qusnrro B cowrie NE OG Ni INTERVALE Di Rea TUITE E SEMIREIE=a =⇒B=o( E) E 0 (E) E OCB ) , ma BE uns o-overseas soo OC B) = B ⇒ 03=01 E) E o (E) =D imma ( A, A ) spszioreisurs.BE, A spszio Campion Akio ( SP. camp. ) -A E A - LA PROBDBI Lirr IPC A) Di A RDPPRESENRD IL GRADO Di FIDVCIS Do 0 A 1 su verifies Rsi Di A, E- UND VSWTAZIONE A PRIORI.ABBI AND 3 POSSIBILI INTERPRETATION I DELLSPROBDBI Lira: • INTERPRET AZIONE SOGCEITIVISTICD : DIPENDE DSL SOWETO CUE careers La PROBAB him, Uri LIZARD PER EVENT NONRIPETIBILI, SPECIE IN FINANZA; •INRERPRErszioNEFREQUENris.rs: RIPETIZIONE DEGLI EVEN ri PER CAL@LARNE CD FREQUENZD- PIA )=¥¥• fnla ) - Qui fnc D) E LA# Casi FAVOREVOLIFREQUENTS RELDTIVD → LEGGE DEIGRANDINUMERO'; • ESRRDZIONE DA POPO LAVON i FINI TE E NON trocar IPCA ) =#Casi Rossi Bili.- ESEM-p.ie : a ESTRAZIONE NON TRUCCDTD DEL LottoIPCESTRDZ. NUMERO PRIMO ) = TO → HESTRDZIONE Ds Popo Lazio NE; • DADO NON TRU ccsro: IPC 3) = 1/6 → [email protected] 3) = O, 15 → FREQUENT isrDg• CADURD DEL LOVERNO NEL 2019 → Soaaefitivdsros- DEF IN 121 ONE : Daro uno spszio miserable Cr,A ) uns I Misora Di)PROBS Bhim E UND Fonzie NE P : A →C0,1 ] the cue : 1) Pls ) =L;2) BE 0-DDDirivs.eu#ro:VsuccEssroNEfAn/ne-NEsu1tssr . me .ee .¥EEri Disaiunri a coppice, OWERO AKA Ae = 01 free ,Vse: ( nAn ) = €7 Plan ) -COMME Nri ALLA DEF: 1) E COME SE SRESSI CALCOLDNDO LD PROBABI Corri 01 ESECUZLONE DEUIESPERRNENRO;2) VOGLIO Calais RE B( Un An ) , CUE E- Forti BILE IN QUANDAi AzUn An E A - Sro Richie sense IPI KAN ) =L Plan ) , Ms Saronno PER Gui Evenki [email protected] An = soundDELLE PROBDBihirb-OCES.ro to Soo Chie DENDO Soho NEL NUNHERABICETEOREMD : Sid Purvis proBDBuirrs-suuoseaziomisuRSB.ie ( r, A ) -sewers: 3) 1/70/1=0 ; 4) BE -abortion, overdo! Dari As,....,An E A Evenri brsawnri a corpse ( Ak Nde =D fate ) use: IPC ! ?Are )=r¥ BLAH - e- Dialosrrnzione: 3) An = 01 Fn =¥s An E A ,Are nAe= 0/101=01 ⇒ Pl IF An ) = ¥? ( An ) ⇒ 1170¥ IF (d) ⇒ 1/70/1=0 , scrrimenri is sound bare BBE un Nou. info Niro-AK SE K= I, ---, honeO 4) At , ---,An E A ; AT =/ d se K> nti - srsppismo ons cue are nose = fate If 1/717 AT )= EITHER )= TERAI ) t.IE/PH-I)orsEssENboEr-=o1pErkzntrY7AT-- LIA ! ⇒ PILE AT ) Plate ) ..mgDEFINIZIONE: AIB = A NBC COROLLDRI: s.is/PuNDPR0BDBiLitA-suuosps2iooldsuRsBuE ( A, CA ) , burrs: 5) A EA ', a. s'E it ⇒ 1PM E Dld ) ; 61 PCA ' )= I- PCA ) ; 71 IPCAIB ) = PCA ) - IPCANB ) ; 8) IPC AUB ) = Platt PCB ) - PCANB ) s RAI t PCB ) iE Ac 30 IPROB.A ADIMOSTRAZIONI: 5) A At = Au (Aha ) ; A' na =L A' AAC ) EA'⇒An IA 't A) =⇒ PCA ' ) = RCA U #A) ) IPC A) t IPC A' not ) > PCA ) 6) Ac = ma =.AVA' = r →I Pls ) = PIA Uac ) # Platt IPA ) ⇒ 1 = IPCAITIPCAC ) ⇒ IPCAC )= I- PCs ) . 7) ( ALB ) U ( AAB ) =A, ( ALBIN ( AAB ) =D ⇒ IPCAI = IPKAIBIUCANB )) 't ' IPCALB ) t IRA NB ) ⇒ DIAB ) = IPCA ) - IPCANB ) > ox'prob. 8) AAB to =a E= ALB ⇒ AT NB = 0 AUB = ( AIB ) UB IP I AUB ) = BHAI B) UB.IE/PCAlB)tlPCB)ElPCAItlPCBl-lPCAnB7slPC A) t PCB ) TEOREMD: sin P UNA PROBABILITD' Su UNO spszio MISURDBIE ( A, A ) , ALLORA, Daro An E A Fn'. 9) An PA, outro An E' uns succession E D1 EVENT: CRE SCENE VERSO A→An E Anti⇒ A E A , ⑥ (A) = HEAD = him ( An ) -§NI: An DECRESCE VERSO A# Acn 10 ) A ntoA- , OWERO An E- uns succession E Dl EVENTiDECRESCENTVERSO A→An3Anti⇒ A E Ct , /P( A) = ( nAn ) = him ( An ) -CRESCE Verse Ac (LEAAE Di DE Morceau ) Direosrrnzione: 9) A = ¥7 An Elt - PREE BE Ai,Bz= DID,, ... ,Bn= And An -,Fn> 2 =p BiUBz=DiUDz ,MD B, nBz = 01 → n¥Bn=nU? An,ma Barbe = a fate ⇒ H7al=/ME?Bn ) IF But Esso ⇐NPlBm¥ Gina "% " * B. upszu.. .ups.v=anfor> ,10) RDGIONANENRO ANALOGAL PRECEDENCE 08/03 LEZIONE 5: Proposition E: /PproBaBuir5 such , A) = .se/PlUnAn ) s I Plan ) → o-sursrosoirivirnOVVEROTOLGOTUITELEINRERSEZIONI* DIMOSTRAZIONE'. An EA th , Are NAE to fete. CONSIDER O AT ,...., In con Ark = AKI ( A, VAN ... . Hn #K It Atte =D Fk #l E IKE At the A. uazu .... = E. UATU ...⇒ RYAN ) = PHAT ) # RATI t Plait .... .ua In ear . te ⇒ ME . ) MAKI ⇒ Ryan ) -191M€ 'RE .lt/PIAaIt....elRA .lt/PCAalt..........=ElPHnl/PpRoBaBiLiTA- so semi CDMPIONARI Discreet (r AL Pio NUMERABLE).DEFINIZIONE: p = DEN Simi DiscernTEOREMD: sirs A Discreet: 1) IP su IA, 27 E CAREER izzsrn Dsi vscori DEAL: Arani lsrwaoci Puri ) IWS , WEIR . Pw = IPIW )INFANT. A) =a Pw z ) sin p :D → IR . Avoid Z! IP : 2 "→ Eat ] probabilism t.cl/7w)=pw ⇒ ( PIZ O Tw ¥p,Pw ⇒ ns-ENRiEop.EE?5EYuiPD7srEirijinsuENrotoo AND iziorre FONDANE NTAE PER PORER INTRODURRE IL MODELED-Dimosrrazione: at Aer ⇒ A= Yea I w ) , Iwlnlw ' 1=0 Fwiw ' ⇒ But )=/P(wUea/wH=¥a Pw 2) ⇐ ) pw=/P4wl ) cue E posrriuswausvropeorssbiu.to. wEsPw=¥r/PHw )) - 1PM -II⇐ I l ! ) sea P⇒ Beat =⇐apw ⇒ Peonies . (t) sin IP : 2%10,13 , Ptak Eea Pw ⇒ BE -una probabilities, infant: 11 Pls )=¥rPw¥ 't ;PCA,) Putz)oo 2) Ane A Tn, AanAe=0 fate, Ply An )=mPw=¥ ,Pw t w¥Pwt --.. = PHN.q.mg ESEMPl_ NON TRUCCATA Di UN ELENE Nro Ds uns POPOCAZIONE DI n INDIVIDUI: 52=11 ,...., n ) , A =P . p su CA DEFINITD DA p : A → IR , Pw =L V-weiNFazmoszioNEiesrerszionENONTRuccr.tn ⇒ Toti Cli Eheuenri resume Lasresss PROBD Bili Thi DiPw> Otw230ESSE RE Esrrsuri: ↳ Pw =L⇒ I hd=1⇒ 2=114 E sempre DI EOUIPROBSBI Lim DEFA AN cue probabilities uniform.⇒ PCA ) = Iea Pw = ¥ , In = In I # A) = #AL =#case Favor.# I#CASI posse Bili =p Forums Varas solo IN ON esse DI SL Fulro, 6N Probs Burris Dlsrri Burris UNIFORMENENRE.cdsipnrriaasri Chios vrsri ) : Rho promo scioto ) = ¥0 ; Pl 3 sounwsncie Di UN arise )= 116 I Resisters ) = 112 -• Distribution Bino Mine: Bcn ,p) HEIN , PECO ,I ] D= 10,1 , -.. .nl ,#Sente. Besancon tenure pw=( It Pw " -P ) ""ssppisreo cue pcwizofw-o.se , .→ n, ⇐ " Pcw ) - ¥ ! I It P" " -P ) ""- Cpt CI- PD "= I → Eons pnorssrsiuirn.• Disrrirsvriove bi Poisson : pen ,-2> o r-- tofu IN , A = 2? p←=E4t I K-- 0,1 ,....,n,... Pk > 0 PERCE Probate Di Funnier, sempre positive, ¥s pre = ¥o% "= e- " ¥•?÷ = e- it et =L→ pie Euns PROBSBicrrb-ATENZIONEPuo-succEDEREC.ae/PC A) =D,ANCRE SE A# 01 : { PCA # 0/1=0 EVEN To IMPROBABLE O FK =D,..., h-'Eso: B I n, 1) , pref I k=n ⇒ 1/749 ..., n- ill = Pk = O =, E- aeitoevenro rmpnerssrsie: PCA # Dl =L EVEN ro Quasi Cerro.TEOREMD: sirs D Discreet: 1) IP su IA, 27 E CAREER izzsrn Dsi vscori DEAL: Arani lsrnaoci Puri ) IWS , WEIR . Pw = IPIW )INFANT. A) =a Pw z ) sin p :D → IR . auors F! IP : I → Eat ] probabilism t.cl/7w)=pw ⇒ ( PIZ O Tw ¥3 Pw =LCOMMENT@ ' . UTILE DA TENERE SOTO MANO PER E AND LOGIE CON ITEORE Mi WE VE DREW IN QUE Sra LEZION E- DEFINIZIONI : 1)SIANO (r, A) une spszio Miserable, Puns probs Bruits suA -→ ( r, A, IP ) E- Denio spszio DI Probabilities; 2) Dsr,UNO spatio Dl PROBS Bili r5 (SL,A, IP ) ED UN EVENT A EA.SE IPCA) =D,A E DENO LMPROBSBIE, -SE INVECE PCA ) =L AE SEKO EVENT Quasi CERRO-TEOREMD! I spszio Csu PIONARIO- ( Cn ) he I PDRRIZIONE Discreet, LA = 0 ( CnIhe I ) :1) Psu ( r, A ) E carsrerizzsrrs DA pen =P (Cn ) NE I, I Discrete IP ( Une,Cn ) = Espn V-J E I;Pn> O Fn EI2) sisp: I-ok, avers I! P : it → lo , I ] t.c. /PCCnl=pn ⇐ I ¥±pn=L COMMEND: PDRTIZIONE Disc RETA → (G) NEI SU QUE Srs FAMIGLIA SIVER ifics UN sold EVENRO so UN' u Nics Parricide (Ce#Cktfetk) E USD0 A VEDERE QUAE SI VERIFIED-I E wsiENENUM.DINRERVA.ciESE! POPOLDZIONE Dl SAN) E MALATI: I MALS ri = LY. I 0,01 → Pesce ON WDNI Due s esse E NE verified 6 Srrsro DI sswre:M s=M'M= "msn.ro", s = ME "sane"- 4=1 M,M' I psrrizrove DID→ 7 ! IP so ( r, A) t.cl/9M)=o,oi,/Pls)= 0,99 . PROBDBILITD 'CONDRZIONARA E INDI PENDENTS- RCA NB ) -DEFINIZIONE: sis lSL,it, P) UNO spszio DI PROB a Bill rni, A. BE A , PCB ) #O ⇒a ↳ probs Bicorn covbrzionsrn DI A Daro B E IPCAIB ) = PCB )" LD PROBSBILOTD' Dl UN EVEN-0 CAMBIO A SECONDA DEUE INFORMATION I DATE LE INFORMATION i NORE Posse No VARDA RE CON IL VARIA RE DEL TEMPO:A PRIORI IN MEDIAS RES A PosterioriVENGONO RISE NOTE ALONE INFO NON ESAUSRIVE SULL' ESTODELL' ESPERINENRO! ESE MPD! LANGO UN DADO DUE VOLTE E VOGLIONOW E- Noro LlEsrro DELL' ESPEN MENTO, NE INFO so Dl ESSOSAPERE QUALE- IPCDVEPDRI) SAPENcue SI EVER ificsro BC#Peddle) AL undo I.Fi Niro LlES PERI MENTO SI Vi ENE s Corvo SEENU DELLE Siro RB ) r IB =L FUNZIONE INDI csrro E DI B ⇒ BVERifics.ro IB = IRA B,--,,---lIIll IRD ) IPCAIB ) IA ) ,B r Indio ⇒ Is =L VIA=OiA ) B\r------'----PREDIZIONE DEI Due Esri ssPENDOCUESE-VERificsroBsaaiorvow.pro BABI cirri Cue srverificui A.si saw verificsro A OPPURE AsESEMPI: •Esrrszione DEL 600: A =" Numero Pari", B="Numero Primo" ⇒ P (AIB) = /PfpfpgyB )= !Yg ,= %g • n Riperiziowi INDI PENDENT' DI UN ESPERINENRO SEARIO.' Fn IA ) a FREQUENTS ASSOLURA DI A so n ESPERIMENRI; for (A) = FREE#NZD Recsrivs DI A sun ESERINE Nri- PCA ) - hot ); PCB ) - faced =. Panisse fncs , =FnCsn , pcsipztnffz-B-F.CA#-fnIzT-~tPfpazTB )Casi ESRREMI: B EA ⇒ IPIAIB ) = pfphgp.gl#=PCB7- =L;PCB)B n A =D ⇒ IPCAIB ) =Plpfgj3)=p÷=0 : sis Ir,A,B) UNO spszio Di probabilism, BE A - sirs PCB )> o, blurs:Los Funaro NE Plo IB): A → E0,1],Dna. PC AIB) E- UND PROBABILITA- so A Dems " (Misuses D.) PROBDBnirb-covrsrzionnr.rs Ds B"Diruosrrszrone: sis a: A- → Co, I],QC A) = PCs 1B ) , swans:1) Q ( D) = IP( RIB) = P (SL, B)=L'posse scsmbrs.EE UNIONE ED INTERLEAGUE#An DrsawnriAkrAe=¢ ! Den B) Cse ⇒ (ARAB)nlDenB) =A⇒'aa--2)An c- A Fn,AK nAe= tree, Ql Yan )=/P/QAn/B)=/P/KAn,B ) PIKCANAB ) ) K Man .B ) Zach immase Pcs, pass > o-=-== Al PCB) PCB ) IRB )AVVERRENZS: Vblaonoruiticrisulrnrinori PER us Probs Bruins riser be Primo Araoaeenro: E sempre: PCA 1131=1 - PCs ' 1B ) , res IPCAIB )# R(AlB4= ppfygg.gg )= TYPE ? ,Ap¥PfA)= ( t- IPCBIA ))PCD )1- PCB)IB: SE IRA ), PCB )> o IPCBIA )= IPCDIB ) PCB )Pcs)ESEMPIO: DADO Truce bro: 52=11, --., 65A- = 2h - p . -_pu=ps=P6= 0,2 , pz= 0,05, Ps = 0,55.A= "ESCE No pose,"; B="E see Numero Prince" -IPCAIB) = Pfp{BB# = P¥/B=gos¥%sto , 1=907 " 41%1=925 A I Az ----..... -. AnB,TEOREMD? SL spszio campions Rio,(An) n PARTI NONE DISCRETD DI Sh, ( Bk ) k UN 'SLTRD PARRIZIONE Discern DID =,Bz:Noro CUE Pn> Oth , %Pn= I; 9km > O FK Fn, € 9km =L th, Aurora:Bk I ! IP so In, At , A = of lad , 113×11=0 ( Ann Biel n,k ) t.ae/PCAnl=pn , MBK Ant = gun the .tn t.c. pn> oBi-0 Din Bip, -9111IFNXHP9111':DIMOSTRDZIONE:A,9"' Br ,→s, ×pa. guy f Ph €9411 =PL9112Bl- saz NB,"ina. " " in →mars. { Epn -IPESCO DA SLAs...B,.DITENZIONE: QUEST GRDFO so SCBERO Risers diorio UNE NEWS Risoluzio NE DEGLI ESEROZI-P3,....iIiBn'PROBABILIRD' Di Fini RE IN WPENP"!gunBi→An AB'P" 9''n/pm ⇐ pan = pmDI Parricide Eph, Inph =LdAn9km'' Bk -sAnn BkPn Grin ESEMPIO ! TEST Clinics: POPOLDZIONE Di Saur E MALATI- SAPP 1AM CHE t. MALATI = 11=0,01- VIENE 6N DOTO UN TEST FRS SANI E Risulrb cue I FDLSI Positive 51=0,05,SMR, positiveED UNO TRA Mauri, DAL OUDE Rishta CHE % Fast N Easrivi = 21=0,02 ( SDDPISMO CUE PIM ) = 0,01 ; Mpls ) =0,05; IP (NIM ) = 0,02 ) N |N=P ' p f -.ssvoMi Cui EDO OUSNROVSLGS IP ( M, P ) SE svoLGESS.tl TEST su UN QUSWNQUE IN Dirr Due Deus Popo LSVENE- Psrriziovi DI SL ⇒AM=50,98PO, 0098M PCM ,p ) = BIPIN )lPlM)= I I- IPINIM ))PIM ! 0,098 got902NGeez0,9g905P9049gSE PERO- Mi INTERESSSSSE u esse molto ireporrosvre IPIMIP )? I s995 No , 9h05ESEOUPIO Molto sianificstivo! UN URNS CONTIENE h> ZPSLLINE WERE,b3 2 PALLI NE BRAN CUE- Fsccio E- STRAW ONE SENZA REIMMLSSI ONE- L 'EVENRO Ak E- L'EVEN TOE srrsaao UND PDLLIND NER sus K-Esiolb ESRRDZRONE-AzA,h.n-Ih (n-I) AE -,- A = old ,, Az ) .PCA,Aa ) =P IAil Az ) PIAA ) = Plata , )PCA . ) - PIA , )= ntb PlAzlAl= ntb-I⇒ PIA , ,Ah= Cntbllhtb -itAz r cCA,AzTEO REMs: FORMULA DELLE PROBS Bilitis Total. (I, A , P) spszio DELLE PROBABI Lirr- ( Cn )hEI Psrriuiowe DID,Cn E CA Fn,AE It ; ACURA:AGGIUNGENDOPaul> OA- IRA )=¥= Plan a) LIE Pcslcnllpccn ) rDrmosrrazione:A, A A = And = An lunch = Klara ) Dove Cancel n Cancel =p ⇒ Rai = Blk launch )) = ZRancnla-ZIPIAIC.nl/Pkn)%aamgag ADCi Cz -.. -. -. CnAGGIUNGENDO PCG )> o Come IPORESI-ESTENSRONE DEL TEO REMA: LA FORMULAE VDLIDD DNCUESE: 1) IPC Cn ) =D PER@valorem;2)a) /P( I↳1=1 K€3 CI =@EVEN ro Quasi CERRO ; b) PICK A Cel =D KFC ⇒ EVEN ro improbable- 12/03 LEZIONE 6: #Dimes Radio NE: L ) ( Cn ) n EI PDRTIZIONE Di A,Cn Ect th,AE A , PICK )=0 KEI CI- SEPRENDO PER CONVENZRONE F IPC At Ck ) ⇒OAK⇒ PCA ).PK#ncnll=EPCAncnl=EalPCAknllPCcn ) t E Blake ) Part .Ch→ Ck 2) (G) HEI In 'E=¢ ,..., -.. I E parrizione air ⇒D= Yee Cn ;Cn ACK#01, IPC Cnnck ) ⇒tenet, FREE ; Piketty Cnt =Laa, Pl G) = (IKeena G) 4=1-1--0 ⇒ presort Ect PCA ) = PILEI I Anon )) FeePlan Cn ) ICiCs----. onb)PRENDO ORS OGNI IN SENE CJ , JEIT E W SOITRDGGO si SOTO ins Emi Cn, NEI Di I ⇒ In = Chick-amy UNIONE Discounts × come no DEF. InOth, Fksoprano we PCI ) =P ( Crick )= IPC Cn ) - Planck ) = Pkn ) . Warre secure (Yet E) UIe G) = r ⇒ Reese AEA PCA ) = Plank # E) u⇐ Call ).PH#=lItnEnDUrI.*lAna.DH=En.=/PCAnEn)tEElPCanaeI TEOREMA: FORMULA DI BITES-(SL,CA,IP ) spszro DI PROBSLirr- ( Cn ) new Posrriziowe Discrete DID, Cn E A th,AE A, PCA )> o,ALLORA: Plata ) Mca ) IPCCKIA ) = En Platen ) Paul Diniosrrazrone: IP ( Ck IA ) = PECK , A)- PCA ) = I (At Cn )lplcn ) per uteorseueprorsrornei ⇒ IPCCKIA ) = Match PICK ) PCH En Platen ) Bcci ESEMP,: sites arnica: PCP ) = IPCPIMIPIMITIPCPIS ) Pls) = 0,0593 ; PIMP ) = BIPIN )PCM ) = 0,1653=16,53't PIPIMIIPCMHIPIPIS )Pls )COMMEND: LA DEREN ruse DI Macari INO DE sue cacao DI IP( MIP ) - E- TACNENTE Bssss LDPROBSB' Lira DIESSERE Nswrl CHE E- Piv-PROBDBILE CREIL TEST SI SBAALICue do srsEFFETIVDMENTEMS.vn2) URNS, 2 Esrrszioni SENA Reimmissrene: 11%27=1/7*21 At Pls , ) = MAID .cl/Pcs.4= n-inuahtb-Ihtbtnib.,my = PCA ,) ! COM MENTO: STO cud EDENDE LL court DELLA SECONDS, NON Mi INTERESSA IL Cohere Deus PWNS- UN ESERINE Nro PUTPENATE SRRSDE DIVERSE, Que STO NON VUOL DIRE Rt SPARE DIVERSE Secs DOWDAE- posts SPNON. Voguish RRMARCARE he Faro cue PCA 1B ) # PCA ,B ) ESE MPD: Pesce UNS CARTA DS UN MAZZO DI TRE cost compose: I n, the ,n,I be , bilbo ) ⇒ b → Bronco; n-a NERO 1,2 identifies no i DUE Lari DO UNA CARRA- N -a Lara SUPERIORE NERO, Be-_N-→ Lara SUPERIORE Branco -A-a Lsro CNFERRORE Bronco- IP ( AIN ) = ? BOBBI And DISRRNLUERE I SEU VENN EVENT i Eh =/ he 1h24 ,Ez= In , lbrb , Ez= I bslbz ) PA RrrziNE or I, Ek Esrrsaae us is Rrs K , R ( Ek) = 1137k -Csn-a verse wires, Ca-on Verde are 2 ( Ck ) = 112 FK→ E ..ci) -116 / N-- Esu l Ezra ) = BIEN ( Ez, c.) TIPI ( E., C.) Ut ,, Chute ,C. D= 412 A ,N= Ez ,C. ⇒Plan N ) = 116 Plan N ) 116-=- PAIN ) -. PCN ) in-- 61=13 ¥ , P aPlate) \ TEOREMS : A,...,Ane A , Play . .→An ) > o ⇒ Play ... ,sm= Plantain .,,... .si/PlAn-ilAn-z....,Ai ) ..... Plans , ) PCA ) . ear nD- DiuosrRAWONE.si/PCAklAk-i ,...,A,)E- BEN DEFRNRRO, cwfsctl Plan , An-i,...., Ail > O,Mrs Arnaz n---NARI A in-..ask n set, FK = 1,.. .in-I ⇒=E lP( An, ..., An-i ) 3 PCA ,,. - An )> 0, East t INTERSEZIONE DI IN SEMI-2) Pls ,...,Ant = Plan Isn-i,... .AT/PlDn-ilAn-r ,...., a) Plan - Ibn -s,...,s, ) .... IPods lays . ) Pearls , )PCs , )= IPCAn.AM i...., si). Plant ,..,s,).Plan. -3€ ... .is,)- PH , s.) Pts ,) 1PM )=&. -.--... @ . P ,I Pls si PCA, s.)-- Pts .) Play = Plan ,.... ,a , ) imma DEFINIZIONE : EVEN ri INDIPENDENTI: Dari A, BEA si DICONO EVEN INDIPENSENR) CE si www.NO 6N AIB) SE PIA , B) = PC A) PCB ) . LD DEFINIZOONE E-ABBASNrs intuitiveWFAN Press A ,B : PCs )> 0, PCB ) SO shops AIB ⇒ Pls B) = PCA ) ⇒ PCB Is ) =PCB), OWEN sooo war PENDENN SE Euro Pro BABICHECONDITION ARE NON SI INF WENDI- Si Nori CHE INDI PENDENTS ED IN@upset Bsu TD Sooo DISANTI E Motto DIFFERENT ID MENO DI DIB AMEND DI B EVEN roB dubs, Ernog¢ImprobableNB: A 1B T PCB )=0 ; AIB ftp.CBI-2 AEB11 AnB=0 ESEMPI: s) Esrrsaao UNA CARRA DA un Mazzo Dl 52;D= I A, -..-. , KBS , # 52=52,- A=2^ ; B=escape 4%1=77 - Zz - fi PCB ' - FILL = 452=73 ⇒ Pcs .BE#AnB---sIz=?lPcs3lPCB7=f-z-f-=tsz A- = ESE UNS Carro DI#SLLa ST AIB 2) LANGO DUE DADI NONTRUCCSTI:Are Risotto K Supreme baseD= I 1, . →65 , A- = 2 ? Posse PERE PRENOERE snare un I Now specifics ro, Ct = 0 ( Ak , Belk ,e ) { Be Risuurbro e sue second babe Allers PRENDO Pt .c. { PAK ) -_lPCBe)= 116 Fk,e=s,..., 6 ⇒ RISK ,Bel= 116-116=1136 Fk,FeAk 1Befk ,Fergo→ AYBTpcsc.hr#IPcBi,IPCAsnB.I-- 116-116=436 - lpcsomus -33=1106 , nBzlulsznB.de/PCA.nBz)tlPCAznB.I=/PCA.IlPCBzItlPCAdlPl Bit = 148 TEO REMs! A 1B ⇒AIB' ⇒ Ac1B ⇒ Ac1B'Diaiosrrnzione: AIB ⇒ Pls , B) = IPCAIIPCBIIHP ,) ⇒ Place )lPCB7= LL - Pca )]PCB )= PCB ) - PCs )PCB ) Ee' PCB ) - PCs ,B)=lPlBlA ) = IPIBNAC ) PCA ' .BE/PCA9IPCB7CHpd=a/PCA)lPCBl=Es-1PCa4IlPCBI-- PCB ) - 1/7137%4 # PIB ) - PCs ', B) = PCB ) -PCBs A) = PCB , s). immaE Stupid! UN URNS CONTIENE NPDLLINE WERE I b PDLLENE Branche-NK → EVENRO Esrreszrowe Duns DDLLIND NERD Aus RESINS ESRRSVOVE K = 1,2.IB! Soriano cowsRSNDO il assess Esrrsriowe car Rerneaeissrone ⇐.N, IN, : PIN ,, Nz ) = PCN , )lPCNz)= ( Itb )! h-l hNEL esse coral NOV AVESSI swore REIMMISSRONE DEUE Pavone NEW'uRNs=f N , #Nz ⇒ PIN ,,Nz) =P (N , IN 2) IPCN 2) = ntb-i htbDEFINIZIONE: UNA FAMIGLIA DI EVENT) ( Ai)i EI (ICON GRAIN surf DELENN, Corrino, PIJ CUE continue vs BENE vause) E- UNA FAMIGLIA DI EVENT'u NuruWENTE INDI PENDENT SE V-J II, JFRN ITO,srsbrauo Pine , Ail = t.IS/PCAi ) Commented: #I E- Quasi as;#J Lte = i¥ IP CAi ) EBEN DEFRN im;SE PRENDIAMO 1=41,43 } ⇒ As,Az, Az FORMAN UNA FAMIGLIA DEVEN PCs . .bz/=lPCA.)lPCAa ) IINTENSE Nri SEE Sao SE: / IPCAz, bad = B (Az)(As ) 2Ls Quadra PROPREtA- Now E- RICSUABRE DAE SIRE 3 ( 1+2+3=44 ) - Nov Basra Donee DARE Pcs ,.az) =P Campus ) 3 Pls , ,Aqbz)=/P( A.) PAL )lPG3)4 U warped@Errs rrs I srrraoli EVENT PER hsrswrrre Ls owns Not PENSE Nb-Arnaz kids so Plains , sad = Csi , Azaz )! PCADIPA.rs/PCAsIIPCA,,szllPGz)/PRoBsBiumi so (IR , BURI ) NB: 52=119 NON Discrete; A = BUR ) NON CENERSTD Ds Parricide Discern-DOMSNDE: s) FIP so (IR , B IIR ))? 2) COME CARAFE Rizzo IP cow Quahogs si Pro SEMPLE?CRRRERRO DI CARSTHEODORY:(I, A ) spsziomisues.BE, B Q PROBDB ruins so it ; E E htt t.c. 1) O (E) = LA j 2) E cuiuss PER inverse zoom Ernie, shoes: D= IQ ←→ Ble = late , overs Peak ⑥ CA tacet ⇐ PCs ) = QCAI freeIN ALFRE PAROLE E- GDRsNrtrD L'unicorn DEUIESENSRENE DI ink pro BA Burnt sucks as SSE WE Rd SPE Rl E opportune iPods'. SE NAOKI BE QI COIN ODOR Svay IN SEMI DEWSUDDEN As SSE Csooo, NSE ME or A) ⇒ corn OAERSN we svruotoA-hee•SE VOGL Cs RANERI 2zsRE Unis Probs Bruins DEN Chie DERE 6uEzreNE GENE Mrs Desai EVEN ri cue sea Chiu Ss Rr SPERO A in CONTROESEMPIO:IBZNostri paid COME l' iporesi Do cuiusurs PER INTERSEZIONI FINITE sis ESSENZIDE-D' AZB'PRENDI sue uns I auscsrssi DELE SUE Parrino Ni E =L Ak Abel K,f- 1,2 ); D= } Ai,Az,B, ,Bz ) , WON Chiu SE PER INRERSEZIONIFINITE (ADIFFE rents PER E SEMPRE DI E = 10 , A. NB., so nBzAznB , ,AznBz )) # 8=4 ;# D= 4- ORS SEPRENSO A = G (D) j IB Q pRoBsBhir5 su A sites:44 '14B,OI12B, D= '14 'k,Br, ="aOB.. /P(A×1=PlBkH It 'I=f=It0= ④ lAH- ④ lBK)TK=1,2, Possesses, RE WE /P/D= ④ ID , ovuero Pls )= (A) TIED,MA ESSEN Do D NONAi AzD, AreChiusi RISEN ALLE INTERSEZIONI FINITE AUREUS CASI (come AOESEMPIO A. nBz EID ) Perai Pt IQ.Definition: si Definisa FUNZIONE Di ripnrrizione Di UNA PROBABILITA- P :B → E 0,1 ] Ls Fonzie WE F: IR → I 0,1 ] , Fcx ):= (( -ee, X] ) TEOREMA! LDFUNZIONE Di RIPARTIZIONE: 1) F : IR -a IR E- FUN NONE or RipsrrizioneDeus Pro Bashir IP ! B → I 0, I ] ⇒Q ) FNONE - DECRESCENRE;b) FE -Conroe Nus DA Desrrs; c) TTX ) *, I,Fix)% O2) Fxcy : Plex , y ] ) = Fly ) -Fox); Plex ,y3)=Fcy , -Fix-I; PHX -I-text; Phx ,yH= Fly -I-text; Plus )=Ftxl -Fini ) - 3¥ Fcrsrsierizzs IP , ONVERO: P1=Pz ⇐ FL=FzDIMOSTRAZI ONE! I =p ) ( surra FREGIA (as) Trappe compress, Now us VED Renee- a) X Ey ⇒ ( -OfX ] E ( -00, y] =p IP ( I-00, X] ) E Pll -00, y ] ) ⇒ FCXIEFCY ) → FNONDECRESENRE,'b) Anbe X successive DECRESCENT AD X- ( -00, Xn ) ¥ . I f- 00, Xn ) = ( -Oo, X] → Ffxn) = ( I-00,2in )) a IPA -00,213 ) = Fcx) ⇒ teohscowrrwurrn DA Desrrs; C) And-oo ,I-oo, xn] to Rt -oo ,xn7=¢ ) = Pll -oo, xn] ) % 1/70/1=0 Xnitoo,I-oo, xn] A Unl -00, xnI=lR⇒ Firm = Pll -oo, Xu] ) EE. 1/91121=1 xsppsrtiENESDO.CN/1NTERusuoDEussvccESSl0NE. 2) PIC x. y ] ) = Pll -oo, y ] ) - IPCC -ee, x] ) = Fey ' -text ; Ex ,yI= An I x- In , 't) .⇒ Plex .gl/=lPlAlx-Hn,yIl=hiu/P4x-4n.yH=liun/Fcyl-FCx-Ion.Y= Fly ) - text ; ix. yl = Ktx , y -th ) ⇒Ii.- I x. y -th. ] ; Iz=ly -tin. ,y .tin. ) ;....;In-- ly - ¥ if - th ) ;... ⇒ Plex ,yH= PIK In )=EPlYInl= him PLT) = linin I Fly -un ) -text ) = Fly - I -Fcxl; [ x. g) = I x ) -- ex ,x ] ⇒ Phx ) ) = Plex ,xN= Fix ) -Fix- I 15/03 LEZIONE 7: * 3) (⇒ ) Find Pll -00,217 ) ⇒ F=lR=Pz=E ;1 21.Y=Fcx)E- Few = IPCC -of x] ) = foxed axa 2) sis Dara us PROBBlurs IP , Discrete so S E IR ,S Depict NUMERABLE t.c. Inf I IX- YI , X. yes ) > O • • • • •a PRENOO ON INSEE Di PONTINONDENSO IN IR, DEVO SEMPRE AVEREVNS Distorts BEN DEFINITA IBENGE MINIMS ) TRS I PUNTI-MODEM PER ES TRA ZIONE Di UN NUMERO REALE CHE PROVERBS DA S-p Cx)> o FreesSia p : S-a [0,17 , f¥spcxl=1 , sneers cuisuiaue p Detosrrniarsceera. sis P : B → I 0,1 ] , PCs )=¥ansP '", Fly ) -_ Pll - orgy ]) = IIs PCH • FfxlI.Ipcxn) Ipcxi ) Piaf AHlIOnX,212--- . -. . .-.. - - .... -... ---- -Xn soroeseaepi-i.at/P=U( 41,... .nl/uNrForoES--/1,-.,h3pcxi--In-VxeS-b)P=BCn,p ) Bonomi ALE, HEIN , PECO,1 ] 5=10,1, -ah ) pCk7= IE) pk CL-p)""FKES;c) P=PCX ions,Iso 5=71+-10,1 ,...,n, -4 pcki-e-t.tn?fkES3)ESEdeProCQUEsroDlVENrERs- SEMPRE PIT GENERALE ) etoo f : IR -on Eo, too), f Now NEG Arius, R-wreaks Bice#, ↳ ¥ " DX =L ( intense R-improper # → f Densrrni cenrrnrua. Sir FAI = I -oo ftxlolx COWERS FauMEIE DENS ith f ) ⇒ FE FUN HONED, Ripnrrizi ONE AVENRE E SEA#Nrl PROP Neris :xa) FE- NOVDECRESCENT (f Even NEGAN Us);b) F court Nus assesses (INFANT F E- Fonzie NE INTEGRAy;c) LYIN F"^^ = ¥76 I -• fctidt = O b¥¥ Fix) = atishoo I! fetid =L Cx crores,)BE- uwspraznesicirni csizsrerizzsrn Ds F ⇒ Pll x. y]I = Fcyi - fcxi-fxfltldt-PIEX.gl/=lPCcx,yD;PClxh=oVxelR;PCQl=lPl # alxlk.EE/PllxH=0lPCBl--?V-BEBFCtl=fCtINB : se f E' E° srrsii ⇒ FE E° (IR ) n E' istratiHR ) → ¥ htt DAE FE continua.• y 1---.--..b- e seroesenpialp-UC-LQ.la ] ) SE fCtI= e) Ice '"⇐",,ax y=fcti -abb) D= E ( X ) ,2> o- SE fit ) =I e- itI eo, too)It)c) D= Nlu , 04 , MEIR ,o'> o se fit , = HIRE ] e- ' VARIA Bili DLEARORIE:(I, It ,P ) SPAZIO DIPROBABILITA-CHE DESCRIVE ON ESPERiah-Nrosu-s.ro Rio; CONSIDER@UNA QUAN rita casus E Di VALORES PRIORI IN CERRO IN QUAN TO DIPENDE Dsu'ESit0DELL' ESPERIELENRO E INDIOCON I 6 SPAZIO DEI vswripossiB.ru A PRIORI- LA QUANrirb-cs.su se DI wi PARUS Me DIPENDENRE DAWESeroE X! SL-aE-PERPOTER FARE PREVISION i so X E- NECESSARio Corsi Erse IN semi come X- LIB ) =IW: XCW) EB ) = (X EB) ersXEAXEBaBEpSERVE CHE (XEBIE A → Posse CALCOLARE IPCXEB ( XEB ) EA FB EE E-Ecctssivo → SELEZIONO E= GIF ) Di INRERESSE.DEFINIZIONE: sis (r, A. P) UNO SPDZIO Di PROBABILIRA', (E, E) u No spswouisorsB.ee =p. uns VARIABLE bestows Ds 52 IN E E- UNA X : I → E TAE cue ( XEB) Ect tfBEE→ X MISURSBIEESEMPI: 1152=11,..., 612;A =/ ; E= IR; E=B ⇒ X:S → R, Xcw) =XCW,, We) = WitwzI 2 3 4 5 6AI• I22X3TNII: A Discrete → Im 1×1=12 ,...,125 Discrete;#Iue CX) E#I4i5• i•.1(X =3 ) = ( X E 13 )) =/ 11,2 ); ( 2,111 ED• Or, A, p1B, B XEB ,B= ( 2K I KE II =/ QIZ; th;... ) (X=34,6,...I Er → IX> IT ) = IXE It, too)) = CX> 4)D••2) D= IR,Ct = B,X:C → E. Xlw) = Fw ) → 'CEIL' ovuero us PARE wrens superiore,, !,•,ygs-2 = E continua; Im (X) = 7L Discrete E = IR, E= B- BEE-B,•MA ( XEB)E ? → E B INFAITI ( X EB) E- ON UNIONE se Phi NVMERSBIEDi INE Rush i → ( K, KTI] EB 19/03 LEZIONE 8: AE⇐MP: N=E=R A = E=B×= "w) E- A CONTINUE' Im 1×1=(9+00) OWN " - × MISURSB't? Xcw ) = w'CX EB) EB FBEB? y=hoX : (A, At,(E, E),(F,F) -X : r → E MISURSBIE,hi E → F ouisursisiu =a Y= hoX= uisuRsBiEYeo→ →cCOMMEND! IN GENERALE use E ETE RE MAiUS6w SEMI Nuevo DAI(X! SL → E, Yin-of), QUEUE minus coin SE Arvo UNS "MAN ipocszrone TRS SPAN Miso RSBN DIVERS loss ( h: E-a F)-Diahsrrsziove: CEF , ( YE C) = Y- ' (c) = X-' tht e AEADEFINIZIONE: INDIO 6N B"LA O-SLAEBRS DI Bora so IR", CUEPER Definition E- B ! 0 II BER" IBAPERRO) ) DEFINIZIONE: h: IR"-a IR"misoRAB.IE Risotto A B" so IR" E B "SO IR"E- SETS BORE LiarsTE0REM→: SIANO (A, Al; (IR",B" ) spbzimisuRDB.in' cow 431, sweeps: 1) X:S → IR VARIABLE senior is ⇒(XE t) Ect ft EIR;2) X=Is: R-a RV.A. ⇒A Ect;3) XK : I → IR V-A- TK =L,...,n ⇒ X= ( Xi, ---, Xn ) : I → IR" V-s- ⇒ LE shade component ri save MisuRdBri; 4) Xk:S → IR v.A. the;h: IR" -AIR Mises BIE =. h ( X=(X,. ,Xn) ) : r → RE V. A-;5) X: r → IRV.A. Y:S → IR v.a- ⇒ Xty ; Xy ; XVY ( →wax IX. tune variable sesame;6) Xk : I → IR V-A- Fk EIN ⇒ sumpXn; inf Xn; limit Xn; limsup Xn Sono Vor- SESRONE;7) Xk: r → IR KA- FREIN,Xk Cw) →XCw)Fw Er ⇒ XIs.8) h: IR"→ IR"continues ⇒ h BoeotiansESEMPL! 1) I =E=lR,A = E = B XCW ) = WZ → Pars Bob E- FUN wore court Nus =p VSRISBIE serious 2) D= IR ,A =B, E= 1132, E=B ',X: IR-AIR'→ Xcwl =/ V-s- arszesueaoeners-3.ee 83) I =E=lR,A- = E = B X'.IR → IR. Xcwl= W't Ie.,,,] Iw) t # ↳ "+ 'w"E' v-A- PER E PRO Premi 5,8 E 2-↳)(I,At sp. mis. ( E, E) = (IR, B I X :S → IR V.A- ⇒ Y = X't TX ) V. A. INFSTI h (x) V-A-E Proprietor 5 mi are cue Souris ol V-s- E- v-s-ESEM-p.io: I spszio DEE composition, A- 2 "⇒ OG Ni X! I → E E Moors BIE F cordoned Nro (E, E) -ESERCIZIO #! I spszio DELLE coupes, zion,A- = 0 ( Ek IK EI), ( Ek ) KEI PDRTIZIONE Di I, I spit nuuERSB.ie- sis (IR, B) SP. Mis- , X:S-0113 Is X Misurs BIE ⇐ X costs NE su OGNI Ek.DEFINIZIONE! Dsro II, A. IP) spatio D, PROB-,CE, E) SP. Mis,X'. I → E V-A.I,A, IPE. E, x→XEB Bx⇒ cteisarismo DISTRIBUTE (o E acne) Di X LD probabilities P× ! E-a Co, I] , P× (B) =P (XEB) FBEE-X ~P×→ X v.A-cow Distribution EP PROPOSE IONE! P× E- UND PROBSBIU to-Drniosrpsziove: 1) P× (E) = PC XEE) =P (D) =LCXEBelnlXEB.ie)=$ triteTo XEBKAXBk-Ma2) Bn E Eth t.c. BenBk=0 Fete → P×(KBn)=lP( XE Un Bn ) =P ( X- 'COn Bull = Plunk - ' cBnN= Ply XEBN) = ↳ IPCXEBN ) = E P×CBn )→ xeBeBeSE (E, E) =4R ,B ) shoes P× E- cars#Rizzstb Daus sus FUN were Diripsrrizione (F.d.R.) FX://3-s.CO , I ] non F×Ctl=P× DI DE DADI 52=11 ,.. -, 612, It =P , IPUNIFORNE . X : I → IR Xcwi-Xcwywd-w.tw ⇒ p×?, F× ? Imad= 42 ,..., 121=5 ns./PCX=k)V-KESX Ptl B) = IPCXEB /P(X=k ) K2 3 a s 6 78 910 in 12 froze _→ THE PROB Abhors so CIR, B) PCX=K GV support § I Dev sirs-p Ck) = /PCX=K ) .ILFAITO CUE IP sis uniform sues Naw DADE NON iNFWENzsPYWFSNPE-proBABiurb-suuvosp.su 'd Csm Provoking MENTEE PX UN's urns Distro Bow@NE-2)r=E=R,A =e=B, D= EH ) ,X: IR- AIR -. Xcwl=rw7 . Pt ? Icu CX) = I Discreet ⇒ P× Discern so S×EIueC× DIX ) - OVVERO PXCB ) = IPCXE PK )→ pck )=PCX=K Discard pcki-KPCX-kl-pclweIRt.cl/cwi=ksl--lPllk..,K ] ) " ¥ I e- "ds =-e- " I = e- " " "-e- ' "= e- t " I e' .e ) . meenrrepcie , ,of Keo xngcs .e- ' I NII: IPCC -one, 031=0 ESEMPIO:r=E=R,A =E=B, D= EH ) ,X: IR-a IR, Xcwi=w - FCB )= IPCXEB : XCWIEBI ) ABE B → X -E 127 → p×=P Y UN's LTRS V-A- Y ! A → IR ⇒ NECESSAN's NENE Y= hcx) PER evacuee h: IR→ IRESEMPIO!A =E=R,A =E=B, D= EH ) ,X: IR -a IR , Xcwl= w' → put sssumERE Turi tuscon foslriui. Iue 1×1=10 , too ) PlX=tl= PH -A; El ) =D htt> co PCx=tl= PCO ) Ft co PX ? Next PAGEa A Fx P×= ? Ero F. d-R. Filth Phet ) _q t> o→ PIC -rt; EH = f! folds = lot 2e- t Sds =L-e-tr # ⇒ FKtl-k-e-M-IIee.ae , HatooPC 0/1=0 o * ecoat- ZE F× continues SOIR E E'istrati,I DENSIRA' continua f×Ctl=¥F×lt7=ft¥ htt> oAITENZIONE: (I,A- , IP ) SP. PROB- ( E, E) SP. Mis.X! I → E V-A-j Y ' -D → EV-A- ALLORA NON so Devone CONFONOERE: yo X Risotto DEL Odo Saito SUVNFOGLIO• Y= X significs Xcwl = YCW ) V-WE I → PER dunes, asI Sir IEsto DELL' ESPERIMENROAT-s.ro Rio- XE Y MIDANNO hostess RISULTDTO → TIROUNDSDO--y Risultbro DEL osooscrirosuuweuoo-n.NO• PX =P "SIGNIFICS PX (B) = PKB ) t BE E → YEsiro DEL Grove brunt base, XESIRO DEL Grove Diw outrobase so so SCRIVE XXO TE STDESEM Pi: LDN Cio uns none to now trocar, X=( , croceI y= f " " " "⇒ Y= I-XO CROCEDEFINIZIONE! DATO Cr, A-, IP) sp-PROBABILITY, CE, E) sp. Mis- X: I-a E V.s- Y : I → E V-A- ⇒ Dr REMO CUE Y=X QUASI CERTDNENRE CE INDI CUE RENO Y=X on.C.) SEPCs) =L F HE A- t.c. ( Kw) =Yew) FWEA , IN Foti(X= 4)2A NI: SE CE, E) = HR ",B")⇒(X-Y ) = ( X-y-o) Ect Eseaepi :S =E=R,A =E=B, D= EH ) - xscwt-wjxzcwl-lwljxzcwl-WI.ua (W ) =o Xi = Xz 9. c. /PlXi=Xz ) = PHO , too )) =L=p Xi = Xz 9. C. P(x, -_xzI= PURI IQ)= I-PC= I-O =L⇒ Xz=X3 9. c. P (X, -_Xz) = IPC IRT, at)= L1B: SE X=Y9. C. A VOLTE si ibenrificswo → si Passed sus Cisse DI EQUIVALENTSEX ] = (Y v,A- t.c.Y= Xpic. ) Proposition: X=Y to X=Y q.c. ⇒ P×=pY Dirhesrrsziowe: X= Y ⇒ X=Yq .c. OVVI'S PER A DEFINIZIONE Dl Uh u Aarons Edens al UAUAGLIDNZS Quasi ⇐ Rrs TRS V-S-SEPCS) =LX=YsuDDAX=Yg. c. =. PIP " : P×CB1= IPCXEB ) = PIYE Bns) = IPCYEB )= P' 'CB) f- BE EATEN LiONE!I)PER csccocs.EE P×Bsstb Cerro SERE X CW), WEA, AE A, PCA)=L.xcw, we,Mi sursB.IE2)PER cavers RE P" Bssrs cue F Xcw), WE A,AE A , PCs ) =I ⑤ Fx E Etc. I =/ X WEA'→ P'TDipenre snobsXcwl, WEAESEMPI: 1) MY ,HO P¥ SEY #O QUASI CERRDMENRE; 2) X= Prints TESRD SO INFINITI LANU → NON Definite su Toro w ⇒ TUITE Croc.SPECIF ictti X: (r,A, IP ) →(IR, B ) V-A. RESE (Posse sector'NrENRE V-s- Novarro we SP. E visors BE)P" E- DETERNINATD Daus Furtive csrsiterisrics F×Ct ) =P"II-agt] ) = IP (Xf t) → Px: B → [o, I].1) SE P't E' UNS PROBSBILITD- DRSCRETD SU SE IR cow DENS ithpx ' . S → E0,1] ( CiOE-Px (B) =Feigns Px't) ) Athens Dicismo cue XE V.A-Discreet CONDENSED Px →→ /PCX€B)=P× (B) = ¥Bns Px't)p Ct) = lPlX=t DiP×= Deus V-S- D, x= It EIRIRIX-t)> of PROPOSE IONE: CONDIZIONE sufficient RE i FX COSTDNTE STRATI =p X v-s- DISCRETD Cow S= Puwri Dr Disco Nri Nurmi D) Fx ; pxlt) = AND#zzs DEL SATO IN t-2) SE P× E- UNA Probs Blurb- "court Nus" cow REN sirs cowrrnvsfi.IR → to, too) (R-wreaks Bre) cree- P"CI) = ¥ f Ct) dt f I EIR bireme WE X E- v-A- ' 'continues"S= supportDi P×= supporter Deus A DIX =L TER I f Ct)> o ) proposition: GONDIZIONE sufficient i F×E IR) nL'srrsuri (IR ) =⑤ f×Lt ) = d/dtF¥NB :SE UNA Pro PNETD' CS wi VAUNTS- DIPENOE DA WED, Vse Fw Es cow AE AE PCA) =L Di REMO CUE Ls profilers USE cuss) CENTMENEEstudio: X- Y 9C. ;X> Of .a.INTEGRDLE RISPEITO AD UND PROB sBiLirA:(D, A, P ) SPAZIO DI PROBSB.-Lirr;X: I → IR V,A- RESE- Vocalism TROVARE TICX] vscore Actes Dl X-a Veere "CENRRAE" RISPETO Arati ipossl Bili X CW), WED . EEX ] =µ NEDRA PESSRADEI Rossi Bili X IW), WER- Eex7=µ= In Xcwllpldw ) =L Xcwidlplw , = In Xdp r.it ,P R ,B A• Xy → 'iDEFINIZIONE: CIN 3 Passi ) .::...PASSO I: DFFINIZIONE! X V-A-REALE si Dice SEND Lice SE X Assume on Numero Finno Divswni, OVVERO #Im IX )= HE IN • Xi Im (X) -_ I Xi, ..., Xn ) jXi LXZL ... (XnSEW C- A,Proposition: X: r , IRe- v.s- Eunice ⇒ X= ¥1 " " Isk ,x×eiR, Are at (Aalii ,psrrizione Dir-Xcw, =AKIA !w ) → Xw "⇐ w.ca.EVEN TO*Dihhsrrszi ONE:(⇒ I SISNOX Dl Iuecxl, Sir AK = ( X-Xk ) Ett → AKE Atk IN Quarto XE an' sunrise, INOLTRE I As, -. ., Ant E- PARTITIONED in X = ¥ "XK Ian,X VALE Xs SE 6 PESO DALLIINSIEME DOVE X Vse X,(DOVE E DEEN→ As ) - ⇐)XV-A- PERCHE-AK EA TK ⇒ Isa Cw) E- uisurs BIE the =* XK Isa E'miso RSBIE Tk ⇒ € """ IDK E'Mi SURSBILE- XE ovvrsiuevre SEMPLEIN QUAND #Iue (X) = n EIN PER Castronovo-DEFINIZIONE:XV-A-Reese see mace ⇒X =" XKIx-xd-a.IE [ × ] = =¥ Xk IP (X=Xk)PASSO II! OSSERVAZIONE! X:S →ED, too] CON O-ALGEBRA E= OCAPERTI) res X V.A.RISE GENERA LIZARD (too Valeri Aeedlissl Bili)E Now Neeasrivs(30)±zn E EX a KII F re Co, u2" -I]SE X us. RESEGENERslizu.rs New NEGAN us =⇒ F Xu : A → Eo, the ) V.a sempcio Fut IN t.a. Xn TX Fw →Xn = In SE X > nA=a Heim ( Xn ) LOO Tn ,Xn Miserable tu → control Musume tissue'RE XD , or BORE Lisa: (Xn=¥/=( KEEF ; )) n x=XnCw)murmur ✓\ •X n T X DAL Basso PER COSRROZIONEDEFINIZIONE :X: r → co, too] v.A. ⇒ E ex], eye EExn ]=hµ Ik¥0 " ¥ t " 4¥ ) th PH > " )) t-c. Xu p ×yw swaps..• FEE X ] ECO , too ] ; • EEX ] I DEFINIZIONE Au Passe I SE XE-SEUPUE;•SE PIX IE Ex ] >, h PIX ) → too - NI : EEXI =too =# PCX : x:r→R ha.. xtpsereposrivoork-xro-auexlx.us ;X- parrENEaarivn.rs, X= - ( X NO ) = - (mint X. d) yo an I I x .NI: X=X+ - K ;lXl=XttX : Xin -a IRV-A Annette rescore sites SE FEExt] A#Ex-JE NON SONO ENTRANTS. PARI Atoo-IN QUESRO esse IEEX. posse I SFX Dei Valero she Passe IIIDEFINIZIONE:X:S → IR u.A- E- INTEGRABLE SE IE Ext) Ltoo E EEK ) Loo ⇒ ECX ⇒ ECIXI : I " l D, A- , p ) =L 'HP ) =L "→ assse DELE v. A. REALI INREARSBILI-Quinol X E L ' ⇒ EEXT ] Ltoo, EEK] Ltoo⇒ ECX ⇒ ECIXI ! PROPHETE Dl HIEX veitazinu ( next BYES ' t Xi YES '; V-e.be/R ) E E: L' → IRE Mapps LRNESRE ( TIE @ Xtby ] =eEEX ]tb E EY ]) E terror Posrrivs I XZ0⇒ EEx]> o, OEXEY , YES '⇒ XE I ', OEEEX ) s ETY ] ) 2) Xs , O, EEX7= O to U.A- X =Dor. c. 3) X. Yd , O, Q,b> o ⇒ EEaXtbY ] = a IEEX RENU's Daus Proprietor I Newswire use per TUNE E v-A-Positive (su cue Now wreanssw, ES too) -41 XEL '←→. KIEL '⇒ IEEXT.lt#ElxD 5) X=Y 9.c. =EIEEX) = EEY ] AITENZIONE! Msr SCRIVERE q -C. Accs Nro so UN VARE ATESO CLUE E- UN #Reset- 6) conuteaenrs movers-No of XNEX a-c- ;XnBXq 7) LEMMA Dl Faroe: Ye Xn Stoop .c. Y E I '⇒ EE liuninf Xn ] E licnninf ECXN9.c. E- UN esse PARTOCOLARE DEL EMMA → VALE us NEDFSRMD Test- 8) co NVERAENZA Dominos I TEOREMS DI LEBESGUE:Xn ,Xp.c- , IXdfyq.ee . Y E £ ' ⇒ Xn E £ ,X EL 'E EIXn]aEEX], E-lh-nmxi-linmE-XD.com#Nr0:6E7pERoUEiT0NO Lo SAMBO Liked RE → INTELRSLE E QUINN pERNET0NO Dl thus RE Uri Li Fondue Dl CALLOW CNTEGRSCE-cerous Rio: ULTERIOR i PROPRIETD- DEL E EX ] !a)X=c f- c- , CER ⇒ XES' E EEX] = e;b) IEEX=IE Ext ) t TICK] j d) Of Xebec . ⇒ Xe L' E QEEEXIE b-Dino srRA2ioNE* : a) X= C a.c. =a Y= c Fw semper→ Y=cIr⇒ IEEX> o ) > o ⇒ Pix> 01>0 = uuexl let ,lbll=c or-c. ⇒ IXIEL 'perche' EEIXDEC Eh XE I '(1)6)Le)Le)=a a sE Ex ] Eb fig . PER PUNTO c) iV-A-SEMPLILI× ELIX:S → Eo, too]X:S-ok 21 /03 LEZIONE 9: ESEMPIOI proposition: r Discrete;A= 2AP ⇒ pw ⇒ EEx] =L XIWIDPCW ) = ¥3 XCW)pw FX:S → Eo, too] txt I' LEAVE : 2163 FEMI I Protestant ; I Discard, A =2r P → pw ⇒ E ex] = fr Xd ) DPW ) = In Xcuspw TX :S → Co , see ] taxes 'I: X v. a. SENNE no Xcw7= AKI # ⇒Cw ) ⇒ EEx3=⇐ ? Xk Plx=Xa ) =¥ "Xk- ¥c×=* , Pw =¥"¥*=± = ? § # wlpw ⇒- terse I ⇒ Ear xcwipw Est :÷ .-• FORMULA A Posse ossie sempreg µ- I X: r → Eo ,too ] was .oe.us. xncw , → ⇒ www.r.im ...." , ⇒ x. can f " " " " I ..W C- I Wun, --., Wurm, --S IT r ! 't cart Fw ...,w., Cus t ' tr wzw,Cour,Mend.SE OE Xu EX ~. X. TX for ⇒ Eex3= E Eliza ]? line LF Exit liar Ew Xncwjpw = linen Ei Xl WH pwn = € r Xcw ) pw II X E £ ' IXnlwll I lxcw )) Fw ⇒ 141 Ella bare XEL '⇒use car. son.- ↳ EF.coresoupAN"CD- HEE - Eex3= Ellis xD = line Ecxi liar Ewxncwjpw = linen Ei Xl WH pwn = Zee Xcw )pw Escape : D= It = to SUN , A=2 " D= PCH .a, pre e " AI K! X :D → IR Xin --k →x- Pla ) EexT=& . ? Ke - ' ¥÷ ex Y : risk KkI=k1 → Ely ] = If ? Kye -a so Yee ter * * ESEMPIO D= IR A =D Pissarro su Saw oersrrs- p . ⇒ EEX ] = § , Xcwspcw 's Tx : RIO , sad txt L 'Pron I = It A=2r D= plz ) Xn : R - ra Xncm=g÷ Ie .sk ) → linus Xn a of . E Exit = ? ! xncklpn . =¥÷e D= EE by Xn ] thin EEK ] =L '→ Ever www.aownsm NEIN . Movements,us x#are o=EIhI# Ely Elin ] -.e- ? DEI : ↳ X DIP = fr Is X dip ta EA na serve Iax > e.dare FAXES , NO . Part fr fad D= f. sdp taels TENENS: ( I , A , IP ) SP. arrobas,wrb ? Xu V. A. for 9. a ) In :D → Lene ) th •⇒ EE -2 K ]= EYE ' Ex ) n= l 9. b) I? E El 'Ll ] Lee=a It Eek cawenaeec.; y ¥? KES '; 3) ⇐ - Ecxif ICE :* ] Did :* Sn=¥" Xk Tn=E ! I Kal ⇒a EET ]= HI E€ilxaD=⇐ # Exit oeTn&r= Ee ? 1kt :L -see, -3⇒ IE Erk EEE lxelI=E[ liz ⇐ " Kal ] = Ellinor ] hah Ecru ] = hug " Elixir D= [email protected] ) Er ? ETCHED ⇒ # Er ] ceo⇒ ACT -_ taeko are EFIXkktoop.c.AE?Xr . as.s.ec . ⇒ Converse e. a.• ⇐ I Xk= line su ee Kev• Sats e. a lsnleltnlst EL ' xtp EFM SEE e E Esu ] 4-Ees ] euros ¥ , Eek ] = EEE ! he ] DEB TEI : Drsvcivscsusnrs DI MARKOV Id Cfcs , ) 8. & props. X : I→R . V. A rise⇒ Paxton ) E ELLI for >0a Dnd : be ) KEL ' are - (b) XE £ '~.on Icixi > e) E txt f wer no TIE a Ic # sell ⇐ERIN ] ⇒ BCLXI I e-L IN Erie × =/ g. c. [ X ] as SSE EAN. X no LYN =L "= carsme see cesssr or Eeuiv. EX ) : X e- f ' Go .Abuse word. - X € L " DEF IE Psoe D £ "= INS RENE SEE V.A. rear X t.cn/XlPef " PL =" I, crass, see .br EX ] : IXIPEFP Xe LP ⇒ & EIXIP ] see- Nba : he g. a ⇒ TE LPV-ps.se TEL Preparers -s, L2 , LP ss ) L' EL', Lt seaweed . ; IEEXYIIET XYEL2⇒ xye L ' ECXLIELYT 4=1 ~. E Ex ) 's E- Ex ' ] s2 ) sep Loo ⇒ LP e LIC L ' thwarts ' DEI : He L ' CE Ext ] Lsee ) .ran Cx7= IE E Cx . *ex I ' ]= Et ( Cx - up ] coo /HIV , / xel'u ✓ XELZ ⇐Rex' ] Lee ⇐ Luanne TEI : Propr . E- us ✓ sworn: sis XELZ : Br ) 6230 b) O' =o⇐, Fct .C. ftp.c . ⇒ Fat -a. X ufc 02 c) lpllx - MI > a ) E Te Drs cueosycuev d) 02 =E Ex ' ] - Fe Ex ] ' Dre : e ) l X - µ ) '>, Fw ⇒ or -- E EH - MI ' ] proper has. I b) ⇒ X= * p. c. ⇒ FeCx7= E Eck e @ Cri =e / ⑤ 62=0 ~,F- EX - ut ) -o-⇒ otw I = EEK - cm aE cod -o⇒ Cx - Miko ee . ⇒ X=µ e. a-Dec.⇐ µ c) [email protected] = ¥1 ? I = of → sorrrursueruusnreov d) quo tonearm. tee :I ?¥ ÷g=k×dPy⇒fif*xdP ' RIB x p pY=px RIO : REGAS vseoEaeEso :Cr, A. P ) steers. IF , E) se. Ms. X: I - E v. s. hit → Klee , norms.A⇒ 13 .e ) hcxl E L' ID ) ⇒ feel ' Cpg xoxo , h ⇒ B. b) Snare E. Esco, -3 f frhlllcw , )PLdw)= →→Saare EE L ' Ipx I = of here ) P×Cdx ) BU ,PE, E.pt RIB LEVEE : 22103 * Dim veins Scarf →FhEcE2.• VAR. Aest. BEAU DISCRETE. ( SL,A, ) spare press. X: R→R V. s.E-arguers se P " E- orsaens , cioE F SECR, S s-pro- Nuu., P :S→ 6,17 t.c. PIB ) = I pees ⇐ F sein,Ssu pro-Nim. f. c. P Cxes ) =L te BnsXSE WE LXES Ich FE S ' I = ( to * we C×¢ to ESNI! SE FX consonance sons N ⇒X Vs nestors hersOFF:Suffern St = f test.c. p I t ) > O ) -REGIS, sore.Brewer.Ap PRoRosE E Ehud ) = feecxsd =fµhdF= Fees htt ) pets -a it e: IR → Ee , - ) mis. ; the I ' Cpe ) ⇒ se XE S '→ FLEX X E £2 - Van Ex ] = Ifs ( t- µTpH ) Escape 2) D= R , D= B , D= E Cx ); X: Ssh , X @ l= TWI → P' 'Issuers so IN car • vs.rs peas = @ - et ) e- " ← " feed ↳ Xd = for EI dkw ) = In rwIIee.no , Hdp Cw ) = £ ¥7 KIK . mind Cw ) ' E? fr K Ice ..,",w) dPM= EIK Pick .yes ) = X= Xfce , too) 9. C-= E- ←Ze- " 'dt-- €71 .-e-7 e- " " " '= Cs -e -7277 , ke- " " ' 'car a- e- 'cart ⇒⇒ Einar - ' '= ÷* - X - e- ' ) .cn#-- ⇐¥ h-I00 Rooms BENE les ) € ,"L "=Z → I a" = 1- F IdlesE- OI-L Joannes § perino →2 ELK " I A 141 o PCH )PCXi=6l= go 1136 46-Plex, -1,4=1) (Y , Xin ? s) Yarsuene su 13,221 , X , ~U l 14,6 )) → (Yak ) discrete so 12 ,.. ,xhxl4 - ,6 ) z ) pck ,ll= ? E- 2. -,12, l -.e. .-,6Kl2 3 h s 67 89 to u 12 plz ,s ) = IP (4=2 , X. =L ) =P ( Xu',X. all It IPC Xvi )Plk=1 ) = 430 I 436 436 436 436 436 " the Oo o O o 46 ZO1/36 . -. - - -.-. 436 O -. O 46 3OO 436 - . . -- - .. -- 1/36 O ---- O " 0 h O'O 1/36 ----- .--. . 436 o o 46 .5O.O0 1/36 . -.... . ---. 436 O46•O O O O O I 136 . ----.- -. . 436¥ ③ → Deems ssrss.EU Pyck ) 436 436 3/36 436 %6 436%6 436 736 436 436 p× ,(e) EDrnias-caows-i-ihsuhinsvx-sluseg.nl Deus TAB.MaeCarr. SUFP. DIAwork PER CX, Y ) arguers: F k.lt .c. pc×,y , ( Kille dis Rock )# o p , (e) to ¥5 Sexy , € Etsy • Xz = risuursre Hasse now rwas.ro E wsre. DAIBEC (Y ,Xs ) ~ ? ⇒ s ) (Y , 'S ) ascend I cows. discrete'2 ( ×, ,×,× , page , . pycmp.ge , a ⇐ ,... .az ;e⇒ , U EATON HEAT, Cssseur. ) CONN Nui.XLA DEF :A www.s/E3EsuvESuClRh,BhJE-E-QuEhsFuNvevE : Mn :B "→ C gree ] t-C Az 1) O-arsoirius; 2) urn (Ac X... XAN )=m( A.) .-. m ( An ) FA ,, - An C- B aA,X, prod : F! Mn Estero : Mz ( Ee ,bIxEc,d ]) =fb - ekd - c) = sins rent.lunch ="venue"DlA Un ( A) =D it Ascot Nun. meant .←⇐ :c ... , / " "" " urn ( lxsx An -11=0 An-if B "'un (A) =D t dim (A ) Eu-I↳ REENA seuivrearsvae pen X :X , A.D) → (IR fB ) his. osNPErErs⇐esE Per h : ( IR ", B ", can ) → (IR , B) BenwareShrove see : 1) Xuan. → XE L' { MA race FUBINI- TONEUI. 2) LIE L' ABBishdavrears.es/cE3EsavEV-C:lRh-sG , too] cars th e C' Can ) Jpg h dm . = fan hcxldx = far hcxi ,... ,Hdx ,... dx . ↳ hcxsdx-ef.ru Iscxlhixdn FAEB " fan head "a,* goes, [ a] e.ae see maC life ) ⇒ → foil " 'd " -- fail # d"- l -- E e. o. mm IINDENE Carouse immersive Is hlxldx =D th > o ours. Th EL 'Carn ) sedan Asm ,or Mrs. nous- DEF : uns IP suck ", B " ) Anne .ie#wsirs-WsPEo0 a Mn SE Ff : IR "→ Coste ) Boutin e. a. Pak fo find dx Fae B "SE lP=P× Diwan AE X E-FA. sear ( Ass. ) corrino EFE Ers.covrrnusol X Fonte a ) f : IR "- SIRE sensors or Prob . sous IP south , Bu ) ⇒ [email protected] >o IR " fond " =L (2) NELcs.se#EfTcsrsrEN2rsPowEUSEf~30,f~=f (3) P so (IR ", B " ) ol oevsirn-fcsrsnvns.CI ] overs se f E f- sensors per P⇒ f = E p . e ES : fext-IT-slx.gs cons =L l set ) Eitel SHEL ) Yes f Benaras 5 SHAUN → f > oeuvre ↳ fcxiysdxdy = Ig fan Is Cx, g) dxdy = ¥ much = IE r7= I DEI :"Supreme"D, P GM. so CIR ", B " )=s= I XEIR "f. C. fool > o ) poxes ) = Ss fcxidx-f.ph Is I x )fcHdx=fµ fcxsdx --2ix'isat,\I\ & Alternate ! X corrino EY corrrrruo ⇒X C X. Y Jlovrrnus -i\,• w ( x ,x2 ) 'n- ESET Y~NCO.sn ) corr. → 4=4 '- X 'll ) Carr.as-a- --- Tw , jaw , as ( X. Y) . - (X. Xl ) macaws. Corrine. SL s -44 , t) e R2 :t= s 't Pax ,yItD=s f / , fcxldx =p Of > o Bon /fdx=e ¥drmS =LTEO: REGAS DEL vswhEsAEse PEN X VET.best-Carrol sews f . h : lRh→R Baelish ⇒is he L' 1PM # h . f e L' I un ) ;D E Eh = fr hcxidp-fxhhdpx-f.ph had fan DA th :B 's cores urs, Feel ' Ipg - ↳ send ( X,X' ) [email protected] . Prod : Sra x= ( X,,... in ) :Cr, A , B) → IIR ? BY UEA seat.corrine censers err f ⇒ 1) Xk :( r , A ,D ) → 4Th , B ) u.A. Cour. assessors fe , feel # = ↳ un f ( x, ----Xu, Xexut, --Xa ) dx , - dxk-idxr.tl ---Ax, f Devoirs cora. Di Xi - la ⇒ sensorsserene; fieofrosirs Msm. DrXk 2) I X ,, -. -- Xn ) E FAH taus wore . ⇒ fcxg .-.,Xu ) = f. Cx , ) .--- final 9.0 , ⇒ fur , -- ,xnl= h , ix. I- lick ) f. o. dare la iR→Rtk . LEZ. 514 VEIT. CONTI Nur NON NEL compere NO → Soo Now NENE EGLE Corsaro Nrs f fcx , - radix .. .da Er Dry is) X= ( X, -. Xu ) ver.send. corr. Dioeosrro f PCXKE = ! ! ↳ - / , flexi . nxn ) dad " - 'd"" --d d " ABED ⇒ Xa v. s.car. Conroe-s. fact ) -_ 1pm fcx ,.. -Xu- strut ,.. -Xn ) da - dxr-idxo-r.dk face ) MIS. INK70 2) X= ( X, -. Xu ) wer.sear. corr. Dioeosrro f I⑤ ⇐ ) fcxi-xnl-f.cm . - Lexie .e → IPCXIEB ,, - ,XnEBn ) -- IPCXE . , xBn ) = fz.IE?j'nd+=/p...?x!jY .. . A " " Ida . .ax. ET = ↳ n- ftp.fdxi .. facxnldx . )dxn= ftp.fccxndxi/.-..ffafacxidxn)--PCxiEBil....lPlXnEBn)VBr-eB ax, -x. wore. ( ⇒ ) confusing . rorsewipeggo rowse Free ⑦ Cheung vavauwsurs E sempre-Ers ⇒ Devo CN Prisms pin X ,.. - Xu wars ⇒ fax . -xBn fhddx.IE?fg.x...xBnffxil-.-facxydx .. . da FB ,. -Bue B Qurna ) Srs gcxi . ..K) = f. 6.7-- face ) → g : IR "→ Co, too) mis. IR " J "^dX=1 Pen FT. FILE f ←Sad DENsrns-age-oens.rs → caosroere IQ :B "→ ⑥ I ] Q CA ) = fsgcxidx tAEB INFANTwrErtF2. PCA ) = IQ I s ) the B , ¥ XBN I BK EB ) ⇒ Pls ) = IQ CA ) TAEB "Per csrsnieeoery IE -- I Bix .. .xBnlBr- EB } → Ole ) -_ B " ecniosseenfffn.net ⇒ D= IQ ⇒ fcxi-gcxsp.ee . TB 1B : X our.nu ⇒ XK Carr tr { Me TEC : Srs x = C X,,=,Xu ) : ( rid ,P ) → CR ", B " ) Veon. sear ⇒ Xie cart tie ¥3 Xcorr .⇒ X Corr. cow Xu Noir. # XK Gnr. KK =L,..., a A X,.. -X-wore.- EDI : ⇐ ) NEO ? ⇐ j P " =P "④ .... ⑤ P " Parrots . P× C Be xbox .. .xBu ) =P "CB, ) .---. Pt " CBS FBK E B FI fz.x.e.xz.fi HD--- fun )dx,. -dx,7Venue Passive as B , ×. -- XBN AD AED CutronaX,... Xn sense IN ar p - Srs Q i B "- sees ] Pros . corr. A DE-s.ro- gcx ,... xn ) =f . .. f- can → PHB ,x... .xBn ) =Q I Bix . - xBa ) FBI . - B- EB x' A → P× (A) I @ CA ) f AEB ' Pericsrsruoesey → PTA ) = dare to fun dx= § fun .. . fcxnldx . dxn FA E B "→ Kerim cont. oioenrsrro f cha . -Xn ) = f. Cx,) - fuk ) B TEC : HE C X,, -,Xu ) : ( bud , P ) → C Rk , Bk ) Veon. sear* E CX ,,=,Xu ) : ( rid ,P ) → CRY BY Veon. searsrsx = C X , ,Xz , # ) → CRY'B" " ' Iver . near⇒ X cont. E [email protected] Ear. Cort, b : IR "→ IRK Beer. Y=h CX ) Eon. sear. E- WNT ? E- Discrete?• SE for Fun. Empire as Ybrsuers (Cerro. www.csreWE#of SPFSF)•SF h ------ ⇒ y continuo• SE h G) = CX,'T ) XNNCO , e ) ⇒Y NE car. NE or screw I .-A-Nero sorrier , is,PSON cowMEDE E VAN2E caraway ×: ( r, A. P ) → CR , B) v. seem. sreaae P "→ swerve P' 'an EEXI , Vonk ) → EEEEgysasjom.ae Cx . Y) : ( r , if D) → (IR ' B ? ) ver. sear. P' " " '→ sure.nu P ""cow E Ex ? # Ey ] ; von varix ) are ero- morass row wsciorxey JER ⇒ Received Cov C Xx ) = EE IX - Efx ] ) (Y - Ely )) T . T --au Nrf :oCov. IX. see as P ""'GV SO ⇒ Measure. X - ECX secy si Y - IE C y] YT Egg #WHY]. an .⇒ um. { - ÷ :÷¥¥¥ .FLY] .ae .am#exn-e-exs.Eexaouco-- ⇒ - ÷ :÷→I⇒ ±⇒⇒o# FEE • CowCx. YI = Cow Chex ) TEC : X. YELZ , XIY ⇒ Covcx ,y ) =DErm: Carex,y ) = EEXY ] - ECX ] - EEX ]Eey ] =D Be AB: CoV ( x. 47=0 =p XIY REFS SE Cow CX , y ) =D as X, Ys row [email protected] ) REM. sent . Corr . cos oe-srro-fcs.tl = # Is 1st ) 5=1 CX. y) EIR ' E. c. I tyres ) ⇒ X # Y Perche app. D) f- S#A XB ( now E- Eas Nhon)INFANT SFX wore. ⇒ fest ) = fees ) felt )bore A-_ srppfz ,Bee sup fz Avner Supp f = A- XB Ms covcxcy ) = # E Cx-E Ex ] )C y -EE 'D )] # qy ] , ✓ FIR'Sfait) dsdt = Igs ¥dsoet= Oxs,muEr~sf. bids Cor Cx ,yI= EEXY ] = fair stfcs.tl dsdt=o 2)CXiY ) Discrete cow Drr. unrrf. Sul CQ1) , ( o, -I ) ,Cs, o ) ;C-I, o) ) ⇒ [email protected] ) If I Vsnlx ) Vsrly ) DIM inn Charmy seuwisrtz APPU cries X~= X-EEx], 5=7 -EETDEF: X. y . VAnew in E , Van C X ) so,Usn C yl ) OCuisine COEFF. Drones 2- KNEE ) f= cq%# ,→ Ifl IL ft 16N . Newusurrsral fee ⇐ KY someoneC-E- est b- ÷ ↳ Dormeuse sees Morrow re. XIY⇒ f =D ↳ r Cxcy ) Did : X. yet vsncx )> o, Vsrcy ) so ⇒ Iff '⇐ s F a to t .c. Y=aXtb s⇒ He x- s T# Eexs VsrxTEE x ] =DVault ) -tcar Cx,yI=0 NI 's XNN C 91 ) Y=X '⇒ 1ft ? If I LL x- user' org suruspsrssas Coasters ) ¥273= IVaux)=L→ lfko BEERS ? Cev. E- BNNEAE : X. Y ,Z, Well , e. b. c. DEIR → CoV ( a Xtby , cttdw )= one calx , 7) tbd Carl Yaw ) tadcarlx.nl/tbccarlyHSrM : CarCex t by ,7 ) LBsm NER. eesroxs.mn Carls, B) = Corliss )) IEE Cextby - E Eextby ) ) I z-Eft ))) = ELI Ce X -a E Ex ]) t Iby - b IE 43417 -E LED = a # E I X- ECD )CZ - EEZ ]) ]t BE C ... ) = a calx , e) tbcorcy.tl Grease : X ,.. . X .EL'⇒Von ( Xist ... # ) = Von (4) t-- t Vo- CK ) -12¥ , GVCX I ÷ " xil.EE (E. Xi - Exit ] .EC/Exil7--ECExi't2-z,xixsI---IEEXi-tExi ] - 2¥ , EEXT -53=2 Varlxi ) t 2€ , Carl .Xi . X ,.. . X , wore. en L2 , E Exit = uti { In =L Ei Xk→ reasons. ES ' E [ In ]=µ ; Usn (In )= . Usa Lxi ) --o' # Si = ? ¥ C xx - IT → von.ws. Else ] .O' sp x=t : Cris .pl - sleigh ) warmer.EEXDE Mcnxn ) DEF : 1) Ver. mesas ECXT Vorenus court . oeuecov. ) ec → Cig = GN (Xi XI ) settee Vtk TEO : X KEL?tk⇒ CE ssdp Cgrown seminar. ,,. corvee sheet ; z ) @ ,, - ed C ) > e t III ) ER " DIM ② owlets IMM. coumarin ② Ca . --- ed CLE ! ) = ÷ ! ai Case ,=,, } ai Gvlxi , Xs )q= Carl ? Riki , } ask )= Van IF ai ti )> o Prof : Hued . SEE XKEL 't ⇒ XEIEEX ) t Im (c) p. C. Ed X VED.SEAT. Corr. ⇒ Cruisers. I Cwensvv. ⇒ Xue conf.X =CX. TEI ! XuEa. Antar. Sverre aeons IE EX ] Msr. rowans G Y - Axtb X : S → R" Ae Mauk ) ⇒ EY=DEXtb BERK Y : r→iR " Cy=A cast LEZI ONE 30/04 FUNWNECARSAERisrlesp.su CR , B ) : 'T , ftp.l h h Psu CR , B ) : , ftp.T Naramore: x= (Fi!) Y= ( Fg ! ) ER"il Prorate some : § =. Naya = X. y = cx. y ) = says = cxiy >DEF: Sis Purrs Props su ( IR", B " ) Ls Funnier csrsrewsrrcscsorrrn.se. or-every Dip E- is Furrier ④ =&: IR "→ Cl, Yum , Ipi ei '"" lpcdx ) eio ⇐ .E" @= cos Oti sing , le 1=1 f@our"is mix>l= Cos Cui x> tism cut x> ,Il""""l =L fu,FXEIR"↳ TE@Ris DEuirwrearsweEPER.hr : CIR",B7p7→ CR, B) mis sniper su h: C IR", Bh, D) →C Q, BZ )mis h CX) = hs Cdt ihr Cx) he : R "-2112mi, tipi hcx ) lPldxl= fpihscxipcdxltilpihzcxipcdxl SIEVE Noone route Economies beware arsene, incase If help It /1h Idp flat fan ei 'n'⇒ lpcdxl -C- L'?icu IN) si-PERCUE-hn.ca = ecorners → his. Fu ; then Ix) 1=1 EL' tie EEF . Sis X: ( r,A, pl → HR ", B " ) VET.sear. Dr ⇐ ace PX - is Fun Ursa. Dr X E- pix = # Fx = ex :pi→e , excuse ↳ n e"""> P' 'Cdt) = fr e""""> lpcdw , = Ec eicuixs ] ESE mpro O.X Nfu mere ⇒ 4×41 = II E ein "' ] = ftp.eintfnlftl-ei~M a> o ⇒ 4×4 - my Cin-d) t±s= X ~ ECM Aso ⇒ fu , Cut LEE ei -× ) = fix e"" fxltsdt-fo-e.int Ae- a←dt = A e I ! =He- in÷I= 1¥ TEI : La Fonz. arson. ④ Di or- preps. Psu UR ", B ")csrsoeair.rs IP , every ④ Cm)= TQ cut their " ⇒ IP C B) =CB ) f Be B "IN front r Couse, Daro unscrew. a Esr X,Ix CARAFE Rizzo P "TEO REMs: Proprietors sroswrrcue or e IP Probes su ( IR",B" ) ⇒ 961=1 19112 ,hour.Erm.. as Kol = Spi ,e "Pldt) =1; by llull = Ifpi ein 't' Platt Is 1*-1 ei 'n't 't Pldt )=L fu.; a, unu Is 4in - it ,4cm,Itt#⇐ a → too u=→n → cnn.it > → cu.it, ⇒ e" 'n"'t'→ e"'m'tI ⑤ Ie""" #Ifeel ⇒ 9cm , #fins pen a. is. TEI: FUN 2. csrsooe. E nueuevrr.Srs X= ( Xi, .. -,Xu): (A,A,P ) → (Rh ,B" ) Vea.sear. t.c. Renege me IN ,Xk E Lm LIP ) t K=s, -, aam- 4cal = in IE EX .,... Xan ei'm"' ] ⇒ 4×-0 -honewort Pr Xtheher.F- Sente' × RE U free,.., n ; ⇐Exa ] =Llid; Nz : ye feel → E Exam =- In ! t9 ; EE Xaxe ] = -27T ! 49 Que aESE Mere:Xm ECA 170 ⇒ IE EX] =L41mi - Fin 7• it 9' co) Y' Iu) =-I A-hi)I inv ein = ÷ # srsossrsF- SEND3 Xv NCµ, 62 I per , 62 > o 4×= ) =iEExein× E Ceinx ]= for eint IF e- "" die = for iz ascent) ed,Ee. DIFF. UN. A°eDwe pi cuz ; E I xeinxy-ifrte.int# e- "" dt =- for Tnf " 'wt' e- " "atIwean penesrrr→ gym ,=-menu,hercows. 4×07--1--8 4×4=5^72 couriers Ferren vs moreenDEI X? ER , BD ) -CR", BY Easier. Furr. CEN. sermon. or x Mx :B "- SIR , µ× in ) = # Ee "'" ] = fire 'm't) Pxcdt ) gun far E.mu#--tEau?E.2ua , trove- Mxhd - Ecac , -xen