Ngā Kaupapa Taketake o te Tūponotanga Tatauranga
Ko te tūponotanga me te tatauranga he wāhanga pūtaiao e rua e tino hira ana mō te mārama ki te koretake me te whakatau i ngā whakataunga e ahu mai ana i ngā raraunga. I roto i te oranga o ia rā, he maha ngā wā ka tūtaki tātou ki ngā pātai pēnei i te "he aha te tūponotanga o te ua i tēnei rā?", "he whai hua te rongoā?", "ka piki ake rānei ngā hoko e te rautaki hokohoko?". Mā te tūponotanga ka whakaratohia he anga pāngarau hei ine i te tūponotanga o tētahi huihuinga, ko ngā tatauranga ia ka āwhina i a tātou ki te tukatuka raraunga, ki te whakatau whakatau, me te matapae. Ka matapakihia e tēnei tuhinga ngā kaupapa matua o te tūponotanga tatauranga, e hanga ana i te tūāpapa o te tātari raraunga hou.
1. Te Mārama ki te Tūponotanga me ngā Tatauranga
Ko te tūponotanga he peka o te pāngarau e ako ana i te tūponotanga o tētahi kaupapa. Ka whakamahia te tūponotanga hei whakatauira i ngā kaupapa matapōkere me te ine i te koretake. Ko ngā uara tūponotanga kei waenga i te 0 ki te 1. Ko te uara o te 0 e tohu ana i te kore e taea, ko te uara o te 1 e tohu ana i te tino pono.
Ko te tatauranga te pūtaiao e ako ana i te kohikohi, te whakarite, te tātari, me te whakamārama raraunga. E rua ngā wāhanga matua o te tatauranga:
1. Ngā tatauranga whakaahua, arā, ngā tikanga mō te whakarāpopoto me te whakaahua i ngā raraunga (hei tauira, te toharite, te tau waenga, ngā kauwhata).
2. Ngā tatauranga whakatau, arā, ngā tikanga mō te whakatau whakatau mō tētahi taupori i runga i tētahi tauira (hei tauira, te whakatau tata, te whakamātautau whakapae).
He hononga tō ngā mea e rua. He maha ngā wā ka whakamahia te tūponotanga hei pūtake ariā mō ngā tatauranga whakatau, nā te mea ina tangohia he tauira mai i tētahi taupori, he matapōkere ngā hua, ā, ka taea te tātari mā te whakamahi i ngā ariā tūponotanga.
2. Ngā Ariā Taketake: Ngā Whakamātautau, Ngā Wāhi Tauira, me Ngā Takahanga
Mō te tūponotanga, ko te taahiraa tuatahi ko te tautuhi i tētahi whakamātautau matapōkere, he tukanga kāore e taea te matapae i mua i te putanga. Hei tauira, ko te whiu i tētahi moni, te huri i ngā mataono, te kōwhiri matapōkere rānei i tētahi tangata mai i tētahi rōpū.
Ko ngā putanga pea o tētahi whakamātautau matapōkere ka kiia ko te wāhi tauira, ā, ko te tikanga ka tohua ko \( S \) ko \( \Omega \ rānei). Hei tauira:
– Whiua he moni: \( S = \{Pikitia, Tau\} \)
– Hurihia ngā mataono: \( S = \{1,2,3,4,5,6\} \)
He huinga iti o te wāhi tauira te takahanga. Hei tauira:
– Te takahanga o te whiwhi tau taurite: \( A = \{2,4,6\} \)
– Te takahanga o te whiwhinga o tētahi tau nui atu i te 4: \( B = \{5,6\} \)
3. Ngā Ture Taketake o te Tūponotanga
Mena he ōrite te tūponotanga o ngā putanga katoa i roto i te wāhi tauira (equiprobable), ka taea te tatau i te tūponotanga o tētahi huihuinga \( A \) penei:
\[
P(A) = \frac{\text{te maha o ngā putanga e tautoko ana i te } A}{\text{te maha o ngā putanga katoa i roto i te } S}
\]
Tauira: te tūponotanga o te whiwhi taurite i runga i ngā mataono:
\[
P(A)=\frac{3}{6}=0,5
\]
Ētahi ture nui:
1. Ngā herenga tūponotanga:
\[
0 \le P(A) \le 1
\]
2. Te tūponotanga o te tāpiritanga (kāore he huihuinga e puta):
\[
P(A^c)=1-P(A)
\]
3. Te ture tāpiri mō ngā takahanga e rua:
– Mēnā he rerekē tetahi i tetahi a \( A \) me \( B \) (kāore e taea te puta i te wā kotahi):
\[
P(A √ B)=P(A)+P(B)
\]
– Ki te kore e whakawehe tetahi i tetahi:
\[
P(A kapu B)=P(A)+P(B)-P(A kapu B)
\]
4. Te Tūponotanga Ā-Whārite me te Mana Motuhake
Ko te tūponotanga ā-tikanga ko te tūponotanga ka puta he huihuinga \( A \), mena kua puta kē taua huihuinga \( B \). I tuhia:
\[
P(A|B)=\frac{P(A \cap B)}{P(B)}
\]
me te \( P(B) \ne 0 \).
He mea nui tēnei ariā i roto i ngā āhuatanga maha, pērā i te tautuhi i tētahi mate i runga i ngā hua o te whakamātautau. Mēnā e mōhio ana tātou kua pāngia tētahi e ētahi tohu, ka rerekē pea te tūponotanga ka tautuhia ia he mate.
E kiia ana he tū motuhake ngā takahanga e rua \( A \) me \( B \) mēnā kāore te putanga mai o \( A \) e pā ki te tūponotanga o te putanga mai o \( B \). Mā te pāngarau:
\[
P(A \cap B)=P(A)\cdot P(B)
\]
he rite rānei ki:
\[
P(A|B)=P(A)
\]
5. Ngā Taurangi Matapōkere me ngā Tohatoha Tūponotanga
I roto i ngā tatauranga me te tūponotanga, he maha ngā wā ka whakamahia e tātou ngā taurangi matapōkere, arā, ngā mahi e hono ana i ngā hua o ngā whakamātautau matapōkere ki ngā tau. Ka wehea ngā taurangi matapōkere ki:
1. Momotu: ka taea te tatau i te uara (hei tauira, te tokomaha o ngā tamariki i roto i te whānau).
2. Tonu: ka taea te uara i roto i tētahi awhe (hei tauira, te teitei, te wā tatari).
Mō ngā taurangi matapōkere motumotu, e mōhio ana tātou ki te mahi papatipu tūponotanga (PMF), mō ngā taurangi tonu ia, e mōhio ana tātou ki te mahi kiato tūponotanga (PDF).
Ngā tauira o ngā tohatoha motuhake nui:
– Te tohatoha Bernoulli: e rua noa ngā putanga, te angitu (1) me te korenga (0).
– Te tohatoha rua-ira: te maha o ngā angitu mai i ngā whakamātautau Bernoulli \( n \)
– Te tohatoha Poisson: ka tatau i te maha o ngā takahanga i roto i tētahi wā/wāhi kua whakaritea.
Ngā tauira o ngā tohatoha tonu:
– Te tohatoha noa: he tohatoha "pere" e puta pinepine ana i roto i ngā āhuatanga taiao me ngā āhuatanga pāpori.
– Te Tohatoha Taupūnga: e whakamahia ana hei whakatauira i te wā i waenga i ngā huihuinga, pērā i te wā i waenga i te taenga mai o te kiritaki.
6. Ngā Inenga o te Whakapūmautanga me te Mararatanga
Ko ngā tatauranga whakaahua e hiahia ana ki ngā mehua e whakarāpopoto ana i ngā raraunga.
Te ine i te pokapū:
– Te toharite (toharite): te tapeke o ngā raraunga ka wehea ki te maha o ngā raraunga.
– Tau waenga: te uara waenga i muri i te whakarōpūtanga o ngā raraunga.
– Aratau: te uara e puta pinepine ana.
Rahi horapa:
– Awhe: te rerekētanga i waenga i ngā uara mōrahi me ngā uara mōkito.
– Rerekētanga: he ine i te rerekētanga tapawhā toharite mai i te toharite.
– Paerewa rerekētanga: te pūtake tapawhā o te rerekētanga, he māmā ake te mārama ki ngā waeine ōrite ki ngā raraunga.
He mea nui ēnei inenga hei tirotiro mēnā kei te tata te raraunga ki tētahi uara kua whakaritea, kei te horapa whānui rānei.
7. Te Ariā o te Tauira, te Taupori, me te Whakatau Tata
I roto i ngā rangahau, kāore e taea te ine i te taupori katoa nā te iti o te utu me te wā. Nō reira, ka tangohia he tauira. Mai i tēnei tauira, ka tatauhia e mātou ngā tatauranga (hei tauira, te toharite tauira) hei whakatau tata i ngā tawhā taupori (hei tauira, te toharite taupori).
Ko te tukanga whakatau tata e kiia ana ko te whakatau tata. Ka taea ngā whakatau tata:
– Whakatau tata: kotahi te uara whakatau tata (hei tauira: toharite tauira).
– Te wā whakawhirinaki: he whānuitanga o ngā uara e whakaponohia ana kei roto te tawhā taupori me tētahi taumata whakawhirinaki (hei tauira, 95%).
8. Te Whakamātautau Whakaaro (Tirohanga Whānui)
Kei roto hoki i ngā tatauranga whakatau te whakamātautau whakapae, he tikanga mō te whakamātautau i ngā kereme mō tētahi taupori. Hei tauira, ka hiahia pea tētahi kamupene ki te mōhio mēnā kua heke te wā whakaputa toharite i muri i te whakatinanatanga o tētahi tikanga hou.
Ko te whakamātautau whakapae ko te tikanga:
– Te whakapae kore (\( H_0 \)): te tauākī tuatahi (hei tauira, "kāore he huringa").
– Te whakapae rerekē (\( H_1 \)): te kōrero hei whakamātau (hei tauira, "kua heke te wā whakaputa").
– Ko te uara-p, te rohenga matua rānei mō te whakatau ki te whakaae, ki te whakakore rānei \( H_0 \).
Ahakoa te āhua uaua o tēnei ariā i te tīmatanga, ko te mea nui ko te whakatau i ngā whakataunga i runga i ngā taunakitanga raraunga me te mōrearea o te hapa ka taea te ine.
Te Katinga
Mā ngā kaupapa matua o te tūponotanga tatauranga ka mārama tātou ki te koretake me te whakatau i ngā whakatau mai i ngā raraunga. Mai i ngā ariā o ngā wāhi tauira me ngā takahanga, ngā ture tūponotanga, te tūponotanga ā-tikanga, ki ngā taurangi me ngā tohatoha matapōkere, he tūāpapa nui katoa ēnei mō te tātari raraunga, te rangahau, me te whakatau kaupapa. I tēnei ao e whakahaerehia ana e ngā raraunga, ehara i te mea he whakaritenga mātauranga anake te mārama ki te tūponotanga me ngā tatauranga, engari he pūkenga mahi hoki e whai hua ana i roto i ngā mara pēnei i te pakihi, te hauora, te hangarau, me ngā pūtaiao pāpori.
Ki te hiahia koe, ka taea e au te hanga i tētahi putanga hangarau ake o tēnei tuhinga (me ngā tauira pātai me te matapakinga) me tētahi putanga māmā ake rānei mō te taumata kura.
