Mafunso a Zitsanzo ndi Kukambirana za Kugawa Mwayi
Kugawa mwayi ndi chimodzi mwa mfundo zazikulu mu ziwerengero ndi kuthekera. Kumagwiritsidwa ntchito kumvetsetsa kuthekera kwa mitengo yosiyanasiyana ya nambala yosasinthika yomwe ikuchitika. Kugawa mwayi kumatha kukhala ndi mitundu yosiyanasiyana kutengera mtundu wa deta yomwe ikuwunikidwa. Mitundu iwiri yodziwika bwino ya kugawa mwayi ndi yosiyana komanso yopitilira. M'nkhaniyi, tiwunikanso mavuto angapo a zitsanzo ndikukambirana za kugawa mwayi kuti timvetsetse bwino nkhaniyi.
Kugawa Kokha
Kugawa kosiyana ndi kugawa komwe kumawerengera mwayi wa kusintha kosiyana kwachisawawa, ndiko kuti, kusintha komwe kungatenge mitengo ina yokha. Zitsanzo zodziwika bwino za kugawa kosiyana ndi Binomial Distribution ndi Poisson Distribution.
Chitsanzo 1: Kugawa kwa Binomial
Kugawa kwa binomial kumafotokoza kuchuluka kwa kupambana mu mndandanda wa mayeso a Bernoulli. Mayeso aliwonse a Bernoulli ali ndi zotsatira ziwiri: kupambana kapena kulephera. Mwayi wopambana umakhalabe wokhazikika panthawi yonse ya mayeso.
Funso:
Kampani yopanga mankhwala ikuyesa mankhwala atsopano pa odwala 10. Kuthekera koti mankhwalawa angagwire ntchito mwa wodwala m'modzi ndi 0.7. Werengani kuthekera koti mankhwalawa angagwire ntchito mwa odwala 7 mwa 10.
Kukambirana:
Chosinthika chosasinthika \(X\) chimatsatira kugawa kwa binomial ndi \(n = 10\) ndi \(p = 0.7\). Ntchito ya binomial probability ndi:
\[ P(X = k) = \binom{n}{k} p^k (1 – p)^{n – k} \]
Kwa \(k = 7\):
\[ P(X = 7) = \binom{10}{7} (0.7)^7 (0.3)^3 \]
Kuwerengera binomial coefficient \(\binom{10}{7}\):
\[ \binom{10}{7} = \frac{10!}{7!(10-7)!} = \frac{10!}{7!3!} = 120 \]
Kuwerengera mwayi:
\[ P(X = 7) = 120 \nthawi (0.7)^7 \nthawi (0.3)^3 \]
\[ P(X = 7) \pafupifupi 120 \nthawi 0.0823543 \nthawi 0.027 \]
\[ P(X = 7) \pafupifupi 0.231 \]
Chifukwa chake, mwayi woti mankhwalawa amagwira ntchito mwa odwala 7 mwa 10 ndi pafupifupi 0.231 kapena 23.1%.
Chitsanzo 2: Kugawa kwa Poisson
Kugawa kwa Poisson kumagwiritsidwa ntchito kutsanzira kuchuluka kwa zochitika zachilendo mkati mwa nthawi kapena malo operekedwa.
Funso:
Sitolo imodzi imapeza makasitomala 4 pa ola limodzi. Kodi pali mwayi wotani woti sitoloyo ipeze makasitomala 5 pa ola limodzi?
Kukambirana:
Chosinthika chosasinthika \(X\) chimatsatira kugawa kwa Poisson ndi parameter \(\lambda = 4\). Ntchito ya Poisson probability mass ndi:
\[ P(X = k) = \frac{\lambda^ke^{-\lambda}}{k!} \]
Kwa \(k = 5\):
\[ P(X = 5) = \frac{4^5 e^{-4}}{5!} \]
Chiwerengero:
\[ P(X = 5) = \frac{1024 \cdot e^{-4}}{120} \]
\[ P(X = 5) \pafupifupi \frac{1024 \cdot 0.0183}{120} \]
\[ P(X = 5) \pafupifupi 0.156 \]
Chifukwa chake, mwayi woti sitoloyo ilandire makasitomala 5 enieni mu ola limodzi ndi pafupifupi 0.156, kapena 15.6%.
Kugawa Kosalekeza
Kugawa kosalekeza kumagwiritsidwa ntchito pamene chisinthiko chosasinthika chomwe chikuyesedwa chikhoza kutenga mtengo uliwonse mkati mwa mtundu wina. Zitsanzo zodziwika bwino za kugawa kosalekeza ndi Normal Distribution ndi Exponential Distribution.
Chitsanzo 3: Kugawa Kwabwinobwino
Kugawa Kwabwinobwino, komwe nthawi zambiri kumatchedwa Kugawa kwa Gaussian, ndi kugawa komwe kumagwiritsidwa ntchito kwambiri m'magawo osiyanasiyana, kuphatikizapo sayansi, uinjiniya, ndi zachuma.
Funso:
Kutalika kwa amuna akuluakulu mumzinda nthawi zambiri kumakhala kofanana ndi masentimita 170 ndipo kusiyana kwapakati pa masentimita 10. Kodi pali mwayi wotani woti mwamuna wosankhidwa mwachisawawa akhale ndi kutalika kwa masentimita 160 ndi 180?
Kukambirana:
Tifunika kuwerengera z-score ya 160 cm ndi 180 cm. Z-score imatanthauzidwa motere:
\[ Z = \frac{X – \mu}{\sigma} \]
Kwa \(X = 160\):
\[ Z_{160} = \frac{160 – 170}{10} = -1 \]
Kwa \(X = 180\):
\[ Z_{180} = \frac{180 – 170}{10} = 1 \]
Tsopano tifunika kuyang'ana ma potential values kuchokera pa -1 mpaka 1 mu tebulo la z. Mtengo wochokera pa z = -1 mpaka z = 1 ndi pafupifupi 0.6826.
Kotero, mwayi woti mwamuna wosankhidwa mwachisawawa ali pakati pa 160 cm ndi 180 cm ndi pafupifupi 0.6826 kapena 68.26%.
Chitsanzo 4: Kugawa Kowonjezera
Kugawa kwa Exponential kumagwiritsidwa ntchito poyesa nthawi pakati pa zochitika mu njira ya Poisson.
Funso:
Nthawi yapakati pakati pa kufika kwa makasitomala awiri m'sitolo ndi mphindi 15. Kodi pali mwayi wotani kuti nthawi pakati pa kufika kwa makasitomala awiri ndi yochepera mphindi 10?
Kukambirana:
Kugawa kwa Exponential kuli ndi parameter \(\lambda\) yomwe ndi yotsutsana ndi mean (\(\mu\)). Ndi avereji ya mphindi 15:
\[ \lambda = \frac{1}{\mu} = \frac{1}{15} = 0.0667 \]
Ntchito yogawa kuchuluka kwa exponential ndi:
\[ P(X \leq x) = 1 – e^{-\lambda x} \]
Kwa \(x = 10\):
\[ P(X \leq 10) = 1 – e^{-0.0667 \times 10} \]
\[ P(X \leq 10) = 1 – e^{-0.667} \]
\[ P(X \leq 10) \pafupifupi 1 – 0.5134 \]
\[ P(X \leq 10) \pafupifupi 0.4866 \]
Chifukwa chake, mwayi woti nthawi pakati pa kufika kwa makasitomala awiri ndi yochepera mphindi 10 ndi pafupifupi 0.4866 kapena 48.66%.
Mapeto
Kugawa kwa kuthekera, kosiyana ndi kopitilira, ndi mfundo zothandiza kwambiri pakukonza ndi kumvetsetsa khalidwe la zosintha zosasinthika. Kugawa kwa binomial ndi Poisson nthawi zambiri kumagwiritsidwa ntchito pa zosintha zosasinthika, pomwe kugawa kwa normal ndi exponential ndi zitsanzo za kugawa kosalekeza.
Kudzera mu zitsanzo zomwe zili pamwambapa, tikukhulupirira kuti mwamvetsetsa bwino momwe mungawerengere ndikutanthauzira mwayi wogawa mwayi. Mukachita zinthu mosalekeza, luso lanu lomvetsetsa kugawa mwayi lidzakula ndipo lingagwiritsidwe ntchito m'magawo osiyanasiyana.
