# Tātai Tohatoha Noa i roto i ngā Tatauranga
Ko te tohatoha noa, e mōhiotia ana ko te tohatoha Gaussian, ko te pihi pere rānei, tētahi o ngā ariā tino taketake o te tatauranga. E kiia ana tōna noho hei tūāpapa mō ngā tātaritanga tatauranga me ngā tātaritanga tūponotanga. Ehara i te mea he maha ngā whakamahinga o tēnei tohatoha i roto i ngā ariā anake, engari i roto anō hoki i ngā tono mahi maha, pērā i te whakahaere mōrearea pūtea, te pūtaiao pāpori, te rongoā, me ētahi atu.
## Whakamāramatanga o te Tohatoha Noa
Ko te tohatoha noa he tohatoha tūponotanga tonu e ōrite ana ki tōna toharite. Arā, ka hangaia he kauwhata whakairoiro o tēnei tohatoha he pihi pere e whānui haere ana i te toharite, e whaiti haere ana i ngā hiku. E rua ngā tawhā matua o tēnei tohatoha: te toharite (μ) me te paerewa rerekētanga (σ).
Mā te toharite ka whakatau i te taunga o te pokapū o te tohatoha, ko te paerewa rerekētanga ia ka ine i te horapa o ngā raraunga huri noa i te toharite. Ka nui ake te paerewa rerekētanga, ka whānui ake, ka poto ake hoki te pihi tohatoha; mā te iti ake o te paerewa rerekētanga, ka whaiti ake, ka pari ake hoki te pihi.
## Te Mahi Tūponotanga o te Kiato
Ko te āhua pāngarau o te mahi kiato tūponotanga (pdf) mō te tohatoha noa koia tēnei:
\[ f(x | \mu, \sigma) = \frac{1}{\sigma \sqrt{2 \pi}} e^{ -\frac{(x-\mu)^2}{2\sigma^2} } \]
Anei:
– He taurangi matapōkere a \( x \).
– Ko \( \mu \) te toharite o te tohatoha.
– Ko te \( \sigma \) te paerewa rerekētanga o te tohatoha.
– Ko \( e \) te pūtake o te logarithm maori, tata ki te 2.71828.
Ka hangaia he kōpiko pere ōrite e te mahi i runga ake nei. Mā te taupū o tēnei mahi i waenga i ngā pūwāhi e rua ka homai te tūponotanga kei waenganui i aua uara e rua te taurangi matapōkere.
## Tohatoha Noa Paerewa
He tohatoha noa te tohatoha noa paerewa me te toharite \( \mu = 0 \) me te paerewa rerekētanga \( \sigma = 1 \). Ko te mahi kiato tūponotanga mō te tohatoha noa paerewa ko:
\[ f(z) = \frac{1}{\sqrt{2 \pi}} e^{ -\frac{z^2}{2} } \]
Anei:
– He taurangi matapōkere a \( z \) e whai ana i te tohatoha noa paerewa.
He maha ngā wā ka whakamahia te tohatoha noa paerewa nā te mea ka taea e tātou te whakataurite i ētahi atu tohatoha noa mā te tukanga e kiia nei ko te "whakataurite". Ko te whakataurite ko te whakawhiti i ngā uara \( x \) o te tohatoha noa \( N(\mu, \sigma) \) ki ngā uara \( z \) o te tohatoha noa paerewa \( N(0, 1) \), mā te whakamahi i te tātai e whai ake nei:
\[ z = \frac{x – \mu}{\sigma} \]
Mā tēnei tukanga ka māmā ake te whakatairite i ngā uara mai i ngā tohatoha noa rerekē mā te mahere i aua uara ki te tauine kotahi.
## Te Whakamahinga me te Hiranga
### 1. Te Ariā Herenga Waenga
He mea tino nui te tohatoha noa i roto i te horopaki o te Central Limit Theorem (CLT). E kī ana te CLT he nui te maha o ngā taurangi matapōkere motuhake ka tata ki te tohatoha noa, ahakoa te āhua o te tohatoha taketake. Ko te tikanga o tēnei ka taea te whakamahi i te tohatoha noa hei whakatau tata i te tohatoha o te toharite tauira, mena he nui te tauira.
### 2. Whakatau Tatauranga
Mā te tohatoha noa ka taea te whakamahi i ngā whakamātautau whakapae, pērā i te whakamātautau-z me te whakamātautau-t. Ka whakamahia e ngā tikanga e rua te tohatoha noa paerewa hei whakatau i te hiranga tatauranga o ngā hua i kitea. Ko te whakamātautau-z te tikanga e whakamahia ana ina nui te rahi o te tauira, ina mōhiotia rānei te paerewa rerekētanga o te taupori, ko te whakamātautau-t ia ka whakamahia ina iti te rahi o te tauira, ina kore rānei e mōhiotia te paerewa rerekētanga o te taupori.
### 3. Tātari Whakamuri
I roto i te tātaritanga whakatauira rārangi, he mea nui te whakaaro he tohatoha noa ngā raraunga hapa. Mā tēnei whakaaro ka taea te tatau i ngā wā whakawhirinaki me te whakamātautau hiranga o ngā tawhā tauira whakatauira. Waihoki, ko te kimi i ngā hapa raraunga, i ngā mea rānei kei waho, ka mahia mā te tirotiro i te tohatoha toenga mō ngā rerekētanga nui mai i te āhua noa.
### 4. Rongoā me te Koiora
I roto i te rongoā, ka whakamahia te tohatoha noa hei whakaahua i te tohatoha o ngā āhuatanga koiora rerekē. Hei tauira, ko te teitei, te toto toto, me ētahi hua whakamātautau taiwhanga he maha ngā wā ka whai i te tohatoha noa. Mā tēnei ka māmā ake te whakatau i ngā uara tapahi mō ngā tātaritanga hauora.
### 5. Pūtea me te Ōhanga
I roto i te pūtea, ka whakamahia te tohatoha noa hei whakatauira i ngā āhuatanga maha, pērā i ngā hokinga mai o ngā hea, ngā reiti huamoni, me ētahi atu. Ahakoa i roto i te mahi, he maha ngā wā ka whakaatuhia e ngā hea he pikinga teitei me te kurtosis, ko te whakaaro he tohatoha noa tonu te pūtake tātari pakari.
## Te Whakatinanatanga me te Tātaitanga
### Te whakamahi i te Python
Mā Python, me ngā whare pukapuka pērā i a NumPy me SciPy, ka taea e ia te whakamahi i te tohatoha noa, ā, ka taea hoki e ia te whakamahi i ēnei whare pukapuka hei whakawhanui me te tuhi i te tohatoha noa:
"`Pitoni
kawemai numpy hei np
kawemai matplotlib.pyplot hei plt
mai i te scipy.stats kawemai tikanga
# Ngā tawhā tohatoha noa
mu = 0 # toharite
tohu = 1 # paerewa rerekētanga
# Raraunga mō te tohatoha noa
x = np.mokowā-ā-ringa(-5, 5, 1000)
y = norm.pdf(x, mu, sigma)
# Kauwhata tohatoha noa
plt.kaupapa(x, y)
plt.xlabel('x')
plt.ylabel('Kiato')
plt.title('Tohatoha Noa N(0, 1)')
plt.whakaatu()
""
I te tauira i runga ake nei, i whakaputahia e mātou ngā raraunga tohatoha noa me te toharite 0 me te paerewa rerekētanga 1, kātahi ka tuhia te mahi tūponotanga kiato.
## Whakamutunga
He mea nui te tūnga noa i roto i ngā tatauranga me te tūponotanga. Nā tōna whakamahinga whānui, mai i te Kaupapa Here Matua ki ngā momo tono mahi pēnei i te tātari whakatautau me te whakamātautau whakapae, koinei tētahi o ngā tūnga tūponotanga tino rongonui me te mea nui. Ko te mārama ki te tātai tūnga noa me te whakamahi pai i tēnei he pūkenga nui mō te hunga e mahi ana i roto i te pūtaiao raraunga, te rangahau, te ōhanga, me te maha atu o ngā mara.
Mā tēnei mōhiotanga, ka taea e tātou te whakatata atu me te whakaoti rapanga tātari maha me te whai hua ake, ā, ka taea ai e tātou te whakatau whakatau pai ake i runga i ngā raraunga me ngā tūponotanga e wātea ana.
