Te Uara e Tūmanakohia ana mō te Tohatoha Binomial
Ko te tohatoha rua-ira tētahi o ngā tohatoha tūponotanga motuhake e kitea whānuitia ana i roto i ngā tatauranga me te tūponotanga. E whakaahua ana tēnei tohatoha i te maha o ngā angitu i roto i tētahi raupapa o ngā whakamātautau rua-ira motuhake (ngā whakamātautau me ngā putanga e rua anake: angitu, kore rānei). Hei mārama ake ki te tohatoha rua-ira, he mea nui kia mārama ki te ariā o te uara e tumanakohia ana, e whakaahua ana i te uara toharite o tētahi whakamātautau e whakahokia ana i roto i te wā roa. Ka arotakehia e tēnei tuhinga te ariā o te uara e tumanakohia ana i roto i te horopaki o te tohatoha rua-ira.
Te Whakamāramatanga o te Tohatoha Binomial
Ka puta te tohatoha rua-ira i roto i ngā horopaki e whakahaerehia ana e tātou he maha o ngā whakamātautau ōrite me te motuhake, ā, e rua ngā putanga motuhake o ia whakamātautau, e kiia ana ko te "angitu" me te "korenga". Hei tauira, te whiu i tētahi moni (ngā pane, ngā hiku rānei), te whakautu i tētahi pātai whakamātautau (pono, teka rānei), tētahi whakamātautau hauora rānei (kua ora, kāore rānei i ora).
E rua ngā tawhā e tautuhia ana te tohatoha rua:
– n , te maha o ngā whakamātautau.
– p , te tūponotanga o te angitu i ia whakamātautau.
I te nuinga o te wā, mēnā he taurangi matapōkere a X e tohu ana i te maha o ngā angitu i roto i ngā whakamātautau n, ka whai a X i tētahi tohatoha rua-ira me ngā tawhā n me p, e tohuhia ana ko X ~ Runga-ira(n, p).
Te Mahi Tūponotanga
Ko te mahi tūponotanga o te tohatoha rua e whai ake nei:
\[ P(X = k) = \binom{n}{k} p^k (1-p)^{nk} \]
kāore i te mana:
– Ko te tauwehenga rua-ira te \( \binom{n}{k} \) , e tatauhia ana ko \( \frac{n!}{k!(nk)!} \).
– Ko te \( k \) te maha o ngā angitu e hiahiatia ana.
– Ko te \( n \) te maha o ngā whakamātautau.
– Ko te tūponotanga o te angitu i ia whakamātautau ko \( p \).
– Ko te \( (1-p) \) te tūponotanga o te korenga i roto i ia whakamātautau.
Uara e Tumanakohia ana
Ko te uara e tumanakohia ana, te toharite rānei o tētahi tohatoha tūponotanga, tētahi o ngā mehua tino nui o te taunga pokapū. Mō te tohatoha rua-ira, ko te uara e tumanakohia ana o tētahi taurangi matapōkere X e whai ake nei i te Runga-ira (n, p) ko:
\[ E(X) = np \]
Te Taunakitanga o te Uara e Tumanakohia ana
Hei mārama he aha te uara e tumanakohia ana o te tohatoha rua-ira he np, ka taea e tātou te whakamahi i te āhuatanga rārangi o te uara e tumanakohia ana, me te tiro me pēhea te tāpiri o ngā taurangi rua ki tā rātou takoha.
Me tautuhi tātou i a \( X \) hei te maha o ngā angitu i roto i ngā whakamātautau rua e n. Mōhio ake, me waiho a \( X_i \) hei taurangi matapōkere e tohu ana i te putanga o te whakamātautau tua-i, me \( X_i = 1 \) mēnā he angitu te whakamātautau tua-i, me \( X_i = 0 \) mēnā he rahua. Kātahi, ka taea e tātou te tuhi i a \( X \) pēnei:
\[ X = X_1 + X_2 + \ldots + X_n \]
I te mea he taurangi rua ia \( X_i \) me te tūponotanga angitu p, ko te uara e tumanakohia ana o \( X_i \) ko:
\[ E(X_i) = 1 \cdot p + 0 \cdot (1-p) = p \]
Ka taea e tātou te whakamahi i te āhuatanga rārangi o te uara e tumanakohia ana mō te uara e tumanakohia ana o X:
\[ E(X) = E(X_1 + X_2 + \ldots + X_n) \]
\[ E(X) = E(X_1) + E(X_2) + \ldots + E(X_n) \]
\[ E(X) = p + p + \ldots + p \]
\[ E(X) = np \]
E whakaatu ana tēnei ko te uara e tumanakohia ana o te tohatoha binomial ko np.
Tauira Whakaahua
Whakaarohia te tauira ina whiua e tātou tētahi moni tika kia tekau ngā wā. Me pātai tātou he aha te uara e tumanakohia ana mō te maha o ngā whiunga ka hua ake he upoko.
I tēnei take:
– n = 10 (te maha o ngā whiunga moni)
– p = 0.5 (te tūponotanga o te whiwhi pane, nā te mea he tika te moni)
Nō reira, ko te uara e tumanakohia ana ko:
\[ E(X) = np = 10 \whakareatia ki te 0.5 = 5 \]
Ko te tikanga o tēnei, mēnā ka whiua e tātou tētahi moni kia tekau ngā wā i roto i te wā roa, i te toharite ka whiwhi tātou i ngā upoko e rima.
Rerekētanga me te Paerewa Rerekētanga
Haunga te uara e tumanakohia ana, he mea nui anō hoki kia mārama ki te rerekētanga me te paerewa rerekētanga o te tohatoha binomial.
Mō te tohatoha rua, ko te rerekētanga \( \sigma^2 \) me te paerewa rerekētanga \( \sigma \) e whai ake nei:
\[ \sigma^2 = np(1-p) \]
\[ \sigma = \sqrt{np(1-p)} \]
Ka ine te rerekētanga i te tawhiti o te horapa o ngā raraunga mai i te uara e tumanakohia ana. Ko te paerewa rerekētanga ko te pūtake tapawhā o te rerekētanga, ā, ka ine hoki i te horapa o ngā raraunga, engari i roto i ngā waeine ōrite ki ngā raraunga taketake.
Whakamutunga
Ko te tohatoha rua-ira he ariā taketake i roto i ngā tatauranga me te tūponotanga, e kitea pinepine ana i roto i ngā tono o te ao tūturu, mai i te pakihi ki ngā pūtaiao pāpori me te koiora. Ko te uara e tumanakohia ana o te tohatoha rua-ira, e tatauhia ana hei np, e whakarato ana i te māramatanga nui ki te maha toharite o ngā angitu i roto i te raupapa o ngā whakamātautau rua-ira. Mā te mārama ki ngā ariā o te uara e tumanakohia ana, te rerekētanga, me te paerewa rerekētanga, ka taea e tātou te mārama ake ki ngā āhuatanga o te tohatoha rua-ira me te whakaahuatanga o ētahi āhuatanga o te oranga o ia rā.
He tino whai hua tēnei mōhiotanga, ehara i te mea mō te tātari raraunga me ngā tatauranga anake, engari mō te whakatau kaupapa hoki e hiahia ana kia aromatawaihia te tūponotanga me te koretake. Mā te mārama ki te uara e tumanakohia ana me te tohatoha rua-ira ka āwhina i a tātou ki te whakatau tika ake, me te whakatau whai whakaaro ake.
