Te Mārama ki te Tohatoha Poisson
I te ao o ngā tatauranga me te tūponotanga, ka whakamahia ngā momo tohatoha hei whakatauira i ngā āhuatanga o te ao tūturu. Ko tētahi tohatoha e whakamahia pinepinetia ana i roto i ngā momo mara ko te tohatoha Poisson. He āhuatanga ahurei tō tēnei tohatoha, ā, he tino whai hua i roto i ngā momo tono, mai i ngā pūtaiao taiao ki te hangarau, te ōhanga, me ngā pūtaiao pāpori. Ka matapakihia e tēnei tuhinga te tohatoha Poisson, ōna āhuatanga, me ōna tono i roto i ngā momo horopaki.
Te Mārama ki te Tohatoha Poisson
Ko te tohatoha Poisson he tohatoha tūponotanga motuhake e whakaahua ana i te maha o ngā wā ka puta he huihuinga i roto i tētahi wā, i tētahi wāhi rānei kua whakaritea. I whakaurua tuatahitia tēnei tohatoha e te tohunga pāngarau Wīwī a Siméon Denis Poisson i te tau 1837. He maha ngā wā ka whakamahia te tohatoha Poisson hei whakatauira i ngā huihuinga matapōkere e puta pinepine ana engari he nui ngā tau i roto i te tapeke o ngā kitenga.
Koinei te tātai tohatoha Poisson:
\[ P(X = k) = \frac{\lambda^ke^{-\lambda}}{k!} \]
kāore i te mana:
– Ko te P(X = k) te tūponotanga kia k ngā takahanga i roto i tētahi wā kua hoatu,
– Ko te toharite o ngā takahanga i roto i te wā ko \( \lambda \),
– Ko te \( k \) te maha o ngā takahanga,
– Ko \( e \) te pūtake o te logarithm maori, arā, tata ki te 2.71828.
Ko te whakaaro taketake o te tohatoha Poisson he motuhake ngā takahanga tetahi i tetahi, ā, he pumau te toharite o ngā takahanga mō ia waeine wā o te wā, o te wāhi rānei.
Ngā Āhuatanga o te Tohatoha Poisson
He maha ngā āhuatanga matua o te tohatoha Poisson e wehewehe ana i a ia mai i ētahi atu tohatoha. Anei ngā āhuatanga matua o te tohatoha Poisson:
1. Momotu, Kore-Kōraro: Ko ngā taurangi matapōkere i roto i te tohatoha Poisson ka taea anake te tango i ngā uara tauoti kore-kōraro (0, 1, 2, …).
2. Te Tū motuhake o ngā Takahanga: Me tū motuhake ia takahanga tetahi i tetahi. Ko te tikanga o tēnei, kāore te putanga mai o tētahi takahanga e pā ki te tūponotanga o te putanga mai o tētahi atu takahanga.
3. Toharite Pūmau: Me pūmau te toharite o ngā takahanga i roto i tētahi wā kua whakaritea. Ko te tikanga tēnei kāore e tika te tohatoha Poisson mēnā ka huri te toharite o ngā takahanga i roto i te wā.
4. Tawhā Kotahi (\( \lambda \)) : Kotahi anake te tawhā o te tohatoha Poisson, arā, \( \lambda \), koia te tau toharite o ngā takahanga i roto i tētahi wā.
5. Te Toharite me te Rerekētanga: I roto i te tohatoha Poisson, he ōrite te toharite (toharite) me te rerekētanga (rerekētanga), arā, \( \lambda \).
Ngā Akoranga Take me ngā Whakamahinga
He maha ngā whakamahinga o te tohatoha Poisson i te ao tūturu. Ko ētahi tauira noa o tēnei tohatoha ko:
1. Te maha o ngā waeatanga waea: Me kī i roto i tētahi pokapū ratonga kiritaki, ko te toharite o ngā waeatanga waea i whiwhihia i ia hāora he 5. Ka taea te whakamahi i te tohatoha Poisson hei whakatauira i te maha o ngā waeatanga i whiwhihia i roto i tētahi hāora kua whakaritea.
2. Ngā Aituā Waka: Me kī ko te toharite o ngā aituā waka e puta ana i tētahi whakawhitinga huarahi motuhake ia marama he 3. Ka taea e te tohatoha Poisson te āwhina i te matapae i te maha o ngā aituā ka puta pea i te marama e whai ake nei.
3. Te Taenga Mai o ngā Kaihoko ki tētahi Wharekai: Mena ko te toharite o ngā kaihoko e haere mai ana ki tētahi wharekai ia hāora he 10, ka taea te whakamahi i te tohatoha Poisson hei whakatauira i te maha o ngā kaihoko ka tae mai pea i roto i tētahi hāora kua whakaritea.
4. Ngā Ira Whakarerekētanga Ira: I roto i te horopaki o te ira, ka taea te whakamahi i te tohatoha Poisson hei whakatauira i te maha o ngā ira whakarerekētanga ira i roto i tētahi rōpū rauropi i roto i tētahi wā kua whakaritea, i te mea he onge engari he motuhake ngā ira whakarerekētanga ira.
Me pēhea te tatau i te tūponotanga mā te Poisson Distribution
Hei mārama ake i te whakamahinga o te tohatoha Poisson, me titiro tātou ki te tatau i te tūponotanga mā te whakamahi i te tātai tohatoha Poisson. Hei tauira:
Me kī ko te toharite o ngā kaihoko e haere mai ana ki tētahi toa i roto i te haora he 4 (\( \lambda = 4 \)). E hiahia ana mātou ki te mōhio ki te tūponotanga ka tae mai ngā kaihoko e 6 i roto i tētahi haora. Mā te whakamahi i te tātai Poisson:
\[ P(X = 6) = \frac{4^6 e^{-4}}{6!} \]
Ka taea e tātou te tatau:
– \( 4^6 = 4096 \)
– \( e^{-4} \tata ki te 0.0183 \)
– \( 6! = 720 \)
Nō reira,
\[ P(X = 6) = \frac{4096 \cdot 0.0183}{720} \approx 0.104 \]
Nō reira, ko te tūponotanga e ono tonu ngā kiritaki e haere mai ana i roto i te haora kotahi he 10.4%.
Ngā Painga me ngā Here o te Tohatoha Poisson
Te taikaha:
1. Māmā, ngāwari hoki: He māmā noa te tātai o te tohatoha Poisson, ā, kotahi noa te tawhā e hiahiatia ana (\( \lambda \)), e māmā ai te whakamahi.
2. Ngā Whakamahinga Whānui: He maha ngā whakamahinga o tēnei tohatoha i roto i ngā mara maha nā te mea he maha ngā huihuinga tūturu ka taea te whakatauira me tētahi tohatoha he huihuinga onge, motuhake hoki ōna.
3. Ngā Whakaaro Tūturu: He tūturu ngā whakaaro mō te tū motuhake me te pumau o te toharite i roto i ngā āhuatanga o te ao tūturu, pērā i te maha o ngā kiritaki e tae mai ana, te maha rānei o ngā waeatanga.
Te Whakatakotoranga:
1. Kāore te Toharite Pūmau i te Rawaka i ngā Wā Katoa: I roto i ngā āhuatanga o te ao tūturu, kāore pea te toharite o ngā takahanga e pumau tonu. Mena ka huri te toharite i roto i te wā, kāore pea te tohatoha Poisson e tika.
2. Te Tū motuhake o ngā Takahanga: Kāore pea i te pono i ētahi wā te whakaaro he tū motuhake ngā takahanga tetahi i tetahi.
3. Mō ngā tauoti anake: Ko te tohatoha Poisson he mea tika anake mō ngā huihuinga ka taea te tatau i roto i ngā tauoti. Kāore e taea te whakamahi mō ngā raraunga tonu.
Ngā Rerekētanga o te Tohatoha Poisson
Ahakoa he tino whai hua te tohatoha Poisson, he maha ngā momo me ngā toronga o tēnei tohatoha hei whakatutuki i ngā āhuatanga uaua ake. Ko tētahi momo rongonui ko te Tohatoha Poisson Ranunga, e whakaae ana ko te maha toharite o ngā takahanga (\( \lambda \)) ka taea hoki te noho hei taurangi matapōkere me tētahi tohatoha motuhake.
Kei reira anō te Tohatoha Poisson Whānui, e whakangawari ana i ētahi o ngā whakapae o te tohatoha Poisson paerewa hei whakatutuki i ngā āhuatanga kāore pea ngā takahanga e tino motuhake, e kore rānei ngā tūponotanga o ngā takahanga tino onge e tau ki te tauira Poisson paerewa.
Whakamutunga
He taputapu kaha te tohatoha Poisson i roto i ngā tatauranga me te tūponotanga e whakamahia ana hei whakatauira i ngā huihuinga matapōkere e puta ana i ngā wā kua whakaritea o te wā, o te wāhi rānei. Mā te whakamahi i tētahi tawhā matua kotahi, \(\lambda\), ka tukuna he huarahi māmā engari he whai hua ki te whakaahua i te whānuitanga o ngā āhuatanga o te ao tūturu, mai i te ratonga kiritaki ki te ira tangata. Ahakoa he whakaaro kei raro iho ka whakawhāiti i tōna tika i roto i ētahi āhuatanga, ko tōna māmā me te whānui o te tono ka waiho hei tetahi o ngā tohatoha tūponotanga tino rongonui me te whai hua. Mā te mārama ki te tohatoha Poisson ehara i te mea ka āwhina noa i te tātari tatauranga engari ka whakarato hoki i te māramatanga ki te mahi a ngā tauira tūponotanga i roto i ngā āhuatanga taiao me ngā āhuatanga i hangaia e te tangata.
