Ukusatshalaliswa kwamathuba ahlukahlukene amamodeli ezibalo achaza ukwenzeka kwezehlakalo ezinamavelu ahlukene, anomkhawulo. Lokhu kusabalalisa kubonakala ngezakhiwo zazo, njengesamba samathuba ayo yonke imiphumela engenzeka elingana no-1 kanye nokuba khona kwepharamitha enquma umumo wokusabalalisa. Kulesi sihloko, sizohlola izici zokusatshalaliswa kwamathuba ahlukene okuvame kakhulu, njengokusatshalaliswa kwe-Bernoulli, ukusatshalaliswa kwe-binomial, ukusatshalaliswa kwe-Poisson, nokusabalalisa kwejometri, kanye nokwethula izivivinyo ezithile ezisebenzayo ukuze uqonde kangcono le mibono.
Ukuqonda umqondo wokusatshalaliswa kwamathuba ahlukene: incazelo elula necacile.
Ukuze uqonde umqondo wokusatshalaliswa kwamathuba ahlukene, kubalulekile ukuqonda ukuthi umsebenzi wezibalo ohlotshaniswa namathuba nomphumela ngamunye ongaba khona wokuhlolwa okungahleliwe. Ngamanye amazwi, ukusatshalaliswa kwamathuba ahlukene kusivumela ukuthi sinqume ithuba lomphumela ngamunye owenzeka kusethi yamathuba anomkhawulo noma angabalwa.
Ukusabalalisa kwamathuba ahlukahlukene kubonakala ngomsebenzi wawo wamathuba, okunikeza umphumela ngamunye inani elingelona inegethivu, nesamba sawo wonke amathuba alingana no-1. Ngaphezu kwalokho, imiphumela engaba khona ihlukile futhi ihlukanisiwe, ngaphandle kokuthi kungenzeka kube khona amanani amaphakathi.
Isibonelo sakudala sokusatshalaliswa kwamathuba ahlukene ukusatshalaliswa kwe-Poisson, okusetshenziswa kabanzi ezinqubweni zokubala, njengenani lezehlakalo ezenzeka esikhathini esithile esinikeziwe. Esinye isibonelo esivamile ukusatshalaliswa kwe-binomial, okuyisibonelo sokuhlola okunemiphumela emibili kuphela engaba khona, njengempumelelo noma ukwehluleka.
Ukuze usebenzise ithiyori yokusatshalaliswa kwamathuba ahlukene, kuyadingeka ukuqonda izici nezici zazo ezithile, futhi ukwazi ukubala okungenzeka futhi utolike imiphumela. Izivivinyo ezingokoqobo zibalulekile ukuze ujulise ukuqonda nokuthuthukisa amakhono kulo mkhakha wamathuba.
Funda mayelana nokusatshalaliswa okuhlukene okuyinhloko okusetshenziswa kuzibalo namathuba.
Funda mayelana nokusatshalaliswa okuhlukene okuyinhloko okusetshenziswa kuzibalo namathuba. Ukusatshalaliswa kwamathuba ahlukahlukene kungamathuluzi abalulekile ekuhlaziyeni kwezibalo, okunika amandla ukumodela nokubikezela izehlakalo ezingahleliwe. Phakathi kokusabalalisa okubalulekile okuyinhloko ukusatshalaliswa kwe-Bernoulli, ukusatshalaliswa kwe-binomial, ukusatshalaliswa kwejometri, ukusatshalaliswa kwe-Poisson, nokusabalalisa kwe-hypergeometric.
A Ukusatshalaliswa kweBernoulli isetshenziselwa ukwenza imodeli yokuhlola enemiphumela emibili kuphela engenzeka, njengempumelelo nokwehluleka. ukusatshalaliswa kwe-binomial Isetshenziswa ezimeni lapho kunenombolo egxilile yokuhlola okuzimele, okunemiphumela emibili kuphela engaba khona esivivinyweni ngasinye, njengokuphumelela nokwehluleka.
A ukusatshalaliswa kwejiyomethri isetshenziselwa ukwenza imodeli inombolo yokuhlola kuze kube yimpumelelo yokuqala ngokulandelana kokuhlolwa okuzimele. Ukusatshalaliswa kwe-Poisson isetshenziselwa ukwenza imodeli yezehlakalo ezingavamile esikhathini esithile noma isikhawu somkhathi.
Ekugcineni, i ukusatshalaliswa kwe-hypergeometric Isetshenziselwa ukwenza imodeli yokuhlolwa lapho kukhona okukhethiwe ngaphandle kokushintshwa kwama-elementi asuka kubantu abanomkhawulo, okunentshisekelo enanini lempumelelo kusampula ethile.
Ukuze uqonde kangcono lokhu kusatshalaliswa okuhlukahlukene kanye nendlela yokukusebenzisa, kubalulekile ukuzijwayeza ngokuzivocavoca. Ukuxazulula izinkinga ezibandakanya lokhu kusabalalisa kungasiza ukuqinisa ulwazi futhi kucije amakhono ezibalo namathuba.
Ngakho-ke, lapho ufunda izibalo namathuba, kubalulekile ukwazi izici nokusetshenziswa kokusatshalaliswa okuhlukene okuyinhloko, njengokusatshalaliswa kwe-Bernoulli, ukusatshalaliswa kwe-binomial, ukusatshalaliswa kwejiyomethrikhi, ukusatshalaliswa kwe-Poisson, nokusabalalisa kwe-hypergeometric.
Izinhlobo zokusatshalaliswa kwamathuba: funda mayelana nezinhlobo ezahlukene zokusatshalaliswa kwezibalo.
Amathuba okusabalalisa amamodeli ezibalo achaza ukuziphatha okungahleliwe kwento ethile. Kunezinhlobo ezahlukene zokusabalalisa okungenzeka, ngayinye enezici zayo kanye nezinhlelo zokusebenza. Kulesi sihloko, sizogxila ekusabalazweni kwamathuba ahlukene, ahlotshaniswa neziguquguqukayo ezihlukene—lezo ezingathatha amanani athile, angabalwa.
Okunye okuvamile kokusabalalisa kwamathuba ahlukene kufaka phakathi ukusatshalaliswa okufanayo, ukusatshalaliswa kwe-binomial, ukusatshalaliswa kwe-Poisson, nokusatshalaliswa kwejiyomethri. Ngayinye yalokhu kusatshalaliswa inezinto zayo futhi isetshenziswa ezimweni zezibalo ezehlukene.
Ukusatshalaliswa okufanayo, ngokwesibonelo, kubonakala ngokunikeza amathuba afanayo kuwo wonke amanani angenzeka wokuhluka okuhlukile. Ukusabalalisa kwe-binomial kusetshenziselwa ukwenza imodeli yenani lempumelelo ngokulandelana kokuhlolwa okuzimele, ngakunye okunethuba elifanayo lokuphumelela. Ukusatshalaliswa kwe-Poisson, nakho, kusetshenziselwa ukumodela inombolo yezehlakalo ezingavamile ngesikhathi noma isikhawu sendawo. Futhi ukusatshalaliswa kwejiyomethri kusetshenziselwa ukwenza imodeli yenani lezivivinyo ezidingekayo kuze kube yimpumelelo yokuqala ngokulandelana kokuhlola okuzimele.
Ukuze uqonde kangcono ukuthi lokhu kusabalalisa kusebenza kanjani, kubalulekile ukuzilolonga ngokuzivocavoca. Isibonelo, singabala amathuba okuthola amakhanda angu-3 ncamashi kumathoni angu-5 ohlamvu lwemali olufanele sisebenzisa ukusabalalisa okubili. Noma singanquma amathuba okungenani emicimbi emi-2 eyenzeka ngesikhathi esithile sisebenzisa ukusabalalisa kwe-Poisson.
Ngokuqonda izici nokusetshenziswa kwalokhu kusatshalaliswa, izibalo kanye nochwepheshe besayensi abahlobene bangenza izinqumo ezinolwazi kakhulu nezinembe ngokusekelwe kudatha engenzeka.
Yikuphi okuguquguqukayo okubhekwa njengokungacacile ngamathuba?
Ngokunokwenzeka, okuguquguqukayo okuhlukile yilawo angathatha inani elilinganiselwe noma elibalekayo lamanani. Lokhu kusho ukuthi okuguquguqukayo okuhlukene yilawo angabalwa, ngokuvamile amelelwa izinombolo. Isibonelo, inani lezimoto endaweni yokupaka, inani labafundi ekilasini, kanye nenani lobuso efashini konke kuyizibonelo zokuhlukahluka okuhlukahlukene.
Lezi ziguquko zihlukile kokuguquguqukayo okuqhubekayo, okungathatha inani elingapheli lamanani phakathi kwebanga elithile. Nakuba okuguquguqukayo okuhlukahlukene kunamanani athile, ahlukene, okuguquguqukayo okuqhubekayo kungase kuthathe noma yiliphi inani ngaphakathi kobubanzi obuqhubekayo. Ngokwesibonelo, ubude bomuntu, isikhathi esisithathayo ukuze uqedele umsebenzi othile, nezinga lokushisa legumbi kuyizibonelo zokuguquguquka okuqhubekayo.
Ngakho-ke, okuguquguqukayo okuhlukene okungenzeka yilawo angabalwa futhi athathe amanani athile, ahlukene, ngokuphambene namaguquguquki aqhubekayo angathatha noma yiliphi inani ngaphakathi kobubanzi.
Ukusatshalaliswa Kwamathuba Ahlukene: Izimpawu, Ukuzivocavoca
As ukusabalalisa kwamathuba ahlukene ziwumsebenzi ohlotshaniswa nento ngayinye ethi X(S) = {x1, x2, …, xi, …}, lapho u-X ewukuhlukahluka okungahleliwe okunikezwayo futhi u-S eyisikhala sesampula, amathuba okuthi lo mcimbi wenzeke. Lo msebenzi f of X(S) ochazwa ngokuthi f(xi) = P(X = xi) ngezinye izikhathi ubizwa ngokuthi umsebenzi wamathuba amaningi.
Le nqwaba yamathuba imvamisa imelelwa ngendlela yethebula. Njengoba u-X ewuhlobo oluhlukile olungahleliwe, u-X(S) unenani elilinganiselwe noma elingapheli lemicimbi. Phakathi kokusatshalaliswa kwamathuba ahlukene okuvame kakhulu ukusatshalaliswa okufanayo, ukusatshalaliswa kwe-binomial, kanye nokusatshalaliswa kwePoisson.
Izici
Umsebenzi wokusabalalisa amathuba kufanele uhlangabezane nezimo ezilandelayo:
Ngaphezu kwalokho, uma u-X ethatha kuphela inani elilinganiselwe lamanani (isb., x1, x2, ..., xn), bese u-p(xi) = 0 uma i > n futhi, ngakho-ke, uchungechunge olungapheli lwemibandela b luba uchungechunge olunomkhawulo.
Lo msebenzi futhi wenelisa izici ezilandelayo:
U-B makabe umcimbi ohlotshaniswa nokuguquguquka okungahleliwe X. Lokhu kusho ukuthi u-B uqukethwe ku-X(S). Ngokuqondile, ake sithi B = {xi1, xi2,…}. Ngakho-ke:
Ngamanye amazwi: amathuba omcimbi B alingana nesamba samathuba emiphumela ngayinye ehlotshaniswa no-B.
Kulokhu singaphetha ngokuthi uma
Izinhlobo
Ukusabalalisa okufanayo kumaphoyinti angu-n
Okuguquguqukayo okungahleliwe okungu-X kuthiwa kulandela ukusabalalisa okubonakala ngokufana kumaphoyinti angu-n uma inani ngalinye linamathuba afanayo abelwe. Amathuba omsebenzi wawo wenqwaba yilokhu:
Ake sithi sinokuhlola okunemiphumela emibili engase ibe khona: kungase kube ukuphenyisisa uhlamvu lwemali imiphumela yalo engaba amakhanda noma imisila, noma ukukhetha inombolo ephelele omphumela wayo ube inombolo eyinqaba noma ngisho; Lolu hlobo lokuhlola lwaziwa ngokuthi ukuhlolwa kwe-Bernoulli.
Ngokuvamile, imiphumela emibili engaba khona ibizwa ngokuthi impumelelo nokwehluleka, lapho u-p engamathuba okuphumelela futhi u-1-p amathuba okuhluleka. Singanquma amathuba okuthi x impumelelo ekuhlolweni okuzimele kwe-Bernoulli ngokusabalalisa okulandelayo.
Ukusabalalisa okubili
Lo msebenzi umele amathuba okuthola impumelelo engu-x kuzivivinyo ezizimele ze-Bernoulli, okungenzeka ukuthi impumelelo yazo ingu-p. Amathuba omsebenzi wawo wenqwaba yilokhu:
Igrafu elandelayo imele amathuba omsebenzi wesisindo samanani ahlukene wepharamitha yokusabalalisa i-binomial.
Ukusabalalisa okulandelayo kunegama lakho kusazi sezibalo esingumFulentshi uSimeon Poisson (1781-1840), owakuthola njengomkhawulo wokusatshalaliswa kwe-binomial.
Poisson Distribution
Okuguquguqukayo okungahleliwe X kuthiwa kunokusabalalisa kwe-Poisson kwepharamitha λ lapho ingathola amanani enombolo ephelele 0,1,2,3, … ngamathuba alandelayo:
Kule nkulumo, u-λ uyisilinganiso senani lezenzeko zeyunithi ngayinye yesikhathi futhi u-x uyinani lezikhathi umcimbi owenzeka ngazo.
Umsebenzi wayo wamathuba amaningi uthi:
Ngezansi kunegrafu emelela amathuba omsebenzi wesisindo samanani ahlukene wamapharamitha wokusabalalisa we-Poisson.
Qaphela ukuthi inqobo nje uma inani lempumelelo liphansi futhi nenani lokuhlolwa okwenziwe ekusabalaliseni kwe-binomial liphezulu, singahlala silinganisela lokhu kusabalalisa, njengoba ukusabalalisa kwe-Poisson kuwumkhawulo wokusabalalisa okubili.
Umehluko omkhulu phakathi kwalokhu kusabalalisa okubili ukuthi, kuyilapho i-binomial incike kumapharamitha amabili - nep -, i-Poisson incike kuphela ku-λ, ngezinye izikhathi okubizwa ngokuthi ukushuba kokusabalalisa.
Kuze kube manje, sikhulume kuphela mayelana nokusatshalaliswa kwamathuba ezimeni lapho ukuhlola okuhlukile kuzimele komunye nomunye; okungukuthi, lapho umphumela womunye ungathinteki umphumela womunye.
Uma izivivinyo zingazimele, ukusatshalaliswa kwe-hypergeometric kusiza kakhulu.
Ukusabalalisa kwe-Hypergeometric
U-N makenze isamba senani lezinto ezisesethi elinganiselwe, esingakhomba u-k ngayo ngandlela thize, sakhe isethi engaphansi K, umphelelisi wayo owakhelwe izakhi ezisele zika-Nk.
Uma sikhetha izinto ezingu-n nomaphi, ukuhluka okungahleliwe okungu-X okubonisa inani lezinto ezingezika-K kulokho kukhetha kuzoba nokusabalalisa kwe-hypergeometric kwamapharamitha N, n, kanye no-k. Umsebenzi wayo wamathuba amaningi uthi:
Igrafu elandelayo imele amathuba omsebenzi wesisindo samanani ahlukene wamapharamitha wokusabalalisa we-hypergeometric.
Izivivinyo ezixazululiwe
Ukuzivocavoca kokuqala
Ake sithi amathuba okuthi ishubhu lomsakazo (elifakwe ohlotsheni oluthile lwesisetshenziswa) lizosebenza amahora angaphezu kwama-500 ngu-0,2. Uma amashubhu angama-20 ehlolwa, yimaphi amathuba okuthi u-k wawo ncamashí uzosebenza amahora angaphezu kwama-500, k = 0, 1,2, 20, …, XNUMX?
Isixazululo
Uma u-X kuyinombolo yamashubhu asebenza ngaphezu kwamahora angu-500, sizothatha ngokuthi u-X unokusabalalisa kwe-binomial. Khona-ke
Manje:
Ku-k≥11, amathuba angaphansi kuka-0,001
Ngakho-ke, singabona ukuthi amathuba okuthi k kulaba abasebenza amahora angaphezu kuka-500 akhuphuka kanjani, aze afinyelele inani lawo eliphezulu (nge-k = 4) bese eqala ukwehla.
Ukuzivocavoca kwe-2
Uhlamvu lwemali luphendulwa izikhathi eziyisi-6. Uma umphumela ungamakhanda, sikubiza ngokuthi impumelelo. Ayini amathuba okuba amakhanda amabili ncamashi?
Isixazululo
Kulokhu, sine-n = 6 futhi amathuba okuphumelela nokwehluleka ngu-p = q = 1/2
Ngakho-ke, amathuba obuso obubili anikezwayo (okungukuthi, k = 2) yi
Ukuzivocavoca kwesithathu
Angakanani amathuba okuthola okungenani ubuso obune?
Isixazululo
Kulokhu, sine-k = 4, 5 noma 6
Ukuzivocavoca kwesithathu
Ake sithi u-2% wezinto ezikhiqizwa embonini zinephutha. Thola amathuba okuthi P kokuthi kunezinto ezintathu ezinesici kusampula yezinto eziyi-100.
Isixazululo
Kulesi simo, singasebenzisa ukusatshalaliswa kwe-binomial kuka-n = 100 kanye no-p = 0,02, sithole njengomphumela:
Nokho, njengoba u-p emncane, sisebenzisa isilinganiso se-Poisson no-λ = np = 2.
Izinkomba
- I-Kai Lai Chung: I-Elementary Probability Theory enezinqubo ze-Stochastic. Inkampani Springer-Verlag New York Inc.
- Kenneth.H. I-Rosen - I-Discrete Mathematics kanye nezicelo zayo. SAMCGRAW-HILL / INTERAMERICANO DE SPAIN.
- Paul L. Meyer Amathuba kanye nezicelo zezibalo. SA ALHAMBRA MEXICANA.
- Seymour Lipschutz Ph.D. 2000 Izinkinga Ezixazululiwe ku-Discrete Mathematics. UMcGraw-HILL
- Seymour Lipschutz Ph.D. Izinkinga Kuthiyori kanye Namathuba. UMcGraw-HILL