- Axiomatic probability izifundo iziganeko random ngokusebenzisa axioms kunye nethiyori.
- Ii-axioms zayo ezintathu ezisisiseko azikho-negativity, ukuba nokwenzeka kwesithuba sesampulu kunye nokongeza.
- Isebenza kwiinkalo ezifana nezemali, amayeza, kunye nokufunda koomatshini.
- Uququzelela ukuthathwa kwezigqibo ngokumisela ukungaqiniseki kunye nokuvavanya umngcipheko.
I-Axiomatic Probability: Inkcazo kunye noMxholo
Axiomatic amathuba, kwaziwa njenge ithiyori yomlinganiselo wokunokwenzeka, yindlela yemathematika esekelwe kwiseti yee-axioms ukuchaza nokufunda ukuba nokwenzeka kweziganeko. Eli sebe lavela ekuqaleni kwenkulungwane yama-20, ngenxa yomsebenzi weengcali zezibalo ezifana no-Andrei Kolmogorov kunye no-Émile Borel, abathe babeka isiseko sethiyori ehambelanayo kunye nengqongqo yokuba kunokwenzeka.
Kulo mxholo, ukuba nokwenzeka kuqondwa njengomlinganiselo oqhelekileyo wethuba lesiganeko esithile esenzeka kwisithuba sesampulu. I-axioms ye-axiomatic probability iqinisekisa ukuba lo mlinganiselo uyanelisa iimpawu ezithile ezibalulekileyo, ezifana nokungabinandaba, ukongezelela, kunye nokuqheleka.
Axiom 1: Akukho Negativity
I-axiom yokuqala yamathuba ithi ukuba nokwenzeka kwaso nasiphi na isiganeko ( A ), esichazwa njengo ( P(A) ), kufuneka sihlale sisikhulu okanye silingana no-zero. Ngamanye amazwi, akukho mathuba angalunganga. Le axiom ichazwa ngokwemathematika ngolu hlobo:
$$ P(A) \geq 0 $$
Lo mgaqo unengqondo, kuba akunakuba sengqiqweni ukuthetha malunga nokuba nokwenzeka okungalunganga kwihlabathi lokwenyani. Umzekelo, asinakutsho ukuba amathuba okufumana iintloko xa uphequlula ingqekembe yemali ngu (-0.5).
I-Axiom 2: Ukubakho kweSithuba seSampuli
I-axiom yesibini ichaza ukuba kunokwenzeka ukuba yonke indawo yesampula, echazwe njenge ( \ Omega ), ihlala ilingana ne-1. Indawo yesampula ibonisa zonke iziphumo ezinokwenzeka zovavanyo olungenangqondo. Ngokwezibalo, le axiom ichazwa ngolu hlobo:
$$ P(\ Omega) = 1 $$
Le axiom ithetha ukuba isimbuku samathuba azo zonke iziganeko ezinokwenzeka kwisampulu yesithuba kufuneka silingane no-1. Umzekelo, xa uqengqeleka i-fair fair side six-side, the sum of the propoabilities of get each of the numbers (1, 2, 3, 4, 5, and 6) ilingana no-1.
I-Axiom 3: Ukongezwa
I-axiom yesithathu ye-axiomatic probability ithi, kulo naluphi na ulandelelwano lweziganeko ezihlukeneyo (( A_1, A_2, \lddots, A_n )), ukuba nokwenzeka komanyano lwezi ziganeko kuyalingana nesimbuku samathuba azo ngamnye. Ngokwezibalo, le axiom ichazwa ngolu hlobo:
$$ P(A_1 \indebe A_2 \indebe \ amachaphaza \ indebe A_n) = P(A_1) + P(A_2) + \ldots + P(A_n) $$
Iziganeko ezibini ziyadityaniswa ukuba azinakwenzeka ngaxeshanye. Umzekelo, xa kuqengqeleka idayizi, iziganeko "ukuqengqeleka inani elilinganayo" kunye "nokuqengqeleka inani elingumnqakathi" zizodwa, kuba inani alinakukwazi ukuba lilingane kwaye libe mnqakathi.
Iithiyori ezisisiseko ze-Axiomatic Probability
Ukusuka kwii-axioms ezisisiseko, i-axiomatic probability ifumana uthotho lweethiyori ezivumela ukubala kunye nokulawula okunokwenzeka kwiimeko ezinzima ngakumbi. Ezinye zeethiyori ezibalulekileyo zezi:
1. Ithiyori enokwenzeka ngokwemiqathango
Imeko enokwenzeka ibhekisa kumathuba okuba isiganeko ( A ) siya kwenzeka, ekubeni esinye isiganeko ( B ) sele senzeka. Le ithiyori ibonakaliswa ngokwemathematika ngolu hlobo:
$$ P(A|B) = \frac{P(A \ ikepusi B)}{P(B)} $$
Apho ( P(A|B) ) imele ukuba nokwenzeka kuka ( A ) kunikwe ( B ), ( P ( A \ cap B ) ) lithuba elinokwenzeka lokuhlangana kwe ( A ) kunye ( B ), kunye ( P ( B ) ) lithuba lokuba ( B ).
Umzekelo wamathuba aphantsi kwemiqathango ingabala amathuba okuba umntu abe nesifo esithile, xa kujongwa ukuba uvavanywe ukuba unayo kuvavanyo lokuxilonga.
2. Ithiyori kaBayes
Ithiyori kaBayes lulwandiso lwemeko enokwenzeka evumela ukuba kunokwenzeka ukuba umcimbi uhlaziywe ngokusekelwe kulwazi olutsha. Le theorem ibonakaliswa ngolu hlobo:
$$ P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)} $$
Apho u-P(A|B) ilithuba lika-A elinikiweyo, u-P(B|A) lithuba lika-B elinikwe uA, uP(A) lithuba langaphambili lika-A, kunye no-P(B) lithuba elinokwenzeka lika-B.
Le theorem isetyenziswa kakhulu kwiinkalo ezifana neyeza, ubukrelekrele bokwenziwa kunye ukwenza izigqibo, ekubeni ivumela ukuhlaziya iinkolelo zokuqala njengoko kufunyanwa ubungqina obutsha.
3. ITheorem enokwenzeka ngokupheleleyo
I-theorem ye-probability iyonke ivumela ukuba sibale ukwenzeka kwesiganeko ( A ), siqwalasela zonke iziganeko ezinokwenzeka ( B_i ) ezenza ulwahlulo lwesampulu yendawo. Ngokwezibalo, ichazwa ngolu hlobo:
$$ P(A) = P(A|B_1) \cdot P(B_1) + P(A|B_2) \cdot P(B_2) + \lddots + P(A|B_n) \cdot P(B_n) $$
Apho ( P(A|B_i) ) lithuba lokuba ( A ) linikwe ( B_i ), kwaye ( P(B_i) ) lithuba lokuba ( B_i ).
Umzekelo wokusebenzisa le theorem inokuba kukubala amathuba okuba umfundi aphumelele iimviwo, kuqwalaselwa iindlela ezahlukeneyo ebenokufunda ngazo (eyedwa, eliqela, nomkhapheli, njl. njl.).
Usetyenziso lwe-Axiomatic Probability
Axiomatic probability inoluhlu olubanzi lwezicelo kwiinkalo ezahlukeneyo zesayensi kunye nokusebenza. Eminye yemimandla apho eli sebe lemathematika linempembelelo ebalulekileyo zezi:
1. IFiziksi yamanani
Kwifiziksi yezibalo, i-axiomatic probability isetyenziselwa ukuchaza kunye nokuqikelela ukuziphatha kweenkqubo eziyinkimbinkimbi ezenziwe ngamasuntswana amaninzi. Imigaqo yamathuba ivumela ukuba senze imodeli yeziganeko ezifana nokuhanjiswa kwesantya kwigesi, imagneti yemathiriyeli, kunye notshintsho lwesigaba.
2. Ezezimali kunye noQoqosho
Kwimimandla yezemali neyoqoqosho, ukubakho kwe-axiomatic ngundoqo kuhlalutyo lomngcipheko, ukuxatyiswa kwe-asethi kunye nokwenziwa kwezigqibo zotyalo-mali. Iimodeli ezinokwenzeka zisetyenziselwa ukufunda ukuguquguquka kweemarike, ukuqikelela ukuziphatha kwamaxabiso kunye nokuvavanya ingeniso yeendlela ezahlukeneyo zotyalo-mali.
3. Ubukrelekrele bokwenziwa kunye nokuFunda koomatshini
I-Axiomatic probability sisixhobo esiphambili ekuphuhliseni ubukrelekrele bokwenziwa kunye ne-algorithms yokufunda koomatshini. Iimodeli ezinokwenzeka, ezifana neenethiwekhi zeBayesian kunye neemodeli zeMarkov ezifihliweyo, zivumela oomatshini ukuba bafunde kwidatha kwaye benze izigqibo ezisekelwe ekungaqiniseki. Obu buchule busetyenziswa kwimimandla efana nokuqondwa kwelizwi, umbono wekhompyuter kunye nengcebiso yemveliso.
4. Amayeza kunye ne-Epidemiology
Kwintsimi yeyeza, i-axiomatic probability isetyenziselwa ukuhlalutya ukusebenza konyango, ukuqikelela ukusasazeka kwezifo, kunye nokuvavanya ukuchaneka kweemvavanyo zokuxilonga. Iimodeli ezinokwenzeka zivumela ukuqikelela umngcipheko wokuphuhlisa iimeko ezithile zonyango, kunye nokuyila amaqhinga okuthintela nokulawula ubhubhane.