Basa reKugovera Binomial: Tsananguro Yakakwana uye Mashandisirwo
Kugoverwa kwebinomial ndeimwe yemigove yeprobability inoshandiswa zvakanyanya muhuwandu hwezviverengero uye probability. Kugoverwa uku kunoratidza huwandu hwebudiriro mumiedzo yakafanana, yakazvimirira, uko bvunzo yega yega ine mhedzisiro mbiri: kubudirira kana kukundikana. Muchinyorwa chino, tichaongorora tsananguro, fomura, hunhu, uye mashandisirwo ebasa rekuparadzira binomial zvakadzama.
Kunzwisisa Kugoverwa kweBinomial
Kugoverwa kwe binomial kunotsanangura huwandu hwe "kubudirira" mumiedzo yakazvimirira ye n, apo:
- Muedzo wega wega unoburitsa mhedzisiro miviri chete: kubudirira kana kukundikana.
- Mikana yekubudirira pamuedzo wega wega i p.
– Mikana yekukundikana ndeye 1 – p.
- Muedzo wega wega wakazvimirira.
Kugoverwa kwebinomial kunoratidzwa seB(n, p), apo n iri nhamba yemiedzo uye p iri mukana wekubudirira mukuyedza kumwe chete.
Fomura Yokugovera Binomial
Kugoverwa kwebinomial kunoverengerwa uchishandisa fomura inotevera:
\[ P(X = k) = \bhinom{n}{k} p^k (1-p)^{nk} \]
Di mana:
– \( P(X = k) \): Mikana yekuwana kubudirira kwe k chaiyo mumiedzo ye n.
– \( \binom{n}{k} \): Musanganiswa wezvinhu n zvakatorwa k.
– \( p \): Mikana yekubudirira pamuedzo wega wega.
– \( n \): Huwandu hwemiedzo yese.
– \( k \): Nhamba yekubudirira inodiwa.
Musanganiswa \(\binom{n}{k}\) unoverengerwa seizvi:
\[ \bhinom{n}{k} = \frac{n!}{k!(nk)!} \]
Hunhu hweBinomial Distribution
1. Kutarisira (Vashoma) uye Kusiyana:
– Tarisiro kana kuti avhareji yekugoverwa kwebinomial ndeye \( \mu = np \).
– Musiyano uri \( \sigma^2 = np(1-p) \).
2. Kuenzana:
– Kugoverwa kwebinomial kwakaenzana kana p = 0.5. Kana p ≠ 0.5, kugoverwa kwacho kunotsamira kurudyi (p < 0.5) kana kuruboshwe (p > 0.5).
3. Kusagadzikana uye Kurtosis:
– Kusarongeka kwekugoverwa kwebinomial ndiko \( \gamma_1 = \frac{1-2p}{\sqrt{np(1-p)}} \).
– Kurtosis iri \( \gamma_2 = \frac{1-6p(1-p)}{np(1-p)} \).
4. Kugoverwa Kunofungidzirwa:
– Kana n yakakura uye p yava kusvika pa0.5, kugoverwa kwebinomial kunogona kufungidzirwa nekugoverwa kwakajairika.
– Kana p iri diki zvikuru uye n iri hombe zvikuru zvekuti np inoramba yakafanana, saka kugoverwa kwebinomial kunogona kufungidzirwa nekugoverwa kwePoisson.
Kushandisa Binomial Distribution
Kugoverwa kwebinomial kunoshandiswa muminda yakaita sebiology, economics, marketing, uye engineering kuratidza zviitiko zvinogona kuratidzwa mumashoko maviri (kubudirira/kukundikana). Heano mimwe mienzaniso chaiyo yekushandiswa kwayo:
Kuedzwa Kwemhando Yechigadzirwa
Ngatitii boka rezvigadzirwa rine mukana we2% wekuva nezvikanganiso. Kana tikaedza mayuniti makumi mashanu ezvigadzirwa, tinogona kushandisa binomial distribution kuverenga mukana wekuwana nhamba yakatarwa yemayuniti asina kururama. Ne n = 50 uye p = 0.02, tinogona kuverenga mukana wekuwana mayuniti asina kururama chaiwo k muboka.
Kuongororwa kweMienzaniso
Semuenzaniso, mukutsvagisa kwemusika, ongororo dzinowanzoitwa nemibvunzo yekuti hongu/kwete. Kana tichida kuziva huwandu hwevanhu vanobvumirana nemashoko ari mumuenzaniso wevanhu zana (tichifunga kuti mukana wekubvumirana we0.7), kugoverwa kwebinomial kunogona kubatsira kufungidzira huwandu hwevanhu vanobvumirana.
Majini
Mukuongorora majini, kugoverwa kwebinomial kunoshandiswa kutevedzera nhaka yehunhu hwakasiyana kubva kuchizvarwa chimwe kuenda kune chinotevera. Semuenzaniso, kana paine mukana we25% wekuti mwana achava nehunhu hwakasiyana hwemajini, tinogona kushandisa kugoverwa kwebinomial kuona mukana wekuti pavana vana, vaviri vachava nehunhu ihwohwo.
Mari neInishuwarenzi
Munyaya dzemari, kugoverwa kwemari mbiri kunogona kushandiswa kuratidza kuitika kwekubhuroka, kubhadhara zvichemo, kana mwero wemubereko pane zvimwe zvinhu zvinozadzisa mamiriro ekubudirira/kukundikana.
Muenzaniso weKuverenga
Ngatitii tinoda kuverenga mukana wekuti, kubva pamari gumi inokandwa, tinowana misoro mitanhatu chaiyo (tichifunga kuti mari dzacho dzakanaka uye p=0.5):
\[ P(X = 6) = \binom{10}{6} (0.5)^6 (0.5)^4 \]
\[ = \frac{10!}{6!4!} (0.5)^{10} \]
\[ = \frac{210}{1024} \]
\[ = 0.205 \]
Saka, mukana wekuwana misoro mitanhatu chaiyo kubva pamari gumi inokandwa i0.205.
Zvishandiso zveKombuta
Munguva yanhasi ye tekinoroji, kugoverwa kwebinomial kunowanzo kuverengerwa uchishandisa software yekuverenga nhamba dzakadai seR, Python, kana maturusi espreadsheet seMicrosoft Excel. Heino muenzaniso wePython script iri nyore uchishandisa raibhurari ye `scipy`:
"'python
kubva ku scipy.stats pinza binom
Semuenzaniso, tinoda kuwana P(X = 6) ye n=10 uye p=0.5
n = 10
p = 0.5
k = zana
prob = binom.pmf(k, n, p)
print(f”Mukana wekuwana misoro chaiyo ye {k} kubva ku {n} coin tosses ndewe {prob:.3f}”)
``
Mhedziso
Kugoverwa kwebinomial chishandiso chakakosha muhuwandu uye mukana, kunyanya pakuongorora zviitiko zvebinary zvakazvimiririra. Kuziva pfungwa iyi kunogona kutibatsira kugadzirisa matambudziko anosanganisira sarudzo dzemari, tsvakiridzo yemusika, mhando yechigadzirwa, majini, uye mamwe mashandisirwo akasiyana-siyana.
Nekunzwisisa basa rekugovera binomial, tinogona kutevedzera nekuverenga mikana yezviitiko nemazvo, uye tobva tasarudza zvichibva pakuongorora kwakasimba kwenhamba. Kufambira mberi muhunyanzvi uye software yenhamba kwakaitawo kuti zvive nyore kuverenga nekuona kugoverwa uku, zvichiita kuti kuve nyore kuwanikwa munzvimbo dzakasiyana-siyana dzekudzidza uye mashandisirwo.
