Mienzaniso yeMibvunzo Inokurukura Basa reKugovera Binomial
Kugoverwa kwebinomial idistribution yekufungidzira yakasiyana inotsanangura huwandu hwekubudirira mukuyedza kunosanganisira miedzo yakazvimirira yakawanda ine mhedzisiro mbiri dzinogoneka: kubudirira nekukundikana. Kuedzwa kwega kwega kunonzi muedzo, uye kugoverwa kwebinomial kunowanzo shandiswa mumamiriro ezvinhu apo huwandu hwekubudirira mumiedzo yakazvimirira yakawanda hunonakidza. Muchinyorwa chino, tichakurukura pfungwa huru dzekugoverwa kwebinomial uye kupa mienzaniso nemhinduro.
Pfungwa Dzekutanga dzeBasa reKugovera Binomial
Tisati tapinda mumibvunzo yemuenzaniso nekukurukurirana, ngatikurukurei dzimwe pfungwa huru dzine chekuita nekugoverwa kwebinomial.
1. Tsanangudzo: Kugoverwa kwebinomial kunotsanangurwa sehuwandu hwebudiriro mumiedzo yakazvimirira ye 'n', uko bvunzo yega yega ine mhedzisiro mbiri dzinogoneka: kubudirira (nemukana wep) kana kukundikana (nemukana wep = 1 - p).
2. Basa reKugona: Basa reKugona kweBinomial Distribution ndeiri:
\[
P(X = k) = \bhinom{n}{k} p^k (1-p)^{nk}
\]
di mana:
– \( P(X = k) \) ndiyo mukana wekubudirira kwe k mumiedzo ye n.
– \( \binom{n}{k} \) musanganiswa we n take k, iyo inotsanangurwa se \( \frac{n!}{k!(nk)!} \).
– \( p \) ndiyo mukana wekubudirira pakuedza kwega kwega.
– \( (1-p) \) ndiyo mukana wekukundikana pakuedza kwega kwega.
3. Kukosha Kunotarisirwa uye Kusiyana:
– Kukosha kunotarisirwa (avhareji) kwekugoverwa kwebinomial ndi \( \mu = np \).
– Musiyano wekugoverwa kwebinomial ndewe \( \sigma^2 = np(1-p) \).
Zvino, ngatishandisei pfungwa idzi mumuenzaniso wedambudziko kuti tinzwisise zvakadzama.
Muenzaniso Mubvunzo 1: Kuverenga Kwekutanga Kwekugoverwa KweBinomial
Mubvunzo:
Kambani inogadzira zvikamu zvemagetsi zvine mukana we0.95 wekuti chikamu chimwe nechimwe chipfuure bvunzo yemhando. Kana zvikamu gumi zvagadzirwa, verenga mukana wekuti zvikamu zvisere chaizvo zvipfuure bvunzo yemhando.
Kukurukurirana:
Tinogona kushandisa fomura yekugovera yebinomial kugadzirisa dambudziko iri. Kutanga, tinoona maparamita anotevera:
– \( n \) (huwandu hwemiedzo yese) = 10
– \( k \) (nhamba yekubudirira) = 8
– \( p \) (mukana wekubudirira) = 0.95
– \( q \) (mukana wekukundikana) = 1 – 0.95 = 0.05
Wobva watsiva izvi zvinhu mufomura yekugovera yebinomial:
\[
P(X = 8) = \binom{10}{8} (0.95)^8 (0.05)^2
\]
Kutanga, verenga musanganiswa \( \binom{10}{8} \):
\[
\binom{10}{8} = \frac{10!}{8!(10-8)!} = \frac{10!}{8!2!} = \frac{10 \times 9 \times 8!}{8! \times 2!} = \frac{10 \times 9}{2 \times 1} = 45
\]
Wobva waverenga mikana \( (0.95)^8 \) uye \( (0.05)^2 \):
\[
(0.95)^8 \inenge 0.6634
\]
\[
(0.05)^2 = 0.0025
\]
Pakupedzisira, wedzera kukosha ikoko kwese kuti uwane:
\[
P(X = 8) = 45 \kawa 0.6634 \kawa 0.0025 \inenge 0.0744
\]
Saka, mukana wekuti zvikamu zvisere kubva pagumi zvipfuure bvunzo yemhando yepamusoro ungangoita 0.0744 kana 7.44%.
Muenzaniso Mubvunzo 2: Kuwanda Kwemukana
Mubvunzo:
Uchiri nekambani imwe chete, verenga mukana wekuti zvikamu zvipfumbamwe kubva pagumi zvinopasa bvunzo yemhando yepamusoro.
Kukurukurirana:
Kuti tigadzirise dambudziko iri, tinofanira kuverenga mukana wekuunganidza. Mikana yekuti zvikamu zvipfumbamwe kubva pagumi zvipfuure bvunzo zvinoreva kuti tinoverenga \( P(X \geq 9) \), izvo zvinogona kunyorwa seizvi:
\[
P(X \geq 9) = P(X = 9) + P(X = 10)
\]
Kushandisa fomura yekugovera yebinomial:
\[
P(X = 9) = \binom{10}{9} (0.95)^9 (0.05)^1
\]
\[
P(X = 10) = \binom{10}{10} (0.95)^{10} (0.05)^0
\]
Kutanga, verenga musanganiswa wenyaya yega yega:
\[
\bhinom{10}{9} = \frac{10!}{9!(10-9)!} = 10
\]
\[
\binom{10}{10} = 1
\]
Wobva waverenga mikana ye \( P(X = 9) \) uye \( P(X = 10) \):
\[
P(X = 9) = 10 \nguva (0.95)^9 \nguva 0.05
\]
\[
(0.95)^9 \inenge 0.6302
\]
\[
P(X = 9) = 10 \kawa 0.6302 \kawa 0.05 \inenge 0.3151
\]
\[
P(X = 10) = 1 \nguva (0.95)^{10} \nguva 1
\]
\[
(0.95)^{10} \inenge 0.5987
\]
\[
P(X = 10) = 0.5987
\]
Mikana yose ye \( P(X \geq 9) \):
\[
P(X \geq 9) = 0.3151 + 0.5987 \inenge 0.9138
\]
Saka, mukana wekuti zvikamu zvipfumbamwe kubva pagumi zvipfuure bvunzo yemhando yepamusoro ungangoita 0.9138 kana 91.38%.
Muenzaniso Mubvunzo 3: Kukosha Kunotarisirwa uye Kusiyana
Mubvunzo:
Verenga kukosha kunotarisirwa uye musiyano wehuwandu hwezvikamu zvinopasa bvunzo yemhando kubva pazvikamu gumi zvakagadzirwa, nemukana wekupasa we0.95.
Kukurukurirana:
Shandisa fomura inotevera:
– Kukosha kunotarisirwa (avhareji) \( \mu = np \)
– Kusiyana \( \sigma^2 = np(1-p) \)
Na \( n = 10 \) uye \( p = 0.95 \):
\[
\mu = 10 \kawa 0.95 = 9.5
\]
\[
\sigma^2 = 10 \kawa 0.95 \kawa 0.05 = 0.475
\]
Saka, kukosha kunotarisirwa kwehuwandu hwezvikamu zvinopasa bvunzo yemhando yepamusoro i9.5, uye musiyano uri 0.475.
Mhedziso
Kuburikidza nematambudziko matatu emuenzaniso ari pamusoro apa, takurukura maverengero ekuti tingaverenga sei mukana tichishandisa binomial distribution mumamiriro akasiyana-siyana: kuverenga mukana chaiwo, mukana wekuunganidza, uye kuverenga kukosha uye kusiyana kunotarisirwa. Ruzivo rwekugoverwa kwebinomial runobatsira muzvikamu zvakasiyana-siyana, zvakaita sekugadzira, tsvakiridzo yezvekurapa, uye nhamba dzevanhu, uko mhedzisiro yekuedza kwakadzokororwa nemhedzisiro miviri inogona kuongororwa kuti ibatsire kuita sarudzo. Tinovimba kuti matambudziko emuenzaniso nehurukuro dzakapihwa zvichabatsira kuwedzera kunzwisisa kwako kwekugoverwa kwebinomial.
