Te Tohatoha Binomial: Te Ariā, Ngā Whakamahinga, me Ngā Tauira
Pendahuluan
He ariā taketake te tohatoha rua-ira i roto i ngā tatauranga me te ariā tūponotanga. Hei tētahi o ngā tohatoha motumotu e whakamahia whānuitia ana, he maha ngā tono a te tohatoha rua-ira i roto i ngā mara pēnei i te rongoā, te ōhanga, te koiora, me ngā pūtaiao pāpori. Ka matapakihia e tēnei tuhinga te tohatoha rua-ira i roto i te hōhonutanga, tae atu ki tōna whakamāramatanga, ngā āhuatanga matua, ngā tātai e pā ana, me ētahi tauira tono mahi.
Te Mārama ki te Tohatoha Binomial
E whakaahua ana te tohatoha rua i te putanga o ngā whakamātautau Bernoulli n, e rua noa iho ngā putanga pea o ia whakamātautau: "angitu" "kore rānei." Hei tauira, i roto i te whiunga moni, ko ngā putanga pea ko "upoko" "upoko" rānei.
Ko ngā tawhā matua e rua i roto i te tohatoha binomial ko:
1. n (te maha o ngā whakamātautau)
2. p (tūponotanga o te angitu i ia whakamātautau)
I roto i te āhua whānui, ka taea te whakaahua i te maha o ngā angitu i roto i ngā whakamātautau n mā te tohatoha rua-ira \( B(n, p) \).
Te Mahi Papatipu Tūponotanga (PMF)
Ko te mahi papatipu tūponotanga o te tohatoha rua ka hangaia penei:
\[ P(X = k) = \binom{n}{k} p^k (1 – p)^{n – k} \]
Kei hea:
– Ko \( \binom{n}{k} \) he huinga o n k kua whiriwhiria,
– Ko te tūponotanga o te angitu i roto i tētahi whakamātautau kotahi te \( p \),
– Ko te \( k \) te maha o ngā angitu,
– Ko te \( n \) te tapeke o ngā whakamātautau.
Ngā Āhuatanga Matua o te Tohatoha Binomial
He maha ngā āhuatanga matua o te tohatoha rua-ira:
1. Toharite (Toharite): I whiwhi mā te whakarea i te maha o ngā whakamātautau ki te tūponotanga o te angitu i roto i ia whakamātautau. Ko te toharite ko \( \mu = np \).
2. Rerekētanga: Ko te rerekētanga o te tohatoha rua-ira ko te hua o te maha o ngā whakamātautau, te tūponotanga o te angitu, me te tūponotanga o te korenga, arā, \( \sigma^2 = np(1 – p) \).
3. Te Āhua ōrite me te Piko: Ina \( p = 0.5 \), he ōrite te tohatoha rua. Mō \( p < 0.5 \), ka piko te tohatoha ki te taha matau, ā, mō \( p > 0.5 \), ka piko te tohatoha ki te taha maui.
4. Ngā Herenga Uara: Ko te uara rua (k) kei waenganui i te 0 ki te n.
Te Tohatoha Binomial me te Ariā Tepe Matua
He mea nui te tūnga rua-ira i roto i te Kaupapa Here Matua. Ina nui rawa te maha o ngā whakamātautau (n), ka tata te tūnga rua-ira ki te tūnga noa me te toharite \( \mu = np \) me te paerewa rerekētanga \( \sigma = \sqrt{np(1 – p)} \).
Tauira Take Mā te Whakamahi i te Tohatoha Binomial
Ka māmā ake te mārama ki ngā kōrero mō te tohatoha rua-ira mā roto i ngā tauira mahi mai i ngā momo mara. Anei ētahi tono o te ao tūturu:
Tauira 1: Whakamātautau Hua
Mehemea he raina whakaputa a tētahi kamupene hikohiko, ko te tūponotanga o tētahi hua hapa he 0.01. Mena ka tirotirohia e te kamupene ngā hua 100, he aha te tūponotanga ka kitea e rua ngā hua hapa?
Mā te whakamahi i te tātai tohatoha rua:
\[ P(X = 2) = \binom{100}{2} (0.01)^2 (0.99)^{98} \]
Mā te tatau i ngā huinga \(\binom{100}{2}\), kātahi ka whakareatia ki ngā tūponotanga e toe ana, ka puta te hua whakamutunga.
Tauira 2: Rangahau Hauora
I roto i tētahi whakamātautau haumanu o tētahi rongoā mō tētahi mate, ko te tūponotanga ka ora te tūroro i te rongoā ko te 0.8. Mēnā ka whakamātauria te 10 tūroro, he aha te tūponotanga ka ora te 8, neke atu rānei, ngā tūroro?
Hei kimi i tēnei tūponotanga, me tāpiri e tātou ngā tūponotanga o te 8, 9, me te 10 tūroro e ora ana:
\[ P(X \geq 8) = P(X = 8) + P(X = 9) + P(X = 10) \]
Mā te whakamahi i te tātai rua-ira mō ia uara o k (8, 9, me te 10), kātahi ka tāpirihia ngā hua.
Tauira 3: Ngā Whakataunga i roto i te Ōhanga
I roto i tētahi rangahau mākete, e 60% o ngā kaihoko i pai ki tētahi hua hou. Mēnā ka tangohia he tauira matapōkere o ngā kaihoko e 20, he aha te tūponotanga kia 15 pea o rātou i pai ki te hua?
Me whakamahi te tohatoha rua-ira hei tatau i te tūponotanga o te maha o ngā pai mai i te 15 ki te 20:
\[ P(X \geq 15) = P(X = 15) + P(X = 16) + P(X = 17) + P(X = 18) + P(X = 19) + P(X = 20) \]
Mā te whakamahi i taua tikanga anō, ka tatauhia, ka tāpirihia hoki ēnei tūponotanga.
Te Whakamahi i te Hangarau i roto i ngā Tātaitanga Tohatoha Binomial
I tēnei ao matihiko, ehara i te mea ka mahia ā-ringa noa ngā tono tatau tohatoha rua engari ka whakamahia hoki ngā pūmanawa pēnei i a R, Python, ētahi atu tātaitai tatauranga rānei.
Anei tētahi tauira o te whakamahi i a Python hei tatau i te tohatoha binomial:
"`Pitoni
mai i te scipy.stats kawemai binom
n = 10 te maha o ngā whakamātautau
p = 0.8 tūponotanga angitu
k = 8 te maha o ngā angitu e tumanakohia ana
te tūponotanga o te 8 angitu
prob_8 = binom.pmf(k, n, p)
te tūponotanga o te 8 angitu neke atu rānei
prob_ge_8 = 1 – binom.cdf(k-1, n, p)
tāia(f”Te tūponotanga o te 8 angitu: {prob_8}”)
tāia(f”Te tūponotanga o te 8 angitu neke atu rānei: {prob_ge_8}”)
""
Whakamutunga
He ariā matua te tohatoha rua-ira i roto i ngā tatauranga me te tūponotanga. Mā te mārama ki te tohatoha rua-ira, ka taea e tātou te whakamahi i ngā tauira tūponotanga maha ki ngā āhuatanga o te ao tūturu e uru ana ki ngā whakamātautau maha me ngā putanga e rua. Mā te kaha ki te whakamahi i ngā taputapu hangarau ka pai ake te mahi tatau me te tika. Ehara i te mea he mea nui noa iho te tohatoha rua-ira i te taha ariā engari he maha hoki ngā tono mahi e tika ana i roto i ngā momo mara pūtaiao me te ahumahi.
Ko te tumanako ka whakaratohia e tēnei tuhinga he māramatanga hōhonu ake mō te tohatoha rua-ira, ā, ka whakaoho ake i te tūhuratanga o ngā mara o te tatauranga me te tūponotanga.
