Ukusatshalaliswa kwe-Poisson amathuba okusabalalisa ahlukene achaza inani lezenzeko zomcimbi ngesikhathi esithile noma isikhawu sesikhala, lapho inani elimaphakathi lezenzeko laziwa. Isetshenziswa kakhulu emikhakheni ehlukahlukene njengezibalo, ubunjiniyela, umuthi, kanye nezezimali.
Ifomula yokusabalalisa i-Poisson inikezwa ngu-P(X = k) = (e^(-λ) * λ^k) / k!, lapho u-λ kuyinani elilinganiselwe lezenzeko zesehlakalo, u-k uyinombolo efiselekayo yezenzeko futhi u-e uhlala njalo ka-Euler (cishe u-2,71828).
Lokhu kusatshalaliswa kunezinto ezithile ezibalulekile, njengokuzimela phakathi kwezenzeko, izinga lokwenzeka njalo, kanye nokungabikho kokwenzeka ngesikhathi esisodwa. Ngaphezu kwalokho, Ukusabalalisa kwe-Poisson kungalinganiselwa Ngokusatshalaliswa Okujwayelekile uma inani elimaphakathi lezenzeko likhulu.
Ithini isibalo esisetshenziswe ekusabalaliseni kwe-Poisson ukuze kubalwe amathuba?
Ukusatshalaliswa kwe-Poisson wukusabalalisa kwamathuba ahlukene achaza inani lezenzeko zomcimbi esikhawulweni sesikhathi esithile noma endaweni ethile. Isibalo esisetshenziswe ekusabalaliseni kwe-Poisson ukuze kubalwe amathuba ami kanje:
P(X = k) = (λ^k * e^(-λ)) / k!
Onde:
- P(X = k) amathuba okuthi ncamashi izehlakalo zika-k zenzeka esikhathini esithile noma endaweni.
- λ isilinganiso sesilinganiso sokwenzeka kwezehlakalo ngeyunithi ngayinye yesikhathi noma indawo.
- e ukufana kwezibalo cishe kulingana no-2,71828.
- k inombolo yezehlakalo esifuna ukubala amathuba azo.
- k! imele ifekthoriyali ka-k, okuwumkhiqizo wawo wonke izinombolo eziphozithivu ezingaphansi noma ezilingana no-k.
Lesi sibalo sisivumela ukuthi sinqume amathuba okuthi ncamashi izehlakalo zika-k zenzeka kumongo othile, ngokusekelwe esilinganisweni sesilinganiso sokwenzeka kwalezi zenzakalo. Ukusatshalaliswa kwe-Poisson kusetshenziswa kakhulu kwizibalo ukwenza imodeli yezimo lapho izehlakalo ezingavamile zenzeka ngokuzimela futhi ngezinga elingashintshi.
Izici eziyisisekelo zenqubo ye-Poisson.
Ukusatshalaliswa kwe-Poisson amathuba okusabalalisa ahlukene achaza inani lezehlakalo ezenzeka kusikhawu esinqunyiwe sesikhathi noma isikhala. Inezici ezimbalwa ezibalulekile eziyenza isebenziseke emikhakheni eyahlukene, okuhlanganisa izibalo, izibalo, ubunjiniyela, kanye nesayensi yemvelo.
Ifomula yokusabalalisa ye-Poisson inikezwa ngu: P(x;λ) = (e^-λ * λ^x) / x!, lapho x imele inani lezehlakalo ezenzeka esikhathini sokuthakasela kanye λ ipharamitha emele isilinganiso esimaphakathi sokwenzeka kwezehlakalo.
Imodeli ye-Poisson ifanele izimo lapho izehlakalo zenzeka ngokuzimela futhi izinga lokwenzeka lihlala njalo ngokuhamba kwesikhathi noma indawo. Isibonelo, ukusatshalaliswa kwe-Poisson kungasetshenziswa ukwenza imodeli yenani lezingcingo ezitholwe isikhungo sezingcingo ehoreni elithile.
Ezinye izici ezibalulekile zokusabalalisa kwe-Poisson zifaka phakathi incazelo nokuhluka, okulingana nepharamitha λ. Ngaphezu kwalokho, ukusatshalaliswa kwe-Poisson akukhonegethivu futhi akunamkhawulo ongaphezulu.
Ngezici zayo eziyisisekelo nezici ezihlukile, idlala indima ebalulekile ekuhlaziyweni kwezibalo nasekuthathweni kwezinqumo kuzimo ezihlukahlukene.
Indlela yokubala ukusatshalaliswa kwe-Poisson: isinyathelo ngesinyathelo nezibonelo ezingokoqobo.
Ukuze ubale Ukusabalalisa kwe-Poisson, kubalulekile ukulandela izinyathelo ezimbalwa futhi usebenzise amafomula alungile. I-Poisson Distribution iwukusabalalisa kwamathuba ahlukene achaza inani lezehlakalo ezenzeka ngesikhathi esithile noma isikhala, uma kubhekwa isilinganiso esimaphakathi sokwenzeka kwalezo zenzakalo.
Ifomula ye-Poisson Distribution ithi:
P(X = k) = (e^(-λ) * λ^k) / k!
Onde:
- P(X = k) amathuba okuthi ncamashi izenzakalo zika-k zenzeka
- e iyisisekelo se-logarithm yemvelo
- λ isilinganiso sesilinganiso sokwenzeka kwezehlakalo
- k inombolo yezehlakalo esifuna ukubala amathuba azo
- k! isizinda se-k
Ukuze ubale i-Poisson Distribution, landela lezi zinyathelo:
1. Thola isilinganiso esimaphakathi sokwenzeka kwezehlakalo (λ)
2. Khetha inombolo yemicimbi ofuna ukubala amathuba okuthi (k)
3. Faka esikhundleni amanani kufomula ye-Poisson Distribution
4. Bala umphumela wokugcina
Isibonelo, uma isilinganiso sezinga lezingozi ekhoneni lomgwaqo singama-2 ngesonto, yimaphi amathuba okuthi kwenzeke izingozi ezi-3 ngeviki?
Ngokusebenzisa ifomula ye-Poisson Distribution, sine:
P(X = 3) = (e^(-2) * 2^3) / 3! = (0.1353) * (8) / 6 = 0.1804
Ngakho-ke, amathuba ezingozi ezi-3 ngqo ezenzeka ngesonto angu-0.1804, noma u-18.04%.
Ungalithola kanjani inani elimaphakathi lemicimbi ngesikhathi esithile?
Ukuthola inani elimaphakathi lemicimbi ngesikhathi esithile, singasebenzisa i-Poisson Distribution. Lokhu kusatshalaliswa kusetshenziswa kakhulu ukwenza imodeli yezehlakalo ezingavamile ngesikhathi esinqunyiwe, njengenombolo yezingcingo ezitholwa isikhungo sezingcingo ngehora.
Ifomula ye-Poisson Distribution inikezwa ngu:
P(X = k) = (e^(-λ) * λ^k) / k!
Onde:
- P(X = k) amathuba okuba izenzakalo zika-k zenzeka ngesikhathi.
- e ukufana kuka-Euler, cishe kulingana no-2.71828.
- λ isilinganiso senani lemicimbi ngeyunithi ngayinye yesikhathi.
- k inani lezehlakalo esifuna ukuzihlaziya.
- k! imele ifekthri ye-k.
Enye yezinto ezibaluleke kakhulu Zokusabalalisa kwe-Poisson ukuthi inani layo elimaphakathi lilingana nesilinganiso salo, okungukuthi, isilinganiso sezehlakalo esikhathini esithile esinqunyiwe sinikezwa ngu-λ.
Ngakho-ke, ukuze uthole inani eliyisilinganiso lezehlakalo esikhathini esithile, vele usebenzise inombolo eyisilinganiso yemicimbi ngeyunithi yesikhathi, emelwe ngu-λ kufomula Yokusabalalisa ye-Poisson.
I-Poisson Distribution: Amafomula, Izibalo, Imodeli, Izakhiwo
A Ukusatshalaliswa kwe-Poisson ukusabalalisa kwamathuba ahlukene, lapho ithuba lokuthi, ngaphakathi kwesampula enkulu futhi phakathi nesikhathi esithile, umcimbi okungenzeka ukuthi amathuba awo amancane enzeke aziwa.
Ngokuvamile, ukusabalalisa kwe-Poisson kungasetshenziswa esikhundleni sokusatshalaliswa kwe-binomial, inqobo nje uma izimo ezilandelayo zihlangatshezwana nazo: usayizi wesampula omkhulu kanye namathuba amancane.
USiméon-Denis Poisson (1781-1840) udale lokhu kusatshalaliswa okunegama lakhe, okuwusizo kakhulu lapho kubhekwana nezenzakalo ezingalindelekile. U-Poisson washicilela imiphumela yakhe ngo-1837, iphepha lokucwaninga mayelana namathuba emisho yobugebengu engalungile.
Kamuva, abanye abacwaningi bavumelanisa ukusatshalaliswa kwezinye izindawo, isibonelo, inani lezinkanyezi ezingatholakala endaweni ethile noma amathuba okuthi isosha libulawe ukukhahlelwa kwehhashi.
Ifomula nezibalo
Ifomu lezibalo lokusatshalaliswa kwe-Poisson limi kanje:
- μ (futhi ngezinye izikhathi kuchazwa ngokuthi λ) isho noma ipharamitha yokusabalalisa
Inombolo ka-Euler: e =2.71828
- Amathuba okuthola u-y = k ngu-P
- k inombolo yempumelelo 0, 1,2,3 ...
- n inombolo yokuhlolwa noma imicimbi (usayizi wesampula)
Okuguquguqukayo okungahleliwe okungahleliwe, njengoba igama lisho, kuncike ethubeni futhi kuthathe amanani ahlukene kuphela: 0, 1, 2, 3, 4 …, k.
Ukusabalalisa okumaphakathi kunikezwa ngu:
Umehluko σ, okala ukusakazeka kwedatha, kungenye ipharamitha ebalulekile. Ngokusatshalaliswa kwe-Poisson, yile:
σ = μ
UPoisson unqume ukuthi uma n → ∞ kanye no-p → 0, incazelo µ – ebizwa nangokuthi inani elilindelekile - ivame ukuhlala njalo:
μ → njalo
Kubalulekile : p amathuba okuthi umcimbi wenzeke kucatshangelwa inani labantu, kuyilapho P (y) isibikezelo se-Poisson sesampula.
Imodeli nezakhiwo
Ukusatshalaliswa kwe-Poisson kunezici ezilandelayo:
-Usayizi wesampula mkhulu: n → ∞.
-Izehlakalo noma izehlakalo ezicatshangelwayo azihambelani futhi zenzeka ngokungahleliwe.
-Amathuba P ukuthi umcimbi othize e kwenzeka ngesikhathi esithile kuncane kakhulu: P → 0 .
-Amathuba omcimbi ongaphezu kowodwa owenzeka phakathi nesikhathi ngu-0.
-Inani elimaphakathi licishe lifane nokunikezwa njalo ngu: μ = np ( n usayizi wesampula )
-Njengoba ukusabalalisa u-σ kulingana no-μ, njengoba kwamukela amanani amakhulu, ukuhlukahluka nakho kuba kukhulu.
-Imicimbi kumele isatshalaliswe ngokulinganayo phakathi nesikhathi esisetshenzisiwe.
-Iqoqo lamanani omcimbi okungenzeka e yilokhu: 0,1,2,3,4….
-Isamba se i okuguquguqukayo okulandela ukusatshalaliswa kwe-Poisson nakho kuyiziguquguquko ze-Poisson. Inani labo elimaphakathi yisamba senani lesilinganiso salokhu okuguquguqukayo.
Umehluko ngokusabalalisa kwe-binomial
Ukusatshalaliswa kwe-Poisson kuyahluka ekusabalaliseni kwe-binomial kulezi zici ezibalulekile ezilandelayo:
-Ukusatshalaliswa kwe-binomial kuthintwa usayizi wesampula kanye namathuba P , kodwa ukusatshalaliswa kwePoisson kuthinteka kuphela μ isilinganiso .
-Esabalazweni se-binomial, amanani angenzeka wokuguquguquka okungahleliwe e zingu-0,1,2, …, N, nokho ekusabalaliseni kwe-Poisson awukho umkhawulo ophezulu walawa manani.
Izibonelo
UPoisson ekuqaleni wasebenzisa ukusatshalaliswa kwakhe okudumile ezinqubweni ezingokomthetho, kodwa ezingeni lezimboni, okunye ukusetshenziswa kwawo kokuqala kwakuwukuphisa ubhiya. Kule nqubo, amasiko emvubelo asetshenziselwa ukuvutshelwa.
Imvubelo iqukethe amaseli aphilayo, inani labantu liyahlukahluka ngokuhamba kwesikhathi. Lapho uphisa ubhiya, udinga ukwengeza inani elidingekayo, ngakho-ke kubalulekile ukwazi inani lamaseli ngevolumu yeyunithi ngayinye.
Phakathi neMpi Yezwe II, ukusatshalaliswa kwePoisson kwasetshenziswa ukuze kunqunywe ukuthi amaJalimane ayeqondise ngempela iLondon esuka eCalais, noma adubula nje ngokungahleliwe. Lokhu bekubalulekile kuma-Allies ukuthi anqume ukuthi ubuchwepheshe bebubuhle kangakanani kumaNazi.
Izicelo ezingokoqobo
Izinhlelo zokusebenza zokusabalalisa kwe-Poisson zihlala zibhekisela ekubalweni kwesikhathi noma isikhala. Futhi ngenxa yokuthi amathuba okuba kwenzeke mancane, aziwa nangokuthi "umthetho wezenzakalo ezingavamile."
Nalu uhlu lwemicimbi ewela kwesinye salezi zigaba:
-Irekhodi lezinhlayiya ekuboleni kwe-radioactive, okufana nokukhula kwamangqamuzana imvubelo, kuwumsebenzi we-exponential.
-Inani lokuvakashela iwebhusayithi ethile.
- Ukufika kwabantu emgqeni wokukhokha noma ukuhlinzekwa (ithiyori yomugqa).
– Inani lezimoto ezidlula endaweni ethile emgwaqeni, phakathi nenkathi ethile yesikhathi.
- Ukuguqulwa kwezakhi zofuzo kuhlupheke ochungechungeni lwe-DNA olunikeziwe ngemuva kokuthola ukuchayeka emisebeni.
-Inani lama-meteorite anobubanzi obungaphezu kuka-1 m awile ngonyaka owodwa.
– Ukonakala nge-square meter yendwangu.
-Inani lamaseli egazi ku-1 cubic centimeter.
-Izingcingo ngomzuzu ekushintshisaneni ngocingo.
– Izingcezu zikashokoledi zikhona ku-1 kg yenhlama yekhekhe.
-Inani lezihlahla ezingenwe i-parasite esinikeziwe ehektheleni elingu-1 lehlathi.
Qaphela ukuthi lezi ziguquko ezingahleliwe zimelela inani lezikhathi umcimbi owenzeka ngazo phakathi nesikhathi esinqunyiwe ( izingcingo ngomzuzu ku-switchboard ) noma indawo ethile yesikhala ( amaphutha endwangu ngemitha yesikwele ).
Lezi zehlakalo, njengoba sezivele zimisiwe, azifani nesikhathi esidlulile kusukela senzeka okokugcina.
Isondela ekusabalaliseni kwe-binomial ngokusabalalisa kwe-Poisson
Ukusatshalaliswa kwe-Poisson kuwukuhlawumbisela okuhle kokusatshalaliswa kwe-binomial kusukela:
-Usayizi wesampula mkhulu: n ≥ 100
-Amathuba unyawo encane: p ≤ 0,1
- μ ibe ngohlelo lokuthi: np ≤10
Kulezi zimo, ukusatshalaliswa kwe-Poisson kuyithuluzi elihle kakhulu, njengoba ukusatshalaliswa kwe-binomial kungaba nzima ukusebenzisa kulezi zimo.
Izivivinyo ezixazululiwe
Ukuzivocavoca 1
Ucwaningo lwe-seismological lwathola ukuthi eminyakeni eyikhulu edlule, kube nokuzamazama komhlaba okukhulu okungu-100 emhlabeni wonke, okulinganiselwa okungenani okungu-93 esikalini se-logarithmic Richter. Cabanga ukuthi ukusatshalaliswa kwe-Poisson kuyimodeli efanelekile kuleli cala. Thola:
a) Isilinganiso sokwenzeka kokuzamazama komhlaba okukhulu ngonyaka.
b) Uma P (y) ngamathuba okuthi kwenzeke e ukuzamazama komhlaba ngonyaka okhethwe ngokungahleliwe, thola amathuba alandelayo:
P (0), P (1), P (2), P (3), P (4), P (5), P (6) no P (7).
c) Imiphumela yangempela yocwaningo imi kanje:
- Iminyaka engu-47 (0 ukuzamazama komhlaba)
- iminyaka engama-31 (ukuzamazama komhlaba oku-1)
- iminyaka engama-13 (ukuzamazama komhlaba oku-2)
- iminyaka engama-5 (ukuzamazama komhlaba oku-3)
- iminyaka engama-2 (ukuzamazama komhlaba oku-4)
- iminyaka engama-0 (ukuzamazama komhlaba oku-5)
- 1 unyaka (6 ukuzamazama komhlaba)
- 1 unyaka (7 ukuzamazama komhlaba)
Injani le miphumela uma iqhathaniswa naleyo etholwe engxenyeni b? Ingabe ukusatshalaliswa kwePoisson kuyisinqumo esihle sokumodela le micimbi?
Isixazululo a)
a) Ukuzamazama komhlaba izehlakalo okungenzeka zibe khona p sincane futhi sicabangela isikhathi esinqunyelwe unyaka owodwa. Isilinganiso sokuzamazama komhlaba sithi:
μ = ukuzamazama komhlaba okungu-93/100/unyaka = ukuzamazama komhlaba okungu-0,93 ngonyaka.
Isixazululo b)
b) Ukubala amathuba aceliwe, amanani ashintshwa kufomula enikezwe ekuqaleni:
y = 2
ngi = 0,93
e =2.71828
Incane kakhulu kune-P(2).
Imiphumela ibalwe ngezansi:
P(0) = 0,395, P(1) = 0,367, P(2) = 0,171, P(3) = 0,0529, P(4) = 0,0123, P(5) = 0,00229, P(6) = 0,000355, P.7, P
Ngokwesibonelo, singase sithi kunamathuba angu-39,5% okuthi kungabikho ukuzamazama komhlaba okukhulu okuzokwenzeka ngonyaka othile. Noma ukuthi kukhona u-5,29% wokuzamazama komhlaba okukhulu okuthathu okwenzeka ngalowo nyaka.
Isixazululo c)
c) Amafrikhwensi ayahlaziywa, aphindaphindeka ngo-n = iminyaka eyi-100:
39,5; 36,7; 17,1; 5,29; 1,23; 0,229; 0,0355 kanye no-0,00471.
Ngokwesibonelo:
- Imvamisa ye-39,5 ibonisa ukuthi eminyakeni engu-39,5 kweyi-100 yokwenzeka kokuzamazama komhlaba okukhulu okungu-0, singasho ukuthi kuseduze nomphumela wangempela weminyaka engu-47 ngaphandle kokuzamazama komhlaba okukhulu.
Ake siqhathanise omunye umphumela we-Poisson nemiphumela yangempela:
Inani lika-36,7 lisho ukuthi kuba nokuzamazama komhlaba okukhulu okukodwa njalo eminyakeni engama-37. Umphumela wangempela wukuthi kwaba nokuzamazama komhlaba okukodwa okukhulu njalo ngemva kweminyaka engama-31, okufana kahle nomfuziselo.
- Iminyaka engu-17,1 kulindeleke ngokuzamazama komhlaba okukhulu kwe-2 futhi kuyaziwa ukuthi eminyakeni engu-13, okuyinani eliseduze, empeleni kwakukhona ukuzamazama komhlaba okukhulu kwe-2.
Ngakho-ke, imodeli yePoisson iyamukeleka kuleli cala.
Ukuzivocavoca 2
Inkampani ilinganisela ukuthi inani lezingxenye ezihlulekayo ngaphambi kokufinyelela emahoreni ayi-100 okusebenza lilandela ukusatshalaliswa kwe-Poisson. Uma isilinganiso senani lokuhluleka kungu-8 ngaleso sikhathi, thola amathuba alandelayo:
a) Ukuthi ingxenye yehluleka phakathi kwamahora angama-25.
b) Ukwehluleka kwezingxenye ezingaphansi kwezimbili emahoreni angama-50.
c) Okungenani izingxenye ezintathu ziyehluleka phakathi namahora ayi-125.
Isixazululo a)
a) Kuyaziwa ukuthi isilinganiso senani lokuhluleka emahoreni ayi-100 siyisi-8; ngakho-ke, emahoreni angama-25, ingxenye yesine yokwehluleka kulindeleke, okungukuthi, ukwehluleka okungu-2. Lokhu kuzoba ipharamitha μ.
Amathuba okwehluleka kwengxenye engu-1 ayacelwa, okuhlukile okungahleliwe “kuyizingxenye ezingaphumeleli ngaphambi kwamahora angu-25” futhi inani lakhona ngu-y = 1. Ngokufaka esikhundleni somsebenzi wamathuba:
Nokho, umbuzo uwukuthi kungenzeka kangakanani lokho izingxenye ezingaphansi kwezimbili ehluleka emahoreni angama-50, futhi hhayi izingxenye ezi-2 ezihlulekayo emahoreni angama-50, ngakho-ke amathuba yilawa:
-Akukho ukwehluleka
- Ukwehluleka 1 kuphela
P(izingxenye ezingaphansi kwezi-2 ziyehluleka) = P(0) + P(1)
P(izingxenye ezingaphansi kwezi-2 ziyahluleka) = 0,0183 + 0,0732 = 0. 0915
c) Lokho okungenani izingxenye ezintathu ziyehluleka emahoreni angu-125, kusho ukuthi u-3, 4, 5 noma ngaphezulu angahluleka ngaleso sikhathi.
Amathuba okuba kwenzeke okungenani esisodwa sezehlakalo ezimbalwa silingana no-1, khipha amathuba okuthi azikho izehlakalo ezenzekayo.
-Isenzakalo esifiswayo ukuthi izingxenye ezi-3 noma ngaphezulu ziyahluleka phakathi namahora angu-125
- Ukuthi umcimbi ungenzeki kusho ukuthi izingxenye ezingaphansi kwezi-3 ziyahluleka, okungenzeka ukuthi: P(0) + P(1) + P(2)
Ipharamitha μ yokusabalalisa kuleli cala ithi:
μ = 8 + 2 = ukwehluleka okungu-10 emahoreni ayi-125 .
P (izingxenye ezi-3 noma ngaphezulu ziyehluleka) = 1 – P(0) – P(1) – P(2) =
Izinkomba
- I-MathWorks Poisson Distribution. Ithathwe ku-: en.mathworks.com
- Mendenhall, W. 1981. Izibalo Zokuphatha kanye Nezomnotho. 3rd edition. I-Iberoamerica Publishing Group.
- I-Stat Trek Zifundise Izibalo. Ukusabalalisa kwe-Poisson Kutholwe ku-: stattrek.com,
- Triola, M. 2012. Izibalo Eziyisisekelo. 11 kwe. U-Ed. Imfundo yePearson.
- Wikipedia Poisson Distribution Kutholwe kusukela: en.wikipedia.org