Introduzione alle Variabili Casuali e ai Modelli Probabilistici, Schemi e mappe concettuali di Statistica

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Synopsis of Statistics Course

STATISTILS DEF : GIVEn A PANDOM EXPENCRENT^ WITH (^) A (^) SAMPLE SPACE (^) C. A (^) FUNCTON X, MOHIEH^ ASSIGONS^ TO^ EALTS ELERENT (^) CEL ONE (^) ANS (^) OWLy ONE REAL NUMBER XIC) =X^ IS^ CALLES^ A^ RANBOM^ VARIABLE.^ THE SPACE OF (^) X IS THE SET (^) OF (^) PEAL NUMBERS (^) : A:{xix-X d,cteS In RONE (^) GENERAL TERMS X EX (^). A (^) CONTINUOUS R (^) .V. XONLM, 82) is arathematicaL moden TO flxs descñibE analatoñY^ phlenorenon funltcon [ XXkin x Plroxoretl Iplnayanytl flx) is^ calleD peobablusty densiTy

EX .A^ DISLPETEAR^.^ V^.^ XN^ BeLT) f(x) = { H+X 11-T)-Y (^) IF X (^) =0, } X (^) is a bernoulli RANBOM VARIABLE W (^) OTHERWWISE fix)

• • (^) - - - - - - - DISCRETE P.V. WS (^) CONTENOOUS M .V. Ä!^ PLX-x):f (^1) x) x 1 Blx 2 x) I f Flx)=Exemf(x)x I (^) Flx): (^) fYf (tsdt
• D l (^2) f(x)=1 (^) JŞflxldxzt BISTRIBOTLON FUNCTION^ FIx }^ XERx^ l By BEF^.^ FIX)^ =^ PrLXIx^ )^ s^ FLx^ ) I Xn (^) Be (^) ( HT (^) ) : FIX)=PF. 1 F xroofxr 1

x iF X 2 1 å

Elx)=SPqXfxbdx if ELxKZXfIxiscot (^) 1=ZxPLXisaisc. xE-Ry (^) XE -2 4 FOR BERNOULLS (^) R .V. X (^) =%; "T ELXIITT { FOR (^) EXPONENTIAL R .V. FIX )=ME-Y EI:fF%e-%dx, X 00 = (^) ű Sõxéňax = (^) re vixJ

• JJdx-flx 15 fixldy =ETIX-ELX3)J I = (^) ELX2) - ZELELX3-X] TETELXSYJ (^) =ELXI-ZELXS-ELXS TELXJEELX?) - EL4) UINEABITY OF^ EC.S^ Elatbk 3 = (^) atbELX 3 ELX,+XL+...+YXu)=ELX,3 +..+^ ELkuI IN GENERAL : ETgIx ]]= J?qgLx 1 fCx)dx FOR BERNOVLI^ R (^) .V. ELX IIT^ ' ELIJET' VLXL-IT--TII-TS A (^) TT11-T5) çi Ę ğt

5 o: / V = = = = "":: """ El (^) 4)^ necuns an if (^) flx} is (^) SYMRETRIC =3ELY) (^) = MedLI IF (^) flx) is (^) SYURETRIS ISEL 4 Y FMeellx 1 LET WS BEFINE MORENT^ GENERATING^ FURLTLON^ LM^ .G.F.) Mylt )= Jet "flyldy = elet"); dt (^) MxLtI =fo at e"Yfexlax = Jutxxfixldy at (^) Mxlt /e. " SXFIIdY =^ ELX)^ INGENERAL^ : aFe MXLH/. =ELX" l. examele (^) For bbernounei (^) SORETIRES ITCAN NOT BELOMPUTEM HE NEEB AT^ LEAST^ FOR^ EEL (^) - G, hl, b 3 o) Esett) (^) = EetYpLX=x =fi-tItetH Idatle =o-eitent multlef. Bet^ 'e'Edi+ fextanHolxilh

• ty^ Do = (^) h Fmty ei 'xlanty f l^ o (^) Exmiexlh FFc
• t^ ?"^
• 
  : Pil^ Eneo^ ) n anJleiortaes (^) amtl " 1- mslie I
• tmul-m dixn [ EtmJlteo--21-Met) t - abl-n)/teo^ =^ Zae";^ VIxl^ = Za-m?: re^ ?

X.tX. nExplie ); X^20 Explas Y (^) =4, + 42 Mylt detEl^ etY^ )^ =Eletx).Eletx.)^ =^ I taes i - lto (^) = El4) =I-zlinta31-aD)l t =o-2m BISTRIBOTLON OF^ RANBOMS^ WARIABLES BINOMESAL :^ N TRIALS OF INBEPENDENT (^) XINBELIT (^) ), 42 NBELTY (^) .... XwnBelT (^) ) BER nowkI R (^) .W. XixX^2 t...+Xn =XNBin(N, (^) IT) Ry =o,2....n PROBABILITY OF^ A^ SPEFICIF^ SEQWENCE^ EX^. O (^) , 1,0,0,0,1, (^) 1,0. .-TX^ 11-^ TJN-x b um BWT IT IS MOT ENOUWH^ X^ SULCESSES W.X (^) FAILS à) ONLEAN (^) ELERENT IS (^) SEWELTED IT CARMOT BE^ SELEETED AGAIN iin) ORDER^ OF^ CHOICE^ DOES^ NOT^ MATTER H 0 w HAnng POSSuBUE LOMBINATLONS^ FOR^ GIWEN PLX

• xJCYJA"/-Tjn-x NANS SOCCESSES NURBER OF MW Y(x) INIXIN - YI! ELXINIT VIIINTLI-T)^ XNBINLITIN)^ WHAT HAPPENS FOR^ N-DO, (^) IT-O, (^) NIT-A3, O 5500? XoPoląs (^) Poissom R (^) .W. CaAo BE WSED E(*) =^ X PLXex) =1Ye", (^) Yeao } TO MOBEL PHENORENA THAT VI^4 I=L^ mal habpen^ oweñ^ tir FOR (^) LARGE (^) NOWRBEMS (^) ( I)^ 'S^ BIFFICULT TO LOMPWTE

PROPERT 9 OF (^) InPolal PIk EWErois (^) in [oBJNPOI3) B=EXPECTES NURBER OF { BIt EVENTs^1 n To,t 33 wPolxE} EVEMOTS EXARPLE (^) : ON AWERAGE (^48) CALLS BER HOuR^. X (^) , nPO (^) 148) WHiCH is PL 3 CALLS^ IN (^) JMINUTES)? XINP.148 (^) 8.) IPLX,=3)=BEY 4 =0. WAITING (^) TIRE (^) PIW,SES: PINOEWENT To,t3) in Iext=PLX:o)i (^170) :1, o!=1) P (^) 1,3 t)^ :1-^ Plwat) =1-Flt=ette 3 FlEI^ =1-ext flttedett (^) M =Y =3 ExP^. proces.^ aññinal^ tires aie indePenDENT - alack of (^) reroring loweR THE (^) PROBABILITY THAT (^) THE EWENT BOES NOT (^) qweetiri sexponentiannng OLLOR THE NORMAL DISTRIBUTION (^) : XNN (^) M.SY (^) FylL - JEN 'EELY-JZIN ROTH (^) JINECTS FY (^) DECREACES (^). FIXI (^) :S!O JEHC 'El G- Pdy } FII (^) HAS woT A CIOSe (^) Foon PLM-2808X 8 M+283) (^) 951, W^ = X - NNLO,1), IFXIY, Z^ =X+Y = enñlnextrçidêtjê ) CHI - SQWARE DISTRIBUTION (^) X' BY BEF^. GET (^) E,72I...E (^) BE :ind. Nauba (^) .Vs. =3 (^2) tiNX; (^) it ofygr0, WX1=g, VIX1= student (^) tbistribOTION BS DEF (^). LET Z BE A^ R .W. S (^) .L. E (^) mNLo ,i} =3 (^) : gNuts i him^ guaso tg =Nlg,i), (^) tyswoo?Nlai) AND (^) Y BEAR .U.^ S.E.^ Yn (^) X gamma bisteibution rial: Soya"e'bdy irlise (^) f 8 esdy a1, rabelarl^ fåga-zesdy - Canir^ lan.)^ Ex. r 14)=Jåyebdy-6i r13): (^) Jågebdyaz^ M14)=3r13)^ rias - SåyeBdy -1^ rlslaarca)^ rlilag^8 e'%s1, rakersn

ASE =ELT-OS I^ INGENERAL^ ASEFV (^) ,^ MSELV^ IFFTIS^ ONBIASED ) V (x)=^ ELX

• ELXIP *rse=rean sqvaden (^) ) error VIXl =WCEVIXå)=WZ'N.S': J =MSELEI tow (^) CAN WE^ COMPUTE (^) PLTX-MISE)? D^ )^ KNOW^ THE DIST. (^23) USE A PåoXY X,... . to i (^). ind (^). N (M,^ 82) MXIEJ =^ ETRTE'S?^ } FOR NORMAL^ M^ , qenemsen :I If XnNlre (^) , 8 ?, =3 =ELeY) EM+;82 j MILE-ELEET) =^ ELETEXI)=ELLREXI) ITELEEXIY
• TTCEMYE?? CENTRAL LIMITTHEOREM^ IL.L.T.) X....Yo ridid^.^ WITH^ ELXI-RRD,^ VIXI-S'OR^ = XMNNDD, Nlo..) IN 330 IS^ ENOUGH^ FOR SIMEM^. DIST.J IN 3100 Is^ EROWGH^ FOR^ NORMALITY^ IN^ GENERALJ LONWERGENLE (^) IN (^) PROBABILITY TO ACONSTANT (^) : By Bef. A^ R^ .W.^ YU^ CONWERGE^ PROBABILITYIN TO A COMSTANT IFF (^) .:. lim (^) BllQu-cl-E) =0 (^) FEsO U-BBO FOR (^) SAMPEREAN X li ND InequauTY LITN - MIEKO. BY CHEB^. CTLEBY EtEEV inEan. ELLY-M 1 R2-p NP^ 0 O I j Pllx-mloks) B PLIx-rlstS)!:^2 or Az Elx1=M,ly]=8W b Lo LONSISER R =CF, ANSX=TO

MIsEFwJE 8 n

KOLMOGOROU THEOREM GIVEN (^) Y .....^ Yo^ rind (^) - Og1.) fonction WEgIXi1-pElg(xi1)^ ca FEXåIT

• BELX) FMCN^ J LIN^ TALS^ CASE^ gl =x^ corollaeies (^) : i) if^ Yn-Bp C =shl^4 ub-pUCC)^ if^ bilc)^ J (^23) if Yu (^) - opC 4 u-pd^ s Xu-pc+d Yu
• 2 " hF 1 rvylIhŞ 1 vivuyrüey"Ex.- 44 ÁF .3XuX =-z xy%

ELX ?1-ELXIEELX?S-Me?: . (^) EE(x?3=retg elsobz *[elŞXå')-FCNX'Y), ELX 2)=M?-

• (^) E., Twlmt 8')-NluE?D) = WN-. INMENS?-NM-S')^ l

, IN-118?: (^82) V 151 = 28 m . i 52= F.EIxi-X)' }

(- (^) o. ' n Mlanbit Canst We AeE^ uwc f rotalY'w-.^ vix ?.-.)=zlo-. =3 (^) V (5) =R.Z-CN-.J= lim N-DVIS) :^ o HOWEVER, TO^ PROVE^ THE^ CONVER^ GENEE^ IN^ PROBABILITY^ OF^52 WWE CAN USE KOEMOGOrOV THEOñEd ANB (^) ITS COROLLANIES

POWER - LAW DISTRIBUTIONS : FIXI =CXY; XOXMIN X (^31) C =LX-14miN with 2=SynxTdx=( E^ ,X4+.)Ymin+) 2=- Eat,tmintt op C^ = (a.) (^) Yñin " flx) =^2 tui (^ *^ mn)" POWER (^) TAILS : I-FLXI-LIXIY'Y (^) WITH LLX) S .t. Ht 3 o 5) lin X-aox) =1 LLxI's a^ slow Ex (^) ,^ liv x-axo legttxllogcx ) =loght? togx T 2 , = VARYING (^) f. in =^0 TO BETECT POWER FLXI=Lx 1 x"wws : IIPFPLOT (^) (logex)i log(i-fixs)) if

• sMean^ wß
  • a (^) ( log^ (^ x); logkixis - a laglxs ) TO A (^) LERTAIN X Contnt (^) movt (^) : (1": ixeewny len M--. INBExJ ZA (^) =GINI A " l I øx CONFIDENCE IOTERVAL BOTTOR (^) VAWOE =L, =2, (^) IX 1...,^ TW)^ ;^ TOP (^) VAWNE =L2=22lx.,.",^ Xa^ ) D, LILL^ TRVE^ BARAMETER ARE SUCH^ THAT 005,00^ PLLTOIL^ )=1-2;^ WITh^ a^ low^ Ex L (^) ,^ ANA^ L 2 ARE^ RANBOM^ WANLABLE^. C^ 6762(.FL,IX,,,X); X,...*.) um (^) BPIVOT SAMPLE WAWES If (^) X (^) .... to minnd. N (^) (m, 8), 8'rnown - a X
• tonNlai) P-EROX-ME 141 x: SYOMETRY OF^ NORRAL^ DIST^. IF X f =0.05 =3 EE=-196-.FJ=-IE=1. X+ -1,96mo 1,56 (^8) n)=0,

IF (^) you don 't bnow 8 2 t=to (-a^ wtw ..^ }^ IIs ANOTHER PINOT)^ MONTECARCO (^) SIRULATION (^) : D) GENERATE (^) XI...XN =S FROM F

2. (^) COMPUTE SAMPE REAN (^) X (^) AM A 3) PEPEATE I^ ANB^ 2) MANY TIRES AND in (^) BETWUEEN (^) , pañaretric YOu^ GET^ THE^ DISTRIBUTION I BOOTSTRaP êlx )=

uêxisvlxtkêlxñ-êlXjp J --^ (M=10k)^ IF (^) JOW WANT THE QUANTILES (^) : TI (^41000) , Ta1"", 9 a^0 z5=T^125 os^1 öp^ LBTHE HIGHEST THE LowEST sampu samplethean REAn BOOTSTRAP (^) LUSER EVAWATE peecision ) FROM (^) EMPIPICAL (^) DISTRIBUTIONI BRAW n lin (^). i.d) (SAMPE WITH (^) SIZE M (^) ) La (^) WITH PEPLALEMENT (^) (THE SARE UNIT LAN BE TAREN MORETH^ AI^ ONCE)^ PAñAMETRIC BOOTSTRAP USES ESTIMATED PARARETER^ BUI assures menoln^ bistribUtIOn EX (^). IFI BNOW THAT (^) BATA CORES FROM AN (^) EXPONENTIAL WITH PANARETER (^) X -^ I USE BATA TO ESTIMATEY WITH X AND (^) I GENERATE RANDOM NOMBERS FROM AN EXPONENTIAL WITH PARARETER X

ba=E.XilogiTt (^) Zie.Iinxiblagint): Ixilogite In Zixi ) begtiu) ^^ =^0 u =ompLEYå-Mu.Eixa)." 7-ß

• AMxiltef)-zFtiLIkEnXILLP :FY. MMPEANFONENy THE (^) ALF (^) .OR? YAL

• Oxå Y EXPONENTIAL (^) CASE : "... (^) Y id Explo); (^) fixloedy, 0 no GorITo ,^ Co^
• layko): oety.^ I uyl IElayo +Z.-Oxå =hlogd-oExi; Od^ =o-p^ F=Ex. ô =[çŻXåJâisTHERAXIMUM^ LIEELLHODD^ ESTIMATOR ' NONMAL LASE (^) : X ..", YaNNLM. S ) iMEIR, S^ 'EIRT Lim,82, (^) : IT s C'?s^2 l he Ii^ 2: Fis^ loght) - mes? (-se =-nlaglns)^ E^ .. PSAMPLEMe |N ŞEmregu =óbm=TäñÅli-mÅtopFxûthûmÇÇxi BIASEDI SAMBLE VARIANCE bEgz =fgFüElxñ-mPs=oshyçzzñèlñmlosslççlxñmŞ SCORE FUNCTION : SOJ :^ JJQTD GRADIENT (^). LETUS (^) CONSIDER ELOGo) = (^) J 2 GofLxIdY E so/JE logh):S 5 o ILIJdY^ = JGo A 1 xDI logflx 1 dy^ Ijff .G roflx ...^ k^ laxanndy w !^ otIlayJf, I (tt'xjodx: Go JfLtdyo vite salel^ since EftGol (^) =o Je jo^ 2= Boclogfix^ folx?=Go(T ): fIxP footxl-(8oJ) i l

fool !^ flyt-tol^ fix , staytlyl zo ' uItGal^ ,-Sl'oP'tleAE=GI I 'tway Iff ?-Goilsflxleleley-ftdaH."M%. I

El $-lag (^1) ou^ J^ : ou)t o Flo, Fisher inforMatioN. ASSUMPTionS. RO: FIX)^ PLXEx)i^ X.... Xn. VOFO'5^ FOFFo' RI (^) : THE SOBPORT OF THE PANDOM VARIABLE IS SUCH (^) THAT SLD. IC.^ Sx:P.^ XNVLa^191 X 130 B^ )^ EX.^ FOR^ FORANIM,SY)^ -^ SSEIR.^ RIISVIDLATED^ .I R (^2) : (^) TRVE PAåAMETERSPACE (^) ) DINt (^) - R. ( (^) INTERIOR POINT OF (^) THE PADARET RO Bö HOLBS (^) ,RI,R 2 IF =3 lim n sLa (^) - sxoPlLo. ) =1. ConSider (^) blag L 112 Q=o^ (^ WELTOR^ OF^ EQWATLONS^ FD=D.,... (^) Ou )IF^ THE^ SOLUTION IS^ UNIQUE^ AND Rd, RI (^) ,R 2 HOLDS^ B^ =n^

• B Do R (^) 3: THE MATRIX (^) jfldo ANB R^ 4: FoJfIxIdx=Jr Gozfixsdx^ RS : I B 3 logL Iaqlere (^) II F DO - CTOOftC ELMI11)-0 (^) EFFICLENT IF (^) ASSWRPILONS PREULOUS How =3 Tn (^ ân-Do) 9 FoN') to (^) , A MLEIS (^) ASYMPTOTICALG {^ a^ 'B) UNBIASED RØ-R4, (^) X..., Xuid^ f^ CONSIER^ Y=^ IX-BELYIIKLDS^ =J^ VARLY^ )I^ MLOL^ PINFO^ , CONSIER AN Y^ AS UNBLASED ESTIMATOR-RELY)=D=S VAR/YI?^ YIUFO , RECAL (^) VARL TLON-DD (^) K =Fo,
• A (^) THAT IS (^) WHY MLE IS EFFICIENT-B IT TAKES THE LOWER (^) BOURR OF (^) VARLY ) CONSIJeR (^) FOlij ; FOEIREXA-s (^) I Fobaj :El-o)ito. MI)IS (^) ASTATISTICS p I 11 D} RAO (^) - CRAMER THEOMEM (^) : XO=ELY) = (^) SULY ) GIIIdY (^) ; KLO)=- go JaelXIf(IJdx ( RG) Srl) (^) Gof (I)dX.^ CONSIJER^ Xå^ iid^ .: ¥ x FYLII IT.FLX;)=J^ WHAT^ IS^ ITS^ DERIATIVE^ WITH^ PESPECT TO (^) O? Ex (^). Golfifilf!ifatfiti GolTfIxis =ElFtiAl f'Li)= I."f 1 x i^1 l.ftittixil } E SMLIT Z (^) AI Ix .,JTyLIJdY ITAKING (^) OUT THE IT AIX:, :)-f") O Smlxtz dento^ JEyLYIdXELY^ ELYJELS.Z)=^ TCOULY^ ,Z)-^ COVlY^ ,E) " qxeTarI.s.T ur (^) varlel = OXE TARIY ). T (^) TEDI E won hT- D 1 KIOL ELYR^ VARIUIF ,O, EVARLYJMFIOL (^) A FIO,^ FVARCY ): ÔN RNLOINILOT^ ) 5 Eo. Clôn - IaIT TFIO, FORQUTERRT TFIOS JEIN^ TO ESTIMATE LONFIDANCE (^) INTERVALS.

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  • Do?tR . now (^) WEWSETHE (^) SOWTIOn OF OPT (^).
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• oo^ }^ tô. (^005) ias) decaldo eiortcooo-o éioos!-ê"ioo?^ à racconloosos^ lan 1 ail Plâx? econ-elopteiool tiool e "loos+? e"10) (^) fetoo e "loosP+R to Ciôr
• tcoos-?(re"a,! EXERCISE - B LATENT (^) WARLABLE SOPPOSE THAT^ THE^ PERSONAL INGORE IS^ A^ R^ .V. XN (^) GANMALR ,B); FIXL-BTX*EBY, x^ 70 so X,B (^) 30, PL.)^ IS^ THE^ EWLER^ GAMMA^ FUNCTION^ : MIZ )^ =SotE-EZtdE^ ECX)=X/B; (^) VIXI=XIB? WEARE INTERESTED COW FINBING AN ESTIMATE OF (^) THE AV. INCORE WE HAVE A^ PROBLER : SAMPLE OF^320 PEOPLE (^) BUI IF (^) Y TS 2 O 0 E-BOUR SAMevalueisOlingore BElow (^8200) IS NOT (^) SUBJECT TO (^) TAXATION (^) ). WE LAN STIL (^) USE MILE THESAMPLE : 120000 How CAN WEGET (^) AN UNBIASED ESTIRATE? 2, Cr,incore) (^) 3, 12000 (1T^ PEMOWE^ O-D^ OUERESTIMATE^ AU^.^ INLoME { 4, o } (^4) I I KEED O (^) - D UNDERESTIMATE au. incore : :^ (3) (^) transfoem o in 8 z 00 - a (^) querestimate au (^). incore 320,1 (^5000) " TRANSFOåd O In 4100 - B? (^) ( TOo SIMPLE AND (^) GENERALLY U o^ lf^ Xi^ =^8 zao wrong (^) ) ià = { (^) Xi lF (^) Xi 3 82 a (^0) CATENT VADIABLE APPROACH ûzâlî IF^ WE^ FIND^ M^ .L.E.^ OF^ (2, B^1 - D^ WEFIND M (^) .L.E.^ Of^ Me. ÁÀ LixBszltîBîlrkllxîéBxå B szoo x^ La^ PROBEM (^) : WE ABE NOT ABWE TO LOMPOTE (^) THS LIKELiHoOD (^).^ UNHAT^ CAN^ WE Do^? FY=P^1 Y^1 y)= if Px?fzoo) or^178200 PLX 8 4) if^ y 38200 Ag l If (^) yoo Jfylysiag =/ 1 f 9 o (^8200) (^7) if ys 820 9=8200^ fyly^ ) if ¤..! 4 o " B še

fylslfFxlSzee )=PLXeszoe)^ if^ 4= fxly) if^ y^38200 fqlelLFxlszo^ 0)^ fylys^ /2-44920b)^ Fy 182001 4 ( y-03+ j413:ob/ (^) exugs/?-433= log LIBI^ :2.

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• khyrtob Ilog fxl^ frel?.? Si+^ R . I (^). fil lay fylbis^ Y tuis car be (^) naxinizes. LINEAR REGRESSION AND (^) ASSUMPTIONS, MAXIMUM LIKELIHODD ESTIMATOR PoP : 4 = Bo+B.X. +BIX2+...+BAXE+E. Y Is THE BEPERBENT (^) WARIABLE - BYEIR K (^) REGRESSORS (^) , ENNLO ,8); YINNL (^) TB ,S 3 HOW LAN WE FINBD AN ESTIMATOR FOR (^) B? LBEIRET); (^) X =T2, 4,, TIBITBOIB.... X2,-.,XE) BEJT. LIKEUHOOD EST (^). FOR O ITlog=IB, 80]. (^) Lo =IT.fs.Lo-Elog fyo - to Yiaid NLXiB , 80); (^) Tagflbi^ )=T e-L-s 2 -^1 eily
• xB) z (y-XBJ LO (^) CHANOE (^) OF WOTATIOO : FY - MLTET SUNLXR SFn )Ps fy =Tags^1 C - EJuly-YPIT13-YP^1 starT FroM^ JC.^ Jeldg=BzEu+EsuLŞ-xBTTYy-XB)= l
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• MR JeAB --YzgzE-zxLs-xBJJ=ğzxTLy-XBko THE OLS EST xTg =xTxB IS ALSO THE MAXIMUM B = IXTXJ'XTY UREUIHOOD EST^. minŞlBlŞŞixîBR (^) bError tslBlldB =oB=LXTXJLxTg), S=XTB+E SIXTB +A-RESIONALS