Variabili aleatorie una variabile aleatoria ( v.a.)

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1 Varabl alator ua varabl alatora ( v.a.) ua applcazo ch assoca u umro ral [0,] ad og rsultato dllo spazo dgl vt gral og sprmto alatoro carattrzzabl tramt ua varabl alatora dscrta o cotua Varabl alator dscrt: ua v.a. dscrta k rapprstata da ua tablla /o da ua formula matmatca, ch spcfca l valor umrco assuto dalla v.a. dscrta la probablta assocata ad og valor umrco assuto dalla v.a. Varabl alatora k assocata al laco d u dado Tablla d dfzo dlla v. a. k Dfzo matmatca dlla v. a. k = 6 pr k =,,...6 u modo grafco d rapprstar ua v.a. dscrta l stogramma u stogramma s grafcao succssv trvall ( bs ) l probablta ( l frquz rlatv ) Attzo: la varabl k ( l cotuo) o smplcmt u umro

2 attzo a o cofodr l coctto d a caso co l da d dstrbuzo uform s. : probablta dlla somma d rsultat l laco smultao d du dad l laco d du dad a caso, ma la somma d rsultat ottut o dstrbuta modo uform Valor mdo Varaza d ua v.a. dscrta pr carattrzzar ua v.a. modo sttco, ma cssaramt approssmatvo, s fa uso d dcator d ctralta d dsprso prcpal dcator soo - dc d ctralta dlla v.a. valor mdo ( < k > o µ ) - dc dlla dsprso dlla v.a. attoro al valor mdo dvazo stadard ( r.m.s. o σ ) σ = ( varaza) Valor mdo d ua v.a. dscrta pr v.a. dscrt k co domo d dfzo k [0, ] ( assumoo ua quatta fta d valor ) /o co k [0, ] ( possoo assumr ua fta umrabl d valor) k = µ < k > = = = k Pk ( ) dato ch pr l assoma d ormalzzazo = =

3 Varaza dvazo stadard d ua v.a. dscrta Varaza: ( k < k> ) Pk ( ) = σ = = ( k k ) Pk ( ) = < > Dvazo stadard: ( k < k> ) Pk ( ) = σ = = ( k k ) Pk ( ) = < > s la v.a. k o foss stsa a tutto l suo domo d dfzo,, ma foss lmtata ad assumr valor comprs tra k = k = < k > ' = = = k σ ' = ( k µ ) Pk ( ) = = tc.

4 alcu tra l prcpal dstrbuzo dscrt soo : v.a. Broullaa [, ] o bomal k [0, ] P ( ) Prob( ) k = k succss prov = q= p k pq k k! k = k!( k)!! = ( ) ( )... ad s. 4! = 4 ( 4 ) ( 4 ) ( 4 3) = 4 3 = 4 < k > = p var = pq v.a. d Posso (vt rar) P ( k) = Prob( k succss quado mda s hao ) = k [0, ] µ µ µ k µ k! < k > = µ var = µ

5 Varabl alator ( v.a.) cotu ua v.a. cotua X ch possa assumr valor umrc, co rapprstata da u fuzo cotua drvabl f() dfta modo ch f ( ) d = Prob{ ch la v. a. X assuma valor [, + d] } f() l grafco dlla f() puo ssr psato com u stogramma d baggo ftsmo f()d pr v.a. cotu X ch hao domo d dfzo [, ] d µ < > = f ( ) d f ( ) d f ( ) d dato ch pr l assoma d ormalzzazo f ( ) d = aalogamt σ ( µ ) ( ) = f ( ) d f d ( µ ) ( ) = f d σ ( µ ) ( ) = f ( ) d f d ( µ ) ( ) = f d s la v.a. X o foss stsa a tutto l suo domo d dfzo, ma foss lmtata ad assumr valor comprs tra X = X = ' µ = f ( ) d f ( ) d σ ' = ( µ ) ( ) f ( ) d f d σ ' = µ ( ) ( ) f ( ) d f d

6 alcu tra l prcpal v.a. cotu soo : v.a. uform ua v.a. uform ( dstrbuzo casual ) ll trvallo [a,b] assum u valor costat [a,b] f( ) = pr [ a, b] ( b a) ( b+ a) < > = var = ( b a) f( ) = 0 altrov ( rms...) = σ = b a f() ( b- a) 0 a b v.a. Gaussaa G(µ,σ ) f( ) πσ µ ( ) σ = pr [, ] G(µ,σ ) µ < > = var = σ ( rms...) = σ 0 µ

7 f( ) = πσ µ ( ) σ µ µ ( ) σ f( µ ) = = πσ πσ G(µ,σ ) πσ FWHM = larghzza a mta altzza FWHM = l σ.36 σ πσ FWHM 0 µ

8 Gaussaa Stadard o Normal s µ = 0 σ = N(0,) N(0,) f( ) = π pr [, ] pr ua gaussaa s ha ch l 68% dlla probablta ( dll ara sotto la curva ) comprsa tra µ σ µ + σ N(0,) f()d = 0.68 tra + pr ua Normal(0,) l 95% dlla probablta ( dll ara sotto la curva) comprsa tra µ σ µ + σ N(0,) f()d = 0.95 tra + pr ua Normal(0,) N(0,) l 99.7% dlla probablta ( dll ara sotto la curva) comprsa tra µ 3σ µ + 3σ f()d = tra 3 +3 pr ua Normal(0,)

9 Fuzo dgl rror la fuzo dfta com l ara da - ad u grco puto z d ua gaussaa stadard, dtta fuzo dgl rror ( rror fucto gls ) ) z rf ( z) = d π altr mportat dsta d probablta soo la Ch Quadrato la t d Studt mportaza dlla gaussaa : torma dl lmt ctral

10 Backup slds