Axiomatic Definition of Probability. Problems: Relative Frequency. Event. Sample Space Examples
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1 Rado Sgals robabl & Rado Varables: Revew M. Sa Fadal roessor o lecrcal geerg Uvers o evada Reo Soe phscal sgals ose cao be epressed as a eplc aheacal orla. These sgals s be descrbed probablsc ers. ose s pcall waed ad o elae we eed o dersad qaavel: we eed probabl heor. Classcal Ive Deo o robabl sse all ocoes are eqall lkel ber o was eve ca occr oal ber o possble ocoes [ ] Relave Freqec Deo o robabl eror a epere es large Co he ber o es ha occrs [ ] 3 4
2 robles: Relave Freqec peres cao be perored a e ver large ber o es sspo ha he relave reqec approaches a cosa l pracce he rao hovers arod a cosa vale oac Deo o robabl ccep a se o aos based o eperece Derve a coplee heor based o aos Rado pere: epere wh odeesc ops. Saple Space: se o ocoes o rado epere s S eleear eves s e coabl e or e 5 6 Saple Space aples pere. Sgle draw ro 5-card deck 5 possble ocoes 5 elees S pere. Vecor Ocoe Throw o wo ar dce observe ber o each de possble ocoes 36 elees S each a -ple ve sse elees o saple space are dso. ve sbse o saple space sbec o cosras e.g. {S 34} S {S 34 5} ves eed o be dso ve Space : se o sbses o S called eves sch ha o erseco coplee o a eve s a eve called sga eld or sga algebra e o s o clded. 8
3 robabl Space Dscree roble: e or coabl e ocoes coplcaos or coos case wh coabl e ocoes. robabl Space: - Saple space - ve space 3- robabl easre os o robabl S U { } dso { } e or coabl e 9 Se Operaos robabl Theor Uo Ierseco Coplee c + aple: Throw o De Saple Space S {3456} 6 elees ve Se {S {} {6}{} {6}.} ower Se o S
4 4 3 Margal robabl Wre arg osde b arra or Codoal robabl robabl o eve gve ha eve has occrred.e. cera eve o deed or possble. Deo agrees wh relave reqec oo. oal ocoes wh oal ocoeswhboh ad 5 aes Theore... all eclsve eves 6 Idepedece Occrrece o oe eve does o aec he lkelhood o he oher
5 Mlple Rado Varables Mlvarae: vecor o rado varables varae: varables Dscree case: o prob. -d. arra Oba argal prob. b addg col. or row p Y Y robabl Dsrbo Fco Deo: The probabl dsrbo co DF o a rado varable s a co deed or each real ber as ollows F < < 7 8 roperes o he DF. F as. F as 3. F s a odecreasg co o. robabl Des Fco For a coos rado varable here ess a oegave co deed o he real le sch ha or ever sbse ab] o he real le he probabl ha akes a vale. The co s called he probabl des co pd a < b d b a 9 5
6 roperes o he pd.. 3. d. df d b a Uor Dsrbo a b elsewhere a b b a robabl o vale a sberval o [ab] s proporoal o s legh. rea o recagle s be. pecao o a Rado Varable peced vale or ea o. d Fco o a Rado Varable Oba he epeced vale o he co sg g g d 3 4 6
7 Moes k h oe Frs oe s he ea k k k k d Varace Secod oe abo he ea Var Var Var { { } } { } { } d 5 6 roperes o he Varace Var a + b a Var Var Var { } { } Var depede Φ oral or Gassa Des [ σ ] ep e d σ Ipora pd becase: Fs a phscal pheoea. Ceral l heore. Copleel descrbed b ea ad varace
8 Rgh Tal robabl robabl o eceedg a gve vale. Copleear clave dsrbo. Q e Φ d Iverse Q s ooocall decreasg ad hece verble. Iverse s pora sgal deeco. Q γ γ Q γ e F d F 9 3 Q rror Fco er erc Φ.5 Q.5 e e d [ + er ] erc erv d er MT slar or Maple >> er% rror co >> erc % Copleear error co >>.5*+ersqr % S. oral < >>.5*ercsqr % S. oral > >> sqr*erv-* % Iverse Q 3 3 8
9 9 33 Ch-Sqare Dsrbo Ceral eger! ~ d e e Y Γ < > Γ degrees o reedo 34 Ch-Sqare oceral degrees o reedo [ ] + Γ < > + λ λ λ λ 4 s cos ep ep ~ d r I I r r r Y 35 Rca Dsrbo θ θ α σ α σ α σ σ d I I + < > + + cos ep ep ~ 36 Mlvarae Dsrbos b vecor < < d F
10 37 Margal Dsrbos Margal pd + d d d d 38 Codoal Dsrbo For codoal des gve ha 39 Correlao ad Covarace Geeralzao o secod oe ad varace o vecor case. { } [ ][ ] { } T T R C C R 4 Mlvarae oral Geeralzao o oral dsrbo o learl depede rado varables. [ ] [ ] T C C de ep
11 Trasorao o Rado Varables g T [ ] [ ] T Wha s he pd o he vecor? How ca we oba ro he pd o? rocedre Solve or a I o solo ess he b I oe or ore solos es g de de [ ] [ ] 4 4 Trasorao o oral RV oral s los he rasorao s olear. oral s preserved he rasorao s lear. T ep C [ ] de C + b b T ~ + b C Reereces row & Hwag Sark & Woods Gra & Davsso M. H. De Groo M. J. Schervsh robabl & Sascs ddso-wesle oso. Ka Fdaeals o Sascal Sgal rocessg: Deeco Theor rece Hall R. D. Hppesel Deeco Theor CRC ress oca Rao
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