Ngā Tauira Pātai me te Kōrero mō te Tohatoha Tūponotanga
Ko te tohatoha tūponotanga tētahi o ngā ariā matua o te tatauranga me te tūponotanga. Ka whakamahia hei mārama ki te tūponotanga o ngā uara rerekē o tētahi tau matapōkere. He maha ngā āhua o ngā tohatoha tūponotanga i runga i te āhua o ngā raraunga e tātarihia ana. Ko ngā momo tohatoha tūponotanga e rua e tino kitea ana ko te motumotu me te haere tonu. I roto i tēnei tuhinga, ka arotakehia e mātou ētahi tauira raruraru, ka matapakihia hoki ngā tohatoha tūponotanga hei āwhina i a mātou ki te mārama ake ki tēnei kaupapa.
Tohatoha Motuhake
Ko te tohatoha motuhake he tohatoha e tatau ana i te tūponotanga o tētahi taurangi matapōkere motuhake, arā, he taurangi ka taea anake te tango i ētahi uara. Ko ngā tauira rongonui o ngā tohatoha motuhake ko te Tohatoha Binomial me te Tohatoha Poisson.
Tauira 1: Tohatoha Binomial
Ko te tohatoha rua-ira e whakaahua ana i te maha o ngā angitu i roto i tētahi raupapa o ngā whakamātautau Bernoulli. E rua ngā putanga o ia whakamātautau Bernoulli: te angitu, te kore rānei. Ka noho pūmau te tūponotanga o te angitu puta noa i te whakamātautau.
Pātai:
Kei te whakamātautau tētahi kamupene rongoā i tētahi rongoā hou ki te 10 tūroro. Ko te tūponotanga ka mahi te rongoā i roto i tētahi tūroro ko te 0.7. Tātaihia te tūponotanga ka mahi te rongoā i roto i te 7 o roto i te 10 tūroro.
Kōrero:
Ko te taurangi matapōkere \(X\) e whai ana i tētahi tohatoha rua-ira me \(n = 10\) me \(p = 0.7\). Ko te mahi tūponotanga rua-ira ko:
\[ P(X = k) = \binom{n}{k} p^k (1 – p)^{n – k} \]
Mō \(k = 7\):
\[ P(X = 7) = \binom{10}{7} (0.7)^7 (0.3)^3 \]
Te tatau i te tauwehenga rua-ira \(\binom{10}{7}\):
\[ \binom{10}{7} = \frac{10!}{7!(10-7)!} = \frac{10!}{7!3!} = 120 \]
Te tatau i ngā uara tūponotanga:
P(X = 7) = 120 \times (0.7)^7 \times (0.3)^3 \]
\[ P(X = 7) \tata ki te 120 \whakaahua 0.0823543 \whakaahua 0.027 \]
\[ P(X = 7) \tata ki te 0.231 \]
Nō reira, ko te tūponotanga ka mahi te rongoā i roto i te 7 o roto i te 10 o ngā tūroro he tata ki te 0.231, arā, 23.1%.
Tauira 2: Tohatoha Poisson
Ka whakamahia te tohatoha Poisson hei whakatauira i te maha o ngā putanga mai o tētahi huihuinga onge i roto i tētahi wā, i tētahi āputa rānei kua whakaritea.
Pātai:
E whā ngā kaihoko i te haora e whiwhi ana te toa i te toharite. He aha te tūponotanga ka whiwhi te toa i te rima tonu ngā kaihoko i roto i te haora kotahi?
Kōrero:
Ko te taurangi matapōkere \(X\) e whai ana i te tohatoha Poisson me te tawhā \(\lambda = 4\). Ko te mahi papatipu tūponotanga Poisson ko:
\[ P(X = k) = \frac{\lambda^ke^{-\lambda}}{k!} \]
Mō \(k = 5\):
\[ P(X = 5) = \frac{4^5 e^{-4}}{5!} \]
Tatau:
\[ P(X = 5) = \frac{1024 \cdot e^{-4}}{120} \]
\[ P(X = 5) \approx \frac{1024 \cdot 0.0183}{120} \]
\[ P(X = 5) \tata ki te 0.156 \]
Nō reira, ko te tūponotanga ka whiwhi te toa i te 5 tonu ngā kiritaki i roto i te haora kotahi he 0.156, arā, 15.6%.
Tohatoha Tonu
Ka whakamahia ngā tohatoha tonu ina taea e te taurangi matapōkere e inehia ana te tango i tētahi uara i roto i tētahi whānuitanga. Ko ngā tauira rongonui o ngā tohatoha tonu ko te Tohatoha Noa me te Tohatoha Taupū.
Tauira 3: Tohatoha Noa
Ko te Tohatoha Noa, e kiia ana ko te Tohatoha Gaussian, he tohatoha e whakamahia whānuitia ana i roto i ngā mara maha, tae atu ki te pūtaiao, te miihini, me te ōhanga.
Pātai:
He rite tonu te horapa o te teitei o ngā tāne pakeke i roto i tētahi tāone, ko te toharite he 170 cm, ā, ko te paerewa rerekētanga he 10 cm. He aha te tūponotanga ka tohua matapōkeretia tētahi tāne kei waenganui i te 160 cm me te 180 cm te teitei?
Kōrero:
Me tatau tātou i te kaute-z mō te 160 cm me te 180 cm. Ko te tautuhi o te kaute-z koia tēnei:
\[ Z = \frac{X – \mu}{\sigma} \]
Mō \(X = 160\):
\[ Z_{160} = \frac{160 – 170}{10} = -1 \]
Mō \(X = 180\):
\[ Z_{180} = \frac{180 – 170}{10} = 1 \]
Me titiro tātou ki ngā uara tūponotanga mai i te -1 ki te 1 i te ripanga z. Ko te uara mai i te z = -1 ki te z = 1 he tata ki te 0.6826.
Nō reira, ko te tūponotanga kei waenga i te 160 cm me te 180 cm te roa o tētahi tangata i tīpakohia matapōkeretia, tata ki te 0.6826, arā, 68.26%.
Tauira 4: Te Tohatoha Taupū
Ka whakamahia te Tohatoha Taupūtanga hei whakatauira i te wā i waenga i ngā takahanga i roto i tētahi tukanga Poisson.
Pātai:
Ko te wā toharite i waenga i te taenga mai o ngā kiritaki e rua ki tētahi toa he 15 meneti. He aha te tūponotanga ka iti iho i te 10 meneti te wā i waenga i te taenga mai o ngā kiritaki e rua?
Kōrero:
Kei te Tohatoha Taupū he tawhā \(\lambda\) koia te whakahurihanga o te toharite (\(\mu\)). Me te toharite o te 15 meneti:
\[ \lambda = \frac{1}{\mu} = \frac{1}{15} = 0.0667 \]
Ko te mahi tohatoha tāpiripiri taupū ko:
\[ P(X \leq x) = 1 – e^{-\lambda x} \]
Mō \(x = 10\):
\[ P(X \leq 10) = 1 – e^{-0.0667 \times 10} \]
\[ P(X \leq 10) = 1 – e^{-0.667} \]
\[ P(X \leq 10) \approx 1 – 0.5134 \]
\[ P(X \leq 10) \tata ki te 0.4866 \]
Nō reira, ko te tūponotanga he iti iho i te 10 meneti te roa o te wā i waenganui i te taenga mai o ngā kiritaki e rua, he tata ki te 0.4866, arā, 48.66%.
Whakamutunga
He tino whai hua ngā tohatoha tūponotanga, arā, te motumotu me te haere tonu, mō te whakatauira me te mārama ki te whanonga o ngā taurangi matapōkere. He maha ngā wā ka whakamahia ngā tohatoha rua me te Poisson mō ngā taurangi motumotu, ko ngā tohatoha noa me te taupūnga he tauira o ngā tohatoha haere tonu.
Mā ngā tauira i runga ake nei, ko te tumanako kua mārama ake koe ki te tatau me te whakamārama i ngā tūponotanga i roto i ngā tohatoha tūponotanga. Mā te mahi tonu, ka pai ake tō pūkenga ki te mārama ki ngā tohatoha tūponotanga, ā, ka taea te whakamahi puta noa i ngā momo marautanga.
