Te Mahi Tohatoha Binomial: Te Whakamārama Katoa me ngā Whakamahinga
Ko te tohatoha rua-ira tētahi o ngā tohatoha tūponotanga motuhake e whakamahia whānuitia ana i roto i ngā tatauranga me te tūponotanga. Ka whakatauirahia e tēnei tohatoha te maha o ngā angitu i roto i tētahi raupapa o ngā whakamātautau ōrite, motuhake, e rua ngā putanga pea o ia whakamātautau: te angitu, te kore rānei. I roto i tēnei tuhinga, ka tūhuratia e mātou te whakamāramatanga, te tātai, ngā āhuatanga, me ngā tono o te mahi tohatoha rua-ira.
Te Mārama ki te Tohatoha Binomial
E whakaahua ana te tohatoha rua-ira i te maha o ngā "angitu" i roto i ngā whakamātautau motuhake e n, ina:
– E rua noa ngā putanga ka puta i ia whakamātautau: te angitu, te kore rānei.
– Ko te tūponotanga o te angitu i ia whakamātautau ko p.
– Ko te tūponotanga o te rahunga he 1 – p.
– He motuhake ia whakamātautau tetahi i tetahi.
Ko te tohatoha rua-ira ka tohua ko B(n, p), ko n te maha o ngā whakamātautau, ā, ko p te tūponotanga o te angitu i roto i tētahi whakamātautau kotahi.
Tātai Tohatoha Binomial
Ka tatauhia te tohatoha rua-ira mā te whakamahi i te tātai e whai ake nei:
\[ P(X = k) = \binom{n}{k} p^k (1-p)^{nk} \]
Kei hea:
– \( P(X = k) \): Te tūponotanga o te whiwhi i te k angitu i roto i ngā whakamātautau n.
– \( \binom{n}{k} \): Te huinga o ngā mea n i tangohia he k.
– \( p \): Te tūponotanga o te angitu i ia whakamātautau.
– \( n \): Te tapeke o ngā whakamātautau.
– \( k \): Te maha o ngā angitu e hiahiatia ana.
Ka tatauhia te huinga \(\binom{n}{k}\) penei:
\[ \binom{n}{k} = \frac{n!}{k!(nk)!} \]
Ngā Āhuatanga o te Tohatoha Binomial
1. Te Tumanako (Toharite) me te Rerekētanga:
– Ko te tumanako, ko te toharite rānei o te tohatoha rua-ira ko \( \mu = np \).
– Ko te rerekētanga ko \( \sigma^2 = np(1-p) \).
2. Te ōritetanga:
– He ōrite te tohatoha rua-ira mēnā ko p = 0.5. Mēnā ko p ≠ 0.5, ka piko te tohatoha ki te taha matau (p < 0.5) ki te taha maui rānei (p > 0.5).
3. Te Piko me te Kurtosis:
– Ko te piko o te tohatoha rua ko \( \gamma_1 = \frac{1-2p}{\sqrt{np(1-p)}} \).
– Ko te kurtosis he \( \gamma_2 = \frac{1-6p(1-p)}{np(1-p)} \).
4. Te Tohatoha Tata:
– Mō te n me te p nui e tata ana ki te 0.5, ka taea te whakatau tata i te tohatoha rua mā te tohatoha noa.
– Mena he tino iti a p, ā, he tino nui a n, kia mau tonu ai te np, ka taea te whakatau tata i te tohatoha rua mā te tohatoha Poisson.
Te Whakamahi i te Tohatoha Binomial
E whakamahia ana te tohatoha rua-ira i roto i ngā mara pērā i te koiora, te ōhanga, te hokohoko, me te hangarau hei whakatauira i ngā kaupapa ka taea te whakaatu i roto i ngā kupu rua-ira (angitu/korenga). Anei ētahi tauira tūturu o tōna whakamahinga:
Whakamātautau Kounga Hua
Me kī, e 2% te tūponotanga o tētahi puranga hua kia hapa. Mēnā ka whakamātauhia e tātou he 50 waeine o te hua, ka taea e tātou te whakamahi i te tohatoha rua-ira hei tatau i te tūponotanga o te kitea o tētahi maha o ngā waeine hapa. Me n = 50 me p = 0.02, ka taea e tātou te tatau i te tūponotanga o te kitea o te k waeine hapa i roto i te puranga.
Aromatawai Tauira
Hei tauira, i roto i ngā rangahau mākete, he maha ngā wā ka whakahaerehia ngā rangahau me ngā pātai āe/kāo. Mēnā e hiahia ana tātou ki te mōhio ki te tokomaha o ngā kaiwhakautu e whakaae ana ki tētahi kōrero i roto i tētahi tauira o te 100 tāngata (me te whakaaro ko te tūponotanga o te whakaae he 0.7), ka taea e te tohatoha rua te āwhina i te whakatau tata i te tokomaha e tumanakohia ana o ngā tāngata e whakaae ana.
Ngā ira
I roto i te ira, ka whakamahia te tohatoha rua-ira hei whakatauira i te tukunga iho o ētahi āhuatanga mai i tētahi whakatipuranga ki tētahi whakatipuranga. Hei tauira, mēnā he 25% te tūponotanga ka whai āhuatanga ira tētahi uri, ka taea e tātou te whakamahi i te tohatoha rua-ira hei whakatau i te tūponotanga ka whai āhuatanga taua uri e rua i roto i ngā uri e whā.
Pūtea me te Inihuarangatia
I roto i te pūtea, ka taea te whakamahi i te tohatoha rua-ira hei whakatauira i te putanga mai o ngā pirau, ngā utu kereme, ngā reiti huamoni rānei mō ētahi taonga e tutuki ana i ngā tikanga angitu/rahunga.
Tauira Tātaitanga
Me kī tātou e hiahia ana ki te tatau i te tūponotanga, mai i ngā whiunga moni e 10, ka whiwhi tātou i te 6 ngā upoko (mehemea he ōrite ngā moni, ā, ko te p=0.5):
\[ P(X = 6) = \binom{10}{6} (0.5)^6 (0.5)^4 \]
\[ = \frac{10!}{6!4!} (0.5)^{10} \]
\[ = \frac{210}{1024} \]
\[ = 0.205 \]
Nō reira, ko te tūponotanga kia 6 tonu ngā upoko i roto i ngā whiunga moni e 10 he 0.205.
Ngā Taupānga Rorohiko
I tēnei ao hangarau o ēnei rā, ka tatauhia ngā tohatoha rua-ira mā te whakamahi i ngā pūmanawa tatauranga pēnei i a R, Python, ngā taputapu ripanga tātaitai pēnei i a Microsoft Excel. Anei tētahi tauira o tētahi tuhinga Python māmā e whakamahi ana i te whare pukapuka `scipy`:
"`Pitoni
mai i te scipy.stats kawemai binom
Hei tauira, e hiahia ana tātou ki te kimi i te P(X = 6) mō n=10 me te p=0.5
n = 10
p = 0.5
k = 6
prob = binom.pmf(k, n, p)
tāia(f”Ko te tūponotanga o te whiwhi i te {k} upoko mai i ngā whiunga moni {n} he {prob:.3f}”)
""
Whakamutunga
He taputapu nui te tohatoha rua-ira i roto i ngā tatauranga me te tūponotanga, inā koa i te tātari i ngā takahanga rua-ira motuhake. Mā te mōhio ki tēnei ariā ka āwhina i a tātou ki te aro atu ki ngā raruraru e pā ana ki ngā whakataunga pūtea, te rangahau mākete, te kounga hua, te ira, me te maha atu o ngā tono.
Mā te mārama ki te mahi tohatoha rua-ira, ka taea e tātou te whakatauira me te tatau i ngā tūponotanga o ngā takahanga me te tika, ā, ka taea hoki te whakatakoto whakatau i runga i te tātari tatauranga pakari. Nā ngā whanaketanga hangarau me ngā pūmanawa tatauranga kua māmā ake te tatau me te whakaatu i tēnei tohatoha, ā, kua māmā ake te uru atu ki roto i te whānuitanga o ngā mara ako me ngā tono.
