Aspectos Generales de Los Elementos de Probabilidad

 

[edvkrsdhfh [edv krsdhfh ^icdtäjedjf ^icdtäjedjf Gksifgkrdjfef.

Demkedkrçf ^ktrickrf ^riofodcdhfh y kstfhçstdjf Fspkjtis mkekrfcks hk cis kckgketis hk priofodcdhfh Fckx Frgfehi Gkehkz ^krkz ckhzkpkimcdeNmgfdc.jig 68 hk @uedi hk ?>?6

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Jietkedhi 6. ^riof ^riofodcd odcdhfh hfh

 

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6.6. Kxpkrdgket Kxpkrdgketi. i.   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.?. Kspfjdi Kspfjdi gukstr gukstrfc. fc.   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.:. Kvketi Kvketis. s.   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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6.8. Fcmuefs Fcmuefs rkcfjdieks rkcfjdieks hk cfs tkirçfs tkirçfs hk jie`ueti. jie`ueti.   . . . . . . . . . . . . . . . . . . . . . .

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?. Fxdi Fxdigfs. gfs.

 

:. Xäjed Xäjedjfs jfs hk jiet jietki. ki.

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:.6. Cf rkmcf hkc prihujti prihujti pfrf pfrf pfrks irhkefhis. irhkefhis.   . . . . . . . . . . . . . . . . . . . . . . . :.?. _kmcf hk hk prihujti prihujti gæs mkekrfc mkekrfc   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . :.:. ^krgutfjdieks ^krgutfjdieks y jigodefjdieks. jigodefjdieks.  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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^rio ^riofo fodc dcdh dhfh fh

Kc tärgdei priofodcdhfh sk rkstkrk fc kstuhdi hk fzfr y cf dejkrtdhugork ke jufcqudkr sdtufjdúe ke cf jufc vfrdis pisdocks sujksis pukhke ijurrdr< cf hdsjdpcdef hk cf priofodcdhfh pripirjdief gätihis hk jufetdstjfr cfs ipirtuedhfhks y priofodcdhfhks fsijdfhfs jie vfrdis sujksis. Kc kstuhdi hk cf priofodcdhfh jigi uef rfgf hk cfs gftkgætdjfs sk rkgietf f gæs hk :>> föis, jufehi efjk ke jiekxdúe jie prkmuetfs dgpcdjfofe hk ks fzfr. Gujbis sk bfe jupfhi kxjcusdvf gketk hk cf priofodcdhfh, pkriquk kc io`ktdvi ke `ukmis kstk jfsi juordr súci cdoris cf pfrtk hk cfiijupfhi gftkrdfkxjcusdvfgketk quk tdkek gæs fpcdjfjdúe hdrkjtf ke priockgfs hk deakrkejdf kstfhçstdjf. 6.6.. 6.6

Kxpkrdg Kxpkrdgke keti. ti.

[e kxpkrdgketi ks jufcqudkr fjjdúe i prijksi juyi rksuctfhi kstæ su`kti f cf dejkrtdhugork. Fuequk cf pfcforf kxpkrdgketi ke mkekrfc sumdkrk uef sdtufjdúe hk prukof judhfhisfgketk jietricfhf ke ue cfoirftirdi, sk ck utdcdzf fquç ke ue sketdhi gujbi gæs fgpcdi. ^ir ci tfeti, kxpkrdgketis quk pukhke skr hk detkräs dejcuyke cfezfr fc fdrk uef giekhf uef vkz i vfrdfs vkjks, skckjjdiefr uef jfrtf i jfrtfs hk ue gfzi, pksfr uef bimfzf hk pfe, kc tdkgpi hk rkjirrdhi hk cf jfsf fc trfof`i ke uef gföfef pfrtdjucfr, iotkekr tdpis hk sfemrk hk ue mrupi hk dehdvdhuis i gkhdr cfs rksdstkejdfs f cf jigprksdúe hk hdakrketks vdmfs hk fjkri. 6.?.. 6.?

Kspfjd Kspfjdi i gukst gukstrfc rfc..

Kc kspfjdi gukstrfc hk ue kxpkrdgketi hkeitfhi pir   θ   , ks kc jie`ueti hk tihis cis pisdocks rksuctfhis hk hdjbi kxpkrdgketi. 6. 6.:. :.

Kvke Kvketi tis. s.

Ke kc kstuhdi hk cf priofodcdhfh, detkrksfe ei súci cis rksuctfhis dehdvdhufcks hk  θ  sdei tfgodäe vfrdfs rkjipdcfjdieks hk rksuctfhis hk  θ . [e kvketi ks jufcqudkr rkjipdcfjdúe (suojie`ueti) hk rksuctfhis jietkedhis ke kc kspfjdi gukstrfc θ . [e kvketi ks sdgpck sd jiesdstk ke kxfjtfgketk ue rksuctfhi y jigpuksti sd jiesdstk ke gæs hk ue rksuctfhi. Jufehi sk rkfcdzf ue kxpkrdgketi, sk hdjk quk ijurrk ue kvketi pfrtdjucfr F sd kc rksuctfhi kxpkrdgketfc iotkedhi kstæ jietkedhi ke F. Ke mkekrfc, ijurrdræ kxfjtfgketk ue kvketi sdgpck, pkri gujbis kvketis jigpukstis ijurrdræe fc gdsgi tdkgpi. 6.8.. 6.8

Fcmuef Fcmuefss rkcfjdi rkcfjdieks eks hk cfs tkirç tkirçfs fs hk jie`ue jie`ueti. ti.

[e kvketi ks sdgpckgketk ue jie`ueti, fsç quk cfs rkcfjdieks y rksuctfhis hk cf tkirçf kckgketfc hk jie`uetis pukhke skr utdcdzfhis pfrf kstuhdfr kvketis. Zk utdcdzfræe cfs sdmudketks ipkrfjdieks pfrf jrkfr kvketis eukvis f pfrtdr hk kvketis hfhis. Kc jigpckgketi hk ue kvketi  Z , hkeitfhi pir  F , ks kc jie`ueti hk tihis cis rksuctfhis ke  θ  quk ei kstæe jietkedhis ke  F . 

Cf uedúe hk his kvketis F y O, hkeitfhis pir   F   ∤  O   y ckçhis –F i O‗, ks kc kvketi quk jiesdstk ke tihis cis rksuctfhis quk kstæe ke   F   i ke   O   i ke fgois kvketis (hk tfc sukrtk :

 

quk cf uedúe dejcuyf rksuctfhis hiehk tfeti  F  jigi  O  ijurrke, fsç tfgodäe rksuctfhis hiehk ijurrk kxfjtfgket kxfjtfgketkk uei), ks hkjdr, hkjdr, tihis cis rksuctfhis rksuctfhis ke pir ci gkeis uei hk cis kvketis. kvketis. Cf detkrskjjdúe hk his kvketis   F  y   O , hkeitfhf pir   F ∠ O  y ckçhf –F y O‗, ks kc kvketi quk jiesdstk ke tihis cis rksuctfhis quk kstæe tfeti ke  F  jigi ke  O . Vuk   ∋  hkeitk kc kvketi euci (kc kvketi sde rksuctfhis). Jufehi  F ∠ O  7 ∋  , sk hdjk quk  F  y  O sie kvketis gutufgketk kxjcuyketks i hds`uetis. ?.

Fxdigfs.

Hfhis ue kxpkrdgketi y ue kspfjdi gukstrfc  θ , kc io`ktdvi hk cf priofodcdhfh ks fsdmefr f jfhf kvketi  F  ue eðgkri  ^ (F), ccfgfhi cf priofodcdhfh hkc kvketi  F , kc jufc hfræ uef gkhdhf prkjdsf hk cf ipirtuedhfh hk quk   F  ijurrf. ^frf mfrfetdzfr quk cfs fsdmefjdieks skræe jiesdstketks jie cfs eijdieks detudtdvfs hk cf priofodcdhfh, tihfs cfs fsdmefjdieks hkokræe sftdsafjkr cis sdmudketks fxdigfs (pripdkhfhks oæsdjfs) hk priofodcdhfh. Fxdigf 6 Fxdigf ? Fxdi Fx digf gf :

^frf jufcqudkr kvketi  F ,  ^ (F)  5  >.   ^ (θ ) 7 6 Zd  F 6, F?, F:,...  ks ue jie`ueti hk kvketis gutufgketk kxjcuyketks,  ∕ ^ (F) ^ (F6   ∠ F?  ∠ F:   ∠ ...) 7 76

ketiejks.

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Kc fxdigf 6 rk﬏k`f cf eijdúe detudtdvf hk quk cf priofodcdhfh hk quk ijurrf F hkokræ skr ei ekmftdvf. Kc kspfjdi gukstrfc ks pir hkstedjdúe kc kvketi quk hkok ijurrdr jufehi sk rkfcdzf kc kxpkrdgketi (θ  jietdkek tihis cis pisdocks rksuctfhis), fsç sk hdjk kc fxdigf ? quk ks cf gæxdgf priofodcdhfh pisdock hk 6 kstæ fsdmefhf f   θ . Kc tkrjkr fxdigf airgfcdzf cf dhkf quk sd sk hkskf cf priofodcdhfh hk quk ijurrf fc gkeis uei hk vfrdis kvketis, y ei ijurrfe his fc gdsgi tdkgpi, ketiejks cf priofodcdhfh hk quk pir ci gkeis uei ijurrf ks cf sugf hk cfs priofodcdhfhks hk cis kvketis kvket is dehdvdhufcks. ^rkpisdjdúe. ^ (∋) 7 >  hiehk   ∋  ks

kc kvketi euci (kc kvketi quk ei jietdkek rksuctfhis ke fosicuti). Ksti f su vkz dgpcdjf quk cf pripdkhfh jietkedhf ke kc fxdigf : ks væcdhf pfrf ue jie`ueti stedti hk kvketis. kvketis. ^rkpisdjdúe.

^frf jufcqudkr kvketi  F ,  ^ (F) + ^ (F ) 7 6, f pfrtdr hk cf jufc  ^ (F) 7 6˒^ (F ). 

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Jigpriofjdúe.

Ke kc fxdigf :, skf  l  7 ?,  F 6 7  F  y  F ? 7  F . Jigi pir hkstedjdúe hk  F ,  F ∠ F 7  θ  ke tfeti F  y  F skfe kvketis hds`uetis,  6 7  ^ (θ ) 7  ^ (F ∠ F ) 7  ^ (F) + ^ (F ). Kstf pripisdjdúe ks sirprkehketkgketk ðtdc pirquk sk prksketfe gujbfs sdtufjdieks ke cfs jufcks ^(F—) ks gæs aæjdc hk iotkekr iotkekr gkhdfetk gkhdfetk gätihis hdrkjtis hdrkjtis quk ^(F). 

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Xäje Xäjedj djfs fs hk jie jietki. tki.

Jufehi cis hdvkrsis rksuctfhis hk ue kxpkrdgketi sie dmufcgketk priofocks (cf gdsgf priofodcdhfh ks fsdmefhf f jfhf kvketi sdgpck), cf tfrkf hk jfcjucfr priofodcdhfhks sk rkhujk f jietfr. Zkf E kc eðgkri hk rksuctfhis ke ue kspfjdi gukstrfc y E(F) kc eðgkri hk rksuctfhis jietkedhis ke ue kvketi F. ^ (F) 7

  E (F) E 

Zd uef cdstf hk rksuctfhis ks aæjdc hk iotkekr y E ks pkquköi, ketiejks E y E(F) pukhke skr hktkrgdefhfs sde utdcdzfr edemðe prdejdpdi hk jietki. Kxdstke, sde kgofrmi, gujbis kxpkrdgketis ke cis jufcks kc ksaukrzi dgpcdjfhi fc kcfoirfr cf cdstf ks pribdodtdvi pirquk E ks ofstfetk mrfehk. Kstfs rkmcfs tfgodäe sie ðtdcks ke gujbis priockgfs quk dgpcdjfe rksuctfhis quk ei sie dmufcgketk priofocks. Zk utdcdzfræe vfrdfs hk cfs rkmcfs hksfrriccfhfs fquç fc kstuhdfr hdstrdoujdieks hk priofodcdhfh ke kc sdmudketk jfpçtuci. :.6.. :.6

Cf rkmcf rkmcf hkc hkc prihujt prihujti i pfrf pfr pfrks ks irhkef irhkefhis his..

Cf prdgkrf rkmcf hk jietki sk fpcdjf f jufcqudkr sdtufjdúe ke cf jufc ue jie`ueti (kvketi) sk jigpiek hk pfrks hk io`ktis irhkefhis y sk hkskf jietfr kc eðgkri hk pfrks. ^ir pfr irhkefhi, sk qudkrk hkjdr quk, sd  I 6   y  I ?  sie io`ktis, ketiejks kc pfr (I6 ,  I ? ) ks hdakrketk hkc pfr (I? ,  I 6 ). ^ir k`kgpci, sd ue dehdvdhui skckjjdief uef cçekf färkf pfrf ue vdf`k hk Cis Æemkcks f Jbdjfmi y (hkspuäs hk rkfcdzfr trfesfjjdieks hk ekmijdis ke Jbdjfmi) ue skmuehi pfrf jietdeufr f Eukvf Tirl, uef pisdodcdhfh ks (Fgkrdjfe, [edtkh), itrf ks ([edtkh, Fgkrdjfe) y itrf gæs ks ([edtkh, [edtkh). Zd kc prdgkr kckgketi u io`kti hk ue pfr irhkefhi pukhk skr skckjjdiefhi hk  e 6  gfekrfs y pir jfhf uef hk kstfs   e6  gfekrfs kc skmuehi kckgketi hkc pfr pukhk skr skckjjdiefhi hk   e?  gfekrfs, ketiejks kc eðgkri hk pfrks ks  e 6 e? . :.?.. :.?

_kmcf _kmcf hk hk prihujt prihujti i gæs gæs mkekr mkekrfc fc

Zd sk cfezf fc fdrk ue hfhi hk skds cfhis jdeji vkjks ke sujksdúe ke cumfr hk súci his vkjks, ketiejks jfhf pisdock rksuctfhi ks ue jie`ueti irhkefhi hk jdeji eðgkris tfc jigi (6, :, 6, ?, 8) i (1, ;, ?, ?, ?). [e jie`ueti irhkefhi hk  l  io`ktis rkjdodræ kc eigork hk  l-tupcf  (pir   (pir tfeti ue pfr ks ue ?-tupcf y ue trdpck ks ue :-tupcf). Jfhf rksuctfhi hkc kxpkrdgketi hkc cfezfgdketi fc fdrk hk kc hfhi ks ketiejks ue ;-tupcf. _kmcf hk prihujti pfrf l-tupcfs Zupúemfsk quk ue jie`ueti sk jigpiek hk jie`uetis irhkefhis hk l kckgketis (l-tupcfs) y quk kxdstke  e!  pisdocks ipjdieks pfrf kc prdgkr kckgketi pir jfhf ipjdúe hkc prdgkr kckgketi, kxdstke  e?  pisdocks ipjdieks hkc skmuehi kckgketi< . . . < pir jfhf pisdock ipjdúe hk cis prdgkris  l ∟ 6 kckgketis, kxdstke  el  ipjdieks hkc kckgketi l ∟ sdgi. Kxdstke ketiejks e 6 e? el pisdocks  l ∟ tupcfs. Kstf rkmcf gæs mkekrfc tfgodäe pukhk skr dcustrfhf pir ue hdfmrfgf hk æroic< sdgpckgketk sk jiestruyk ue hdfmrfgf gæs kcfoirfhi föfhdkehi uef tkrjkrf mkekrfjdúe hk rfgfs quk kgfefe hk cf puetf hk jfhf skmuehf mkekrfjdúe, cukmi rfgfs hk jufrtf mkekrfjdúe, y fsç sujksdvfgketk, bfstf quk pir ðctdgi sk fmrkmfe rfgfs hk l-äsdgf mkekrfjdúe.

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^krgu ^krgutfjd tfjdieks ieks y jigodef jigodefjdiek jdieks. s.

Jiesdhärksk ue mrupi hk e dehdvdhuis u io`ktis hdstdetis (–hdstdetis‗ sdmedstjf quk kxdstk fcmuef jfrfjtkrçstdjf quk hdakrkejdf f jufcqudkr dehdvdhui u io`kti hk jufcqudkr itri). ¹Juæetfs gfekrfs kxdstke hk skckjjdiefr ue suojie`ueti hk tfgföi l hkc mrupi9 ^ir k`kgpci, sd ue kqudpi hk cdmfs gkeirks tdkek 6; `umfhirks rkmdstrfhis, ¹juæetfs gfekrfs kxdstke hk skckjjdiefr 4 `umfhirks pfrf uef fcdekfjdúe dedjdfc9 I sdvfjfjdieks ke su cdorkri tdkek 6> cdoris hkgfekrfs gdstkrdikxdstke ei ckçhis hkskf skckjjdiefr : pfrf cckvfrcis jiesdmi ke uefs jirtfs, ¹juæetfs hkybfjkrci9 [ef rkspukstf f cf prkmuetf mkekrfc quk sk fjfof hk pcfetkfr rkqudkrk hdstdemudr ketrk his jfsis. Ke fcmuefs sdtufjdieks, tfc jigi kc ksjkefrdi hkc oädsoic, kc irhke hk cf skckjjdúe ks dgpirtfetk. ^ir k`kgpci, jie Æemkcf jigi cfezfhir y Oke jigi rkjkptir sk iotdkek uef fcdekfjdúe hdakrketk hk fqukccf jie Æemkcf jigi rkjkptir y Oke jigi cfezfhir. F gkeuhi, sde kgofrmi, kc irhke ei ks dgpirtfetk y f efhdk ck detkrksf quä dehdvdhuis u io`ktis skfe skckjjdiefhis, jigi skrçf kc jfsi ke kc ksjkefrdi hk skckjjdúe hk cdoris. [e suojie`ueti irhkefhi sk ccfgf pkrgutfjdúe. Kc eðgkri hk pkrgutfjdieks hk tfgföi  l  quk sk pukhk airgfr jie cis  e  dehdvdhuis u io`ktis ke ue mrupi skræ hkeitfhi pir  ^ ll,e ,e . [e suojie`ueti ei irhkefhi sk ccfgf jigodefjdúe. [ef airgf hk hkeitfr kc eðgkri hk jigodefjdieks ks  J ll,e ,e , pkri e ke su cumfr sk utdcdzfræ uef eitfjdúe quk ks ofstfetk jigðe ke cdoris hk priofodcdhfh0 l , quk sk ckk –hk e sk kcdmke l‗. Kc eðgkri hk pkrgutfjdieks sk hktkrgdef utdcdzfehi cf prdgkrf rkmcf hk jietki pfrf l-tupcfs. Zupúemfsk, pir k`kgpci, quk ue jickmdi hk demkedkrçf tdkek sdktk hkpfrtfgketis, hkeitfhis pir f, o, h, k,sdktk a y m.rkprksketfetks, Jfhf hkpfrtfgketi tdkekquk ue skr rkprksketfetk ke prksdhketk, kc jiesk`i hkitri kstuhdfetks hkc jickmdi. Hkj,kstis uei tdkek kckmdhi jigi jigi vdjkprksdhketk y ue tkrjkri jigi skjrktfrdi. ¹Juæetfs gfekrfs kxdstke pfrf skckjjdiefr cis trks istjdfcks9 Ks hkjdr, ¹juæetfs pkrgutfjdieks hk tfgföi : pukhke skr airgfhfs jie cis = rkprksketfetks9 ^frf rkspiehkr kstf prkmuetf, bforæ quk pkesfr ke airgfr uef trdpcktf (:-tupcf) ke cf jufc kc prdgkr kckgketi ks kc prksdhketk, kc skmuehi ks kc vdjkprksdhketk y kc tkrjkri ks kc skjrktfrdi. [ef trdpcktf ks (f, m, o), itrf ks (o, m, f) y itrf gæs ks (h, a, o). Fbirf odke kc prksdhketk pukhk skr skckjjdiefhi ke jufcksqudkrf hk  e 6  7 = airgfs. ^ir jfhf airgf hk skckjjdiefr kc prksdhketk, kxdstke  e ?  7 1  airgfs hk skckjjdiefr kc vdjkprksdhketk y pir jiesdmudketk  = x1 7 8?  (pfrks hk prksdhketk, vdjkprksdhketk). ^ir ðctdgi, pir jfhf airgf hk skckjjdiefr ue prksdhketk y vdjkprksdhketk, kxdstke   e:   7 ;  airgfs hk skckjjdiefr kc skjrktfrdi. Ksti hf ^ :,=  7 (=)(1)(;)

jigi kc eðgkri hk pkrgutfjdieks hk tfgföi : quk sk pukhke airgfr jie = dehdvdhuis hdstdetis. [ef rkprksketfjdúe hk hdfmrfgf hk æroic gistrfrçf trks mkekrfjdieks hk rfgfs. Cf kxprksdúe pfrf :,= ^   pukhk skr rksjrdtf jie cf fyuhf eitfjdúehksjkehketk afjtirdfc. _kjuärhksk =‗) ks uef eitfjdúe jigpfjtf pfrf kchkprihujti hk ketkris

quk =! (sk ckk –afjtirdfcGæs hk (=)(1)(;)(8)(:)(?)(6). mkekrfcgketk, pfrf jufcqudkr ketkri pisdtdvi   g, g! 7   g(g  ∟  6)(g  ∟  ?)(?)(6). Ksti hf  6! 7 6   y tfgodäe sk hkstek  >! 7 6. Ketiejks ^ :,=  7

 (=)(1)(;)(8!)   =!   7 (8!) 8!

Mkekrfcgketk0 ^ ll,e ,e  7  e (e ∟ 6)(e ∟ ?)....(e ∟ (l ∟ ?))(e ∟ (l ∟ 6))

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Guctdpcdjfehi y hdvdhdkehi pir   (e ∟ l)!  sk iotdkek uef kxprksdúe jigpfjtf pfrf kc eðgkri hk pkrgutfjdieks. ^ ll,e ,e  7

  e! (e ∟ l )!

=