Scarica Probability theory(math2, bocconi, second partial) e più Sintesi del corso in PDF di Matematica Applicata solo su Docsity! - (R):number of Consider a set de elementscontaineri 2: POWER ser of er:collection of all susers ofe 124=number of SUWSPACES I cert e de Ariescaneallacrossestesamade use (20) =2 - - PROPERTIES ⑦MONOrnic1M:VA,82rs128 =M(A)=M(6) ② finireabortivi:consider A1,Az.. Ames sr AilAr= fits. Ier =>M(,Ai) =E,(ti) ③ measureof a finiteset 8A i FINIE ser:A ={wn,..,w-3=M(A) =M (Swe..., wm3) =M(E {wi3) =E,M(wi) ↳ -08A00:M(AV8) +M(Ang) =M(A) +M(8) - i A18=b- 100. PROBABILIM:any setfunction Piz S: P: 2*- [0,1] ⑦ èa MEASURE ② NORMAL12E0:P(-) =1 I PROPERMESMens:PA =1 - PCA) vAca! - NAMES: ① wE-:STAROf THEWORLD ②1:STAR SPACE ③ A subspace ofe:EVEN 8 2*:SPACEof de EVEN ⑲ (e, ):PROGAII SPACE Exof FINANCIAl Asses ↑ . PROBABILI:Gameiden;fixa statewotre ↳ r ={H,T3=Siexfare cor 2=54, E13,5 T3,}E Letf={wo3 ② UNIFORM Probabili: let(na = m (frNE) P(u)= VwE ex fair orce, fare corn etc. I=N hetA finire - P(A) =em FA?r I A=x=N + ile [ becomes SERIES ③ POSSON PROBABIL:when ro:N Pm=({m3) =eUmer (P(n):m= I PCA) =EVA=N ↑ ④ GEOMETRIC PROBABILIM:When =N Pn =P({n3) =qr(1 -9) kreN RANDOM VARIABLE Sin- to every elementinsa,Sassociato real number PROPERTIES poo of all orwhere leis equal (P -a.el e 2 random variables I,gines are equal ALMOSTEVERYWHER Under is P(wer:f(w) =g(w)=1 - the can be differentinall thewo mich P(w) =0 PSME-S,gineR ase-a.e Fweruppp,f(w) =g(w) + (leye equalan the support (uppi containedinthe netof porre where leie equal EXPECE VALUE (ne consider only simplepros Ep(8) =[f(u)P(w) - rum of theVALVES relatedtothewof thesupp weigle mich itsProAl,ins etrup FAR set:EV=0 & tley is notLINKEO: ex u can have a fare seand Nora faim ber and viceverse FAR OIE:all outcomes have SAMEPROBABILIM prop of EV (ascomides only simple proces Dif 8,g:re are ?- a.e =Ep(8)=Ep(y) ② LINEARI:Ep(68+3) =dEp(8)+Ep(8) Va,e 8,gine ③ MONOTONICIMiff(w)[g(u) Fwer =Ep(8)?Ep(el ② VASinite ssupp IA:Ep(8) =[f(w)P(w) wtA if i finire FAcm, PCA):eflui the wall Plui)-P: ellgi allto(_Protin:1=(a.....,(n) e=v:Ep(s) =22 re ={w,, ..., Wa I and Deirrum=- ↓VER :=(8,...,fr)e Expectesvalue of an affinefunction of a rand, vos Consider (r,) will simple. Then:Vin ka,ER, Ep(x8 +B) =2Ep(8)+ SLIDEG elingon Er to see letter Expectesvalue of an affinefunction of a rand, vos Consider (r,) will simple. Then:Vin ka,ER, Ep(x8 +B) =2Ep(8)+ PROOF How to measurevariabilit Consider. 20:(H, T) -P=(z,z) It's litetoring a Saircoin. Consider 8 ={ -Ep(8) =Ep(e) =0 villeg differ in VARIABILI 8:Ecco ↳mise elinee REMARK: IS8,9 ase RANO VAR 2 =[b(r)- Er (8)J2 isa RANO VAR z=[f(x) - Ep(8)][y(n)-Ep(z) ia.V. VARANCE:Vp (8) =Er[2) =Eme)-Ep8](a) &>ROPERTIES ⑦ if2 RANO vam ose EQUAL P-a.e-stheyave same variance ①TH19844VE-Secere un so STANDARD DEVation:Op(e):Fot PROPERTIES (2,00p(8 +) =V8) =(ape =(610p(8) COVANCE:Cowp(8,2) =[ (f(n) - Ep(8)] [f(n)- Eo(e)) (a)-8,8 ennovo (fla)-Ercel] [ga)-Eoal) i a maror. i8=g - VARIANCE wzzp 2 RANO. VAR. CANOE =Ep((f(u) - Er(e)](g(r) - E-(y)] Positively LINEARLY CORRELaren Cowp(8,9)- 3.8 move mopposidirection war expector values NEGATIVELY LINEARLY Correlated Cowp(8,8)=0 - 8,8 more inan Openen Way war expecter values LINEARLY UNCORRELATED Corp(8,8)0- e,g moreinsame direction wir expecter values computaronal FORMULA; Cowp(8,8):Ep (89) - Ep(8) Ep (9) - proof Affinefunctions:car (28+B, ug +S) =aucoup (8,9) LES S VARIANCEof a SUM:Vp(8+g) =Vp(8) +Vp(y) +2cow(8,9) - proof varlanceof a linear combination: %(28 +39) =aV(8) +pV(z)+rapow(8,g) =(6)e) (i) =[(849) ia2x2 sym. matincoller VARIANCE-COVAMANE Matrixof 8,9 COVArlanceis gounoe:(au (8,9)) =op(e)Op (9) eproof correlation coefficiens:f(8,9) =Cowo(8,2) -> became ofH. lefre - - - (p(8,9)1 -p(8) .0p(y) d museleforVo(8).Voles to ferr MEANING: ⑧contant on a set Ac Hatisfinireand-0,3-sor NEG.CORR. ha PROBABIrry 1) fo(8,2) Faeewe**e 0.9 - STRONO sos cOR corcorf of a (n.comwi, (28 +p,ug +6) =15p(8,8) 2. 2,00 - = (8,8) GRAPHICAL REP uf POSITIVEL. CORRELATION NEGALEL CORRELATION gr -9 ↑ &siti is · ↑ ... 38 ... & ...... ... perfect lIN. correlation:19 (8,8)1=1 eIcto,e s.r. g=a +almart ergreen under -Ef.pos cor,480 (8,9) =E210 ↓ PERE. NEccor. Sp (8,8) = - 1 =20 GRAPAICAl MEANING POSITIVE NEGATILE ny ..... ....e . L - ...........8....... ↳ Simpledensity function: p:R- [0,7) osiriveand to only alfinitelymany points3-....,3m is a simple densitySunction of Bif (x) =(3:) Xxe =Pp(Al & LESSON 8 1) I stepfunction) SPeimple density - :sum of mah till x 2) "generalized"stepfunction Ipcountable demity=seines of Muae till x 3) I has INEGRALEdensity E54 sr =SPlt) it=> Econtinuous II notcorte can'thave int. Densi 2) èhas a continuous densis an carrier> Ii(([a.27) ep =j' ↳adistribution function &(x)=[afae ! I a / Con itle retinedas an INESPAL P esFUNCRON. [a,b) istsminimum conier Let q(x)=() =Saatheleubre let's compute (t):1)xca SPEd=Sd = 4gare purodeto[at) 3)ix-SP(t)d =p(td =0+ 0 =1 exponential distribution function &()={ex ......... NO CARRIER a ICORNER fas an integrale, density function graphically: i &(x)={ 7 ifcontinuous -> Bcambeiwortose densi su can findemilyrivingand decktheINTEGRAL Standard daussian distriumon &()= i THEOREM consider Iwillcent ispcour outside cannier (a,l)= P(x)=0 Fx4(a,l) PROOF &()={p(t) dE VXER - nice has canneruix=udx= Six z,zbu willz,zz Ier:*Pax =0 Since consin[,zu]e Man P(x)=0 an [z,zz). Since zn,zzb are architary -> P(x) =0 an(b, +a Same fa (-0,a) =P(x) =0 Xx=(a.2] c"(sa,e)):function (((Sa,e)) tat startwillg(a) =0 BARROW-TORRICELLI tate ge(((a.es) => 8 diff, with a core. g'an(a,)5!U:[a,e]esrg(x) =0xd vx=(a,l) He lijectivefunction:T:g((((a,2))- (=(((a,b)) ⑤has a continuous wensis an carrier i(([a.27) -p =I' ↳prolecon torricell gc(o((a,a))-(=(((a,a)) 4 =W Ep(8) =mp(a)P(w) Exof distribution function - auvifor IK)=[bee -.... i spes. Sene iè -> d ② exponeNAL E(x):{o·yo - osranarosaussiante date TH Consider I withINEGRALEP. 18 Ihas a carrier [a,2) and iscour autside i carrie =>P(x) =0 Fx4[a,2) BARROW-ORRICELLI g:[a,2)-R mithg(a) =0 -diff. Wircont. DER ge(((,e))=5!U:[a,2)esrg(x)=S(E)de Vxe(a,2) T:g=(Ö([0,2)) -WE([r.e)) igiren bT(y)(x) =g(x),Fxe[a,2) rervareunrior TWE([r.13) - gE(([r,e)) i givenby To)(A) =SUSAdUxe[a,e) IreGRAL fummor APPLIARON: Iwillcavier [r,1) has a con par [a.)Èis(Ca,es EU Consider (o,will SIMPCE Tate biet and:1(0,) =Ep(8) =xd(x) uder [a,2) isan arren of
