Zitsanzo za Mafunso Okhudza Ntchito Yogawa Binomial
Kugawa kwa binomial ndi kugawa kwa mwayi wosiyana komwe kumafotokoza kuchuluka kwa kupambana mu kuyesa komwe kumakhala ndi mayeso angapo odziyimira pawokha omwe ali ndi zotsatira ziwiri zomwe zingatheke: kupambana ndi kulephera. Kuyesa kulikonse kumatchedwa kuyesa, ndipo kugawa kwa binomial nthawi zambiri kumagwiritsidwa ntchito pazochitika zomwe kuchuluka kwa kupambana m'mayesero angapo odziyimira pawokha ndikofunikira. M'nkhaniyi, tikambirana mfundo zoyambira za kugawa kwa binomial ndikupereka zitsanzo ndi mayankho.
Malingaliro Oyambira a Ntchito Yogawa Binomial
Tisanayambe kukambirana za mafunso ndi kukambirana, tiyeni tikambirane mfundo zina zoyambira zokhudzana ndi kugawa kwa binomial.
1. Tanthauzo: Kugawa kwa binomial kumatanthauzidwa ngati kuchuluka kwa kupambana mu mayeso odziyimira pawokha a 'n', pomwe mayeso aliwonse ali ndi zotsatira ziwiri zomwe zingatheke: kupambana (ndi kuthekera p) kapena kulephera (ndi kuthekera q = 1 - p).
2. Ntchito Yotheka: Ntchito yotheka ya kugawa kwa binomial ndi:
\[
P(X = k) = \binom{n}{k} p^k (1-p)^{nk}
\]
Kumene:
– \( P(X = k) \) ndi mwayi wokhala ndi chipambano cha k mu mayeso a n.
– \( \binom{n}{k} \) ndi kuphatikiza kwa n take k, komwe kumatanthauzidwa kuti \( \frac{n!}{k!(nk)!} \).
– \( p \) ndi mwayi wopambana pa yesero lililonse.
– \( (1-p) \) ndi kuthekera kwa kulephera pa yesero lililonse.
3. Mtengo Woyembekezeredwa ndi Kusintha:
– Mtengo woyembekezeredwa (avereji) wa kugawa kwa binomial ndi \( \mu = np \).
– Kusiyana kwa kugawa kwa binomial ndi \( \sigma^2 = np(1-p) \).
Tsopano, tiyeni tigwiritse ntchito mfundo izi mu chitsanzo cha vuto kuti timvetse bwino.
Chitsanzo Funso 1: Mawerengedwe Oyambira a Kugawa kwa Binomial
Funso:
Kampani imapanga zigawo zamagetsi zomwe zili ndi mwayi wa 0.95 kuti gawo lililonse lipambane mayeso a khalidwe. Ngati zigawo 10 zapangidwa, werengani mwayi woti zigawo 8 zenizeni zipambane mayeso a khalidwe.
Kukambirana:
Tingagwiritse ntchito njira yogawa ya binomial kuti tithetse vutoli. Choyamba, tikupeza magawo otsatirawa:
– \( n \) (chiwerengero chonse cha mayesero) = 10
– \( k \) (chiwerengero cha zomwe zapambana) = 8
– \( p \) (kuthekera kwa kupambana) = 0.95
– \( q \) (kuthekera kwa kulephera) = 1 – 0.95 = 0.05
Kenako sinthani mfundo izi mu fomula yogawa ya binomial:
\[
P(X = 8) = \binom{10}{8} (0.95)^8 (0.05)^2
\]
Choyamba, werengerani kuphatikiza \( \binom{10}{8} \):
\[
\binom{10}{8} = \frac{10!}{8!(10-8)!} = \frac{10!}{8!2!} = \frac{10 \nthawi 9 \nthawi 8!}{8! \nthawi 2!} = \frac{10 \nthawi 9}{2 \nthawi 1} = 45
\]
Kenako, werengani mwayi \( (0.95)^8 \) ndi \( (0.05)^2 \):
\[
(0.95)^8 \pafupifupi 0.6634
\]
\[
(0.05)^2 = 0.0025
\]
Pomaliza, chulukitsani mfundo zonsezo kuti mupeze:
\[
P(X = 8) = 45 \nthawi 0.6634 \nthawi 0.0025 \pafupifupi 0.0744
\]
Chifukwa chake, mwayi woti zigawo 8 mwa 10 zipambane mayeso a khalidwe ndi pafupifupi 0.0744 kapena 7.44%.
Chitsanzo Funso 2: Kuthekera Kochuluka
Funso:
Ngakhale ndi kampani yomweyi, werengani mwayi woti zinthu zosachepera 9 mwa 10 zipambane mayeso a khalidwe.
Kukambirana:
Kuti tithetse vutoli, tifunika kuwerengera kuthekera kokwanira. Kuthekera koti osachepera 9 mwa 10 zigawo zipambane mayeso kumatanthauza kuti timawerengera \( P(X \geq 9) \), zomwe zingalembedwe motere:
\[
P(X \geq 9) = P(X = 9) + P(X = 10)
\]
Kugwiritsa ntchito njira yogawa ya binomial:
\[
P(X = 9) = \binom{10}{9} (0.95)^9 (0.05)^1
\]
\[
P(X = 10) = \binom{10}{10} (0.95)^{10} (0.05)^0
\]
Choyamba, werengani kuphatikiza kwa mlandu uliwonse:
\[
\binom{10}{9} = \frac{10!}{9!(10-9)!} = 10
\]
\[
\binom{10}{10} = 1
\]
Kenako, werengerani mwayi wa \( P(X = 9) \) ndi \( P(X = 10) \):
\[
P(X = 9) = 10 \nthawi (0.95)^9 \nthawi 0.05
\]
\[
(0.95)^9 \pafupifupi 0.6302
\]
\[
P(X = 9) = 10 \nthawi 0.6302 \nthawi 0.05 \pafupifupi 0.3151
\]
\[
P(X = 10) = 1 \nthawi (0.95)^{10} \nthawi 1
\]
\[
(0.95)^{10} \pafupifupi 0.5987
\]
\[
P(X = 10) = 0.5987
\]
Kuthekera konse kwa \( P(X \geq 9) \):
\[
P(X \geq 9) = 0.3151 + 0.5987 \pafupifupi 0.9138
\]
Kotero, mwayi woti zinthu zosachepera 9 mwa 10 zipambane mayeso a khalidwe ndi pafupifupi 0.9138 kapena 91.38%.
Chitsanzo Funso 3: Mtengo Woyembekezeredwa ndi Kusiyana
Funso:
Werengerani mtengo woyembekezeredwa ndi kusiyana kwa chiwerengero cha zigawo zomwe zapambana mayeso a khalidwe mwa zigawo 10 zomwe zapangidwa, ndi mwayi woti zidutse 0.95.
Kukambirana:
Gwiritsani ntchito njira iyi:
– Mtengo woyembekezeredwa (wapakati) \( \mu = np \)
– Kusiyana \( \sigma^2 = np(1-p) \)
Ndi \( n = 10 \) ndi \( p = 0.95 \):
\[
\mu = 10 \nthawi 0.95 = 9.5
\]
\[
\sigma^2 = 10 \nthawi 0.95 \nthawi 0.05 = 0.475
\]
Kotero, mtengo woyembekezeredwa wa chiwerengero cha zigawo zomwe zikupambana mayeso a khalidwe ndi 9.5, ndipo kusiyana kwake ndi 0.475.
Mapeto
Kudzera mu zitsanzo zitatu zomwe zili pamwambapa, takambirana momwe tingawerengere mwayi pogwiritsa ntchito kugawa kwa binomial pazochitika zosiyanasiyana: kuwerengera mwayi weniweni, mwayi wochuluka, ndikuwerengera mtengo ndi kusiyana komwe kukuyembekezeka. Kudziwa za kugawa kwa binomial ndikothandiza m'magawo osiyanasiyana, monga kupanga, kafukufuku wazachipatala, ndi ziwerengero za anthu, komwe zotsatira za kuyesa mobwerezabwereza ndi zotsatira ziwiri zomwe zingatheke zitha kusanthulidwa kuti zithandize kupanga zisankho. Tikukhulupirira kuti mavuto ndi zokambirana zomwe zaperekedwa zikuthandizani kumvetsetsa bwino kugawa kwa binomial.
