Kuparadzirwa kweBinomial: Dzidziso, Mashandisirwo, uye Mienzaniso
Pendauluan
Kugoverwa kwebinomial ipfungwa huru muzvidzidzo zvehuwandu hwezvinhu uye mukana wekuwana ruzivo. Seimwe yenzira dzinoshandiswa zvakanyanya dzekugoverwa kwebinomial, kugoverwa kwebinomial kune mashandisirwo akasiyana-siyana muminda yakaita semishonga, economics, biology, uye social sciences. Chinyorwa chino chichakurukura zvakadzama nezvekugoverwa kwebinomial, kusanganisira tsananguro yayo, hunhu hwakakosha, mafomula ane chekuita nayo, uye mienzaniso yakati wandei yekushandisa.
Kunzwisisa Kugoverwa kweBinomial
Kugoverwa kweBinomial kunotsanangura mhedzisiro yemiedzo yeN Bernoulli, uko muedzo wega wega une mhedzisiro mbiri chete: "kubudirira" kana "kukundikana." Semuenzaniso, mukukanda mari, mhedzisiro inogona kunge iri "misoro" kana "misoro."
Ma parameter maviri makuru mukugoverwa kwe binomial ndeaya:
1. n (nhamba yemiedzo)
2. p (mukana wekubudirira mukuyedzwa kwega kwega)
Zviri pamutemo, huwandu hwebudiriro mumiedzo ye n hunogona kutsanangurwa nekugoverwa kwebinomial \( B(n, p) \).
Basa reKuwanda Kwemukana (PMF)
Basa rekuti mukana wehuwandu hwe ...
\[ P(X = k) = \binom{n}{k} p^k (1 – p)^{n – k} \]
dimana:
– \( \binom{n}{k} \) musanganiswa we n akasarudzwa k,
– \( p \) mukana wekubudirira mukuyedzwa kamwe chete,
– \( k \) ndiyo nhamba yekubudirira,
– \( n \) ndiyo nhamba yese yemiedzo.
Hunhu Hukuru hweBinomial Distribution
Kugoverwa kwe binomial kune zvinhu zvakakosha zvakasiyana:
1. Avhareji (Avhareji): Inowanikwa kubva pakuwedzera huwandu hwemiedzo nekukwanisa kubudirira mumuedzo wega wega. Avhareji ndeye \( \mu = np \).
2. Kusiyana: Kusiyana kwekugoverwa kwebinomial ndiko kunobva muhuwandu hwemiedzo, mukana wekubudirira, uye mukana wekukundikana, ndiko kuti, \( \sigma^2 = np(1 – p) \).
3. Kuenzana uye Kutsveyama: Kana \( p = 0.5 \), kugoverwa kwebinomial kwakaenzana. Kune \( p < 0.5 \), kugoverwa kwacho kwakakombamiswa kurudyi, uye kune \( p > 0.5 \), kugoverwa kwacho kwakakombamiswa kuruboshwe.
4. Miganhu Yekukosha: Kukosha kwebinomial (k) kunotangira pa0 kusvika pa n.
Kugoverwa kweBinomial uye Theorem yeCentral Limit
Kugoverwa kwebinomial kunoita basa rakakosha muCentral Limit Theorem. Kana huwandu hwemiedzo (n) hwava hukuru kwazvo, kugoverwa kwebinomial kuchasvika pakugoverwa kwakajairika neavhareji \( \mu = np \) uye standard deviation \( \sigma = \sqrt{np(1 - p)} \).
Muenzaniso weChiitiko Uchishandisa Kugoverwa kweBinomial
Kukurukurirana pamusoro pekugoverwa kwebinomial kuchave nyore kunzwisisa kuburikidza nemienzaniso inoshanda kubva muminda yakasiyana-siyana. Heano mamwe mashandisirwo chaiwo:
Muenzaniso 1: Kuedzwa Kwechigadzirwa
Ngatitii kambani yemagetsi ine mutsetse wekugadzira apo mukana wekuti chigadzirwa chive chisina kunaka uri 0.01. Kana kambani ikaongorora zvigadzirwa zana, mukana wekuwana zvigadzirwa zviviri zvisina kunaka ndeupi?
Kushandisa fomura yekugovera yebinomial:
\[ P(X = 2) = \binom{100}{2} (0.01)^2 (0.99)^{98} \]
Nekuverenga misanganiswa \(\binom{100}{2}\), tobva tawedzera neasara mikana, tinowana mhedzisiro yekupedzisira.
Muenzaniso 2: Tsvakurudzo yezvekurapa
Mukuyedzwa kwekiriniki kwemushonga wechirwere chakati, mukana wekuti murwere arapwe nemushonga uyu i0.8. Kana varwere gumi vakaongororwa, mukana wekuti varwere vanosvika vasere varapwe ndeupi?
Kuti tiwane mukana uyu, tinofanira kuwedzera mikana yekuti varwere 8, 9, uye 10 vari kupora:
\[ P(X \geq 8) = P(X = 8) + P(X = 9) + P(X = 10) \]
Nekushandisa fomura yebinomial yehuwandu hwese hwe k (8, 9, uye 10), wobva wasanganisa zvabuda.
Muenzaniso 3: Sarudzo muEconomics
Muongororo yemusika, 60% yevatengi vakafarira chigadzirwa chitsva. Kana vanhu 20 vakatorwa vasingatarisirwi, mukana wekuti vanhu 15 vavo vakafarira chigadzirwa chacho ndeupi?
Tinoda binomial distribution kuti tiverenge mukana wekuti vanhu vangada kubva pa15 kusvika pa20:
\[ P(X \geq 15) = P(X = 15) + P(X = 16) + P(X = 17) + P(X = 18) + P(X = 19) + P(X = 20) \]
Tichishandisa nzira imwecheteyo, tinoverenga uye tinosanganisa mikana iyi.
Kushandisa Tekinoroji muKuverenga Kugoverwa kweBinomial
Munguva yedhijitari, mashandisirwo ekuverenga mabhinomial distribution haangoitwe nemaoko chete asiwo achishandisa software yakaita seR, Python, kana mamwe ma statistical calculator.
Heino muenzaniso wekushandisa Python kuverenga kugoverwa kwebinomial:
"'python
kubva ku scipy.stats pinza binom
n = nhamba gumi yemiedzo
p = 0.8 mukana wekubudirira
k = 8 nhamba inotarisirwa yekubudirira
mukana wekubudirira kasere chaizvo
prob_8 = binom.pmf(k, n, p)
mukana wekubudirira kanenge ka8
prob_ge_8 = 1 – binom.cdf(k-1, n, p)
print(f”Mwero wekubudirira 8 chaidzo: {prob_8}”)
print(f”Mwero wekubudirira kanosvika ka8: {prob_ge_8}”)
``
Mhedziso
Kugoverwa kwebinomial ipfungwa inokosha muhuwandu hwezviverengero uye mukana. Nekunzwisisa kugoverwa kwebinomial, tinogona kushandisa mamodheru akasiyana-siyana emukana kumamiriro ezvinhu chaiwo anosanganisira kuyedza kwakadzokororwa nemhedzisiro mbiri. Kugona kushandisa maturusi etekinoroji kunoita kuti maitiro ekuverenga ave anoshanda uye akarurama. Kugoverwa kwebinomial hakungori kukosha chete mudzidziso asiwo kune mashandisirwo akawanda akakodzera muzvikamu zvakasiyana zvesainzi neindasitiri.
Tinovimba kuti chinyorwa chino chinopa kunzwisisa kwakadzama kwekugoverwa kwebinomial uye chinokurudzira kutsvaga kwakawanda muminda yehuwandu hwezviverengero uye mukana.
