Ngā Ariā Taketake o ngā Taurangi Matapōkere
I roto i ngā tatauranga me te ariā tūponotanga, ko ngā taurangi matapōkere tētahi o ngā ariā tino taketake, e hono ana i te āputa i waenga i ngā takahanga matapōkere me te tātari pāngarau ka taea te ine. Mā roto i ngā taurangi matapōkere, ka taea e tātou te "whakamāori" i ngā hua o tētahi whakamātautau matapōkere—e tito ana i ngā takahanga, i ngā kāwai rānei i te tīmatanga—ki ngā tau ka taea te tukatuka: te tatau i ō rātou tūponotanga, te whakarāpopoto i a rātou me ngā toharite, te ine i ō rātou horapa, tae noa ki te whakatauira i a rātou mā te whakamahi i ngā tohatoha motuhake. Ka matapakihia e tēnei tuhinga ngā ariā taketake o ngā taurangi matapōkere, ō rātou momo, me ngā ariā matua pēnei i te mahi tūponotanga, te mahi tohatoha whakaemi, te uara e tumanakohia ana, me te rerekētanga.
1. He aha te taurangi matapōkere?
I ngā kupu māmā noa iho, he mahi te taurangi matapōkere e hono ana i ia putanga mai i te wāhi tauira ki te tau tūturu. Ko te wāhi tauira te kohinga o ngā putanga katoa ka taea o tētahi whakamātautau matapōkere.
Hei tauira, me kī ka hurihia e tātou he mataono taha-ono. Ko te wāhi tauira ko {1, 2, 3, 4, 5, 6}. Ka taea e tātou te tautuhi i te taurangi matapōkere \(X\) hei "te tau e puta ana i runga i te mataono." Kātahi ka taea e \(X\) te whai uara mai i te 1 ki te 6, me te tūponotanga ōrite mēnā he tika te mataono.
Tētahi atu tauira: ka whiua e tātou ngā moni e rua. Ko te wāhi tauira ko {HH, HT, TH, TT}. Mēnā ka tautuhia e tātou te taurangi matapōkere \(Y\) hei "te maha o ngā pane (H) e puta mai ana", kātahi:
– HH → \(Y = 2\)
– HT → \(Y = 1\)
– TH → \(Y = 1\)
– TT → \(Y = 0\)
I konei ka kite tātou kāore e hiahiatia kia "whakaata" tika ngā taurangi matapōkere i te putanga taketake; he huarahi ēnei ki te tuku uara tau ki ngā putanga matapōkere i runga i ngā hiahia o te tātari.
2. Ngā momo taurangi matapōkere: motuhake me te tonu
I te nuinga o te wā, ka wehea ngā taurangi matapōkere ki ngā momo matua e rua:
a) Ngā taurangi matapōkere motuhake
Ko te taurangi matapōkere motuhake he taurangi matapōkere ka taea te tatau i ōna uara takitahi (tātai), i te nuinga o te wā he tauoti, he huinga uara motuhake rānei.
Tauira:
– Te maha o ngā tamariki i roto i te whānau (0, 1, 2, 3, …)
– Te maha o ngā waka e haere ana i te pou utu i roto i te 1 meneti
– Te maha o ngā taonga hapa i roto i ngā hua 10 i tirotirohia
Mō ngā taurangi matapōkere motumotu, ka taea te whakaatu tika i te tūponotanga o ia uara i roto i te āhua o tētahi pānga papatipu tūponotanga.
b) Ngā taurangi matapōkere tonu
Ko te taurangi matapōkere tonu he taurangi matapōkere ka taea te tango uara i runga i te wā tonu i runga i te rārangi tau tūturu (kāore e taea te tatau), hei tauira, ko ngā uara katoa i waenga i te 0 me te 1, ko ngā uara tūturu pai katoa rānei.
Tauira:
– Te teitei o te tangata
– Te roa o te tatari a te kiritaki ki te kaute
– Te pāmahana o te hau i tētahi hāora kua whakaritea
Mō tētahi taurangi matapōkere tonu, ko te tūponotanga i tētahi pūwāhi he kore noa iho. Nō reira, ka tatauhia te tūponotanga i runga i tētahi whānuitanga o ngā uara (hei tauira, i waenga i te 10 me te 12 meneti), mā te whakamahi i te mahi kiato tūponotanga.
3. Ngā mahi tūponotanga: PMF me PDF
Ko te ariā nui e whai ake nei ko te pēhea e "honoa" ai te tūponotanga ki te uara o tētahi taurangi matapōkere.
a) Te Mahi Papatipu Tūponotanga (PMF)
Mō tētahi taurangi matapōkere motuhake \(X\), ko te PMF te mea i tautuhia penei:
\[
p(x) = P(X = x)
\]
me te whakarato i:
1. \(p(x) \ge 0\) mō ngā \(x\) katoa
2. \(\tapeke_x p(x) = 1\)
Tauira māmā: mataono tika
\[
P(X=k)=\frac{1}{6}, \quad k=1,2,3,4,5,6
\]
b) Te Mahi Tūponotanga o te Matū (PDF)
Mō tētahi taurangi matapōkere tonu \(X\), ka whakamahia e mātou te PDF \(f(x)\) kia rite ai te tūponotanga i te wā \([a,b]\):
\[
P(a \le X \le b) = \int_a^bf(x)\,dx
\]
me te whakarato i:
1. \(f(x) \ge 0\)
2. \(\int_{-\infty}^{\infty} f(x)\,dx = 1\)
He mea tika kia whakanuia: mō tētahi taurangi matapōkere tonu, \(P(X=x)=0\) mō ia uara o \(x\). He mea nui tonu te tūponotanga ina kōrerohia ngā awhe.
4. Te mahi tohatoha tāpiri (CDF)
Ahakoa he motumotu, he tonu rānei, ka taea te whakaahua i ngā taurangi matapōkere mā te mahi tohatoha whakaemi (CDF), e tautuhia ana penei:
\[
F(x) = P(X \le x)
\]
He maha ngā āhuatanga nui o te CDF:
– Kei waenganui i te 0 me te 1 te uara o \(F(x)\)
– Kāore a \(F(x)\) e heke (kāore e heke)
– \(\lim_{x\to -\infty}F(x)=0\) me \(\lim_{x\to\infty}F(x)=1\)
Mō ngā taurangi motuhake, he āhua "arapae" te CDF (e piki ana i ētahi wāhi). Mō ngā taurangi tonu, he maeneene te CDF, ā, koia te wāhanga matua o te PDF:
\[
F(x)=\int_{-\infty}^{x} f(t)\,dt
\]
5. Te ine i te ia matua: te uara e tumanakohia ana (te tumanakohanga)
Kia mōhio tātou ki te tohatoha tūponotanga, he maha ngā wā ka hiahia tātou ki te whakarāpopoto i te taurangi matapōkere me tētahi tau kotahi e tohu ana i tōna "uara toharite mō te wā roa." Koinei te uara e tumanakohia ana, te tumanako rānei.
a) Ngā tumanakohanga taurangi motuhake
Mena he motumotu a \(X\):
\[
E[X] = \tapeke_x x\,p(x)
\]
b) Te tumanako ki ngā taurangi tonu
Mena he tonu te \(X\):
\[
E[X] = \int_{-\infty}^{\infty} x\,f(x)\,dx
\]
Kāore te tumanako i te rite tonu ki te "uara e tino puta pinepine ana" (aratau), ā, ehara i te mea ko te uara tino tūpono ka puta tonu, engari he tino whai hua mō te whakatau kaupapa, te matapae, me te tātari mōrearea.
Tauira tono: I roto i te pakihi, ka taea te whakamahi i ngā tumanakohanga hei tatau i te hua toharite e tumanakohia ana o tētahi rautaki, me te whai whakaaro ki ngā horopaki me ō rātou tūponotanga.
6. Ngā inenga o te marara: te rerekētanga me te paerewa rerekētanga
E rua ngā taurangi matapōkere e whai ana i te tumanako kotahi engari he rerekē ngā taumata o te koretake. Nō reira, me whai inenga tātou mō te marara, arā, te rerekētanga me te paerewa rerekētanga.
Ko te rerekētanga o \(X\) e tautuhia ana ko:
\[
Var(X)=E[(XE[X])^2]
\]
Ko te paerewa rerekētanga ko te pūtake tapawhā o te rerekētanga:
\[
\sigma = \sqrt{Var(X)}
\]
Ngā tātai whai hua e whakamahia pinepinetia ana:
\[
Var(X) = E[X^2] – (E[X])^2
\]
Ka nui ake te rerekētanga, ka nui ake te horapa o ngā uara \(X\) mai i te toharite, ko te tikanga he nui ake te koretake.
7. Ngā tohatoha tūponotanga e whakamahia pinepinetia ana
I roto i te mahi, he maha ngā taurangi matapōkere e whai ana i ētahi tauira tohatoha. Ko ētahi tohatoha rongonui ko:
– Bernoulli: e rua ngā putanga (angitu/korenga), hei tauira, pono-hē, ora-mate.
– Binomial: te maha o ngā angitu mai i ngā whakamātautau \(n\) Bernoulli, hei tauira te maha o ngā ākonga i puta i te 20 tāngata.
– Poisson: te maha o ngā takahanga i roto i tētahi wā/wāhi, hei tauira te maha o ngā waeatanga taumai ia meneti.
– Tonu ōrite: he ōrite te tūponotanga o ngā uara katoa i roto i te wā.
– Noa (Gaussian): he maha ngā āhuatanga taiao me ngā āhuatanga pāpori e whakatata ana ki tēnei tohatoha, pērā i te teitei, te hapa ine rānei.
Mā te whiriwhiri i te tohatoha tika ka pai ake te whakatauira me te tātari.
8. He aha te hiranga o ngā taurangi matapōkere?
Ko ngā taurangi matapōkere te pūtake mō:
– Ngā tatauranga whakatau tata: te whakatau tata i ngā tawhā taupori i runga i ngā tauira
– Te whakamātautau whakapae: te whakatau mēnā kei te tautokona tētahi kereme e ngā raraunga
– Te ako mīhini: te whakatauira i te koretake me te tūponotanga matapae
– Te whakahaere mōrearea: te ine i te tūponotanga o ngā mate me ngā āhuatanga tino kino
– Te hangarau me te pūtaiao: te tukatuka tohu, te pono o te pūnaha, te ariā rarangi
Mō ngā taurangi matapōkere, he reo pāngarau tā tātou hei kōrero pūnaha mō te koretake.
Whakamutunga
Ko te taurangi matapōkere he ariā matua i roto i te ariā tūponotanga e mahere ana i ngā putanga o ngā whakamātautau matapōkere ki ngā uara tau. Ka taea e ngā taurangi matapōkere te noho motuhake, te haere tonu rānei, ā, he rerekē te huarahi o ia taurangi ki te whakaatu i ngā tūponotanga mā te PMF, PDF rānei. Hei tāpiri, ka whakaratohia e te CDF he huarahi noa hei tiro i te kohikohinga o ngā tūponotanga. Hei whakarāpopoto i tētahi tohatoha, ka whakamahia te tumanako hei ine i te ia matua me te rerekētanga/paerewa rerekētanga hei ine i te marara. Mā te mārama ki ēnei ariā taketake ka āwhina i te ako i ngā kaupapa matatau ake pēnei i ngā tohatoha tūponotanga, te whakatau tatauranga, te whakatauira, me te whakatauira mōrearea me te tātari raraunga hou.
Ki te hiahia koe, ka taea hoki e au te tāpiri i ngā tauira pātai me ā rātou matapakinga (motuhake me te haere tonu) kia māmā ake ai te mārama ki te ariā o ngā taurangi matapōkere.
