Mibvunzo yeMienzaniso neKukurukurirana kweBinomial Distribution
Kugoverwa kwebinomial ndeimwe yenzvimbo dzinonyanya kushandiswa dze discrete probability distributions. Inobatsira pakuenzanisa huwandu hwebudiriro mumiedzo yakafanana, yakazvimirira, imwe neimwe ichiburitsa kubudirira kana kukundikana. Muchinyorwa chino, tichanyatsoongorora kugoverwa kwebinomial nekupa mienzaniso yakati wandei nekukurukurirana kwakadzama.
Nhanganyaya yekugoverwa kweBinomial
Hunhu hukuru hwekugoverwa kweBinomial:
1. n : Huwandu hwemiedzo kana kudzokorora.
2. p : Mikana yekubudirira mukuyedzwa kwega kwega.
3. q = 1-p : Mikana yekukundikana mukuyedza kwega kwega.
Basa rehuwandu hwemukana wekugoverwa kwebinomial nderinotevera:
\[ P(X = k) = {n \sarudza k} p^k (1-p)^{nk} \]
di mana:
– \( {n \choose k} = \frac{n!}{k!(nk)!} \)
– \( X \): Chinoshanduka chinomiririra huwandu hwebudiriro.
– \( k \): Huwandu hwebudiriro dzinotsvakwa.
Mibvunzo yemuenzaniso nekukurukurirana
Ngatitangei nemibvunzo yemuenzaniso kuti tinzwisise pfungwa yekugoverwa kwebinomial zvakadzama.
Muenzaniso 1: Kusarudza kubva kuBoka reVadzidzi
Semuenzaniso, ngatitii tine boka revadzidzi gumi, uye mukana wekuti mudzidzi wega wega asarudzwa kuti atore chikamu mumakwikwi i0,3. Tinoda kuziva mukana wekuti vadzidzi vana chaivo vakasarudzwa.
Danho 1: Ziva maparamita ekugoverwa kwebinomial.
– \( n = 10 \)
– \( p = 0.3 \)
Danho rechipiri: Shandisa binomial distribution kuverenga mukana we \( X = 4 \).
\[ P(X = 4) = {10 \sarudza 4} (0.3)^4 (0.7)^6 \]
Kuverenga \( {10 \sarudza 4} \):
\[ {10 \sarudza 4} = \frac{10!}{4!(10-4)!} = \frac{10!}{4!6!} = 210 \]
Zvino verenga \( (0.3)^4 \) uye \( (0.7)^6 \):
\[ (0.3)^4 = 0.0081 \]
\[ (0.7)^6 = 0.117649 \]
Saka,
\[ P(X = 4) = 210 \cdot 0.0081 \cdot 0.117649 \inenge 0.20012 \]
Saka, mukana wekuti vadzidzi vana chaivo vasarudzwe ungangoita 0.20012 kana 20.012%.
Muenzaniso 2: Mikana Isina Kuenzana kana Kuti Yakaenzana na2
Semuenzaniso, tinobvunzwa nezvemukana wekuti vadzidzi vari pasi kana kuti vakaenzana nevaviri vachasarudzwa.
Danho 1: Tinofanira kuverenga \( P(X = 0) \), \( P(X = 1) \), uye \( P(X = 2) \).
– Kune \( P(X = 0) \):
\[ P(X = 0) = {10 \sarudza 0} (0.3)^0 (0.7)^{10} \]
\[ {10 \sarudza 0} = 1 \]
\[ (0.7)^{10} = 0.0282475 \]
\[ P(X = 0) = 1 \cdot 1 \cdot 0.0282475 = 0.0282475 \]
– Kune \( P(X = 1) \):
\[ P(X = 1) = {10 \sarudza 1} (0.3)^1 (0.7)^9 \]
\[ {10 \sarudza 1} = 10 \]
\[ (0.3) \cdot (0.7)^9 = 0.1210608 \]
\[ P(X = 1) = 10 \cdot 0.3 \cdot 0.1210608 = 0.3631824 \]
– Kune \( P(X = 2) \):
\[ P(X = 2) = {10 \sarudza 2} (0.3)^2 (0.7)^8 \]
\[ {10 \sarudza 2} = 45 \]
\[ (0.3)^2 \cdot (0.7)^8 = 0.2334744 \]
\[ P(X = 2) = 45 \cdot 0.09 \cdot 0.2334744 = 0.2334744 \]
Danho rechipiri: Wedzera mikana iripo.
\[ P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2) \]
\[ P(X \leq 2) = 0.0282475 + 0.3631824 + 0.3826372 = 0.7740671 \]
Saka, mukana wekuti vadzidzi vaviri vasarudzwe uri pasi kana kuti wakaenzana nevadzidzi vaviri ungangoita 0.7740671 kana 77.41%.
Muenzaniso 3: Mikana yekuti zvingangoita 8
Kana kuyedza kukaitwa kagumi nembiri, uye mukana wekubudirira mukuyedza kwega kwega uri 0.5, mukana wekuti kubudirira kanokwana kasere kuitike ndeupi?
Danho 1: Isa ma parameter e binomial: \( n = 12, p = 0.5 \).
Danho rechipiri: Tsvaga mukana wekuti \( X \geq 8 \).
Izvi zvinoda kuverenga mikana yakati wandei uye kuwedzera:
\[ P(X \geq 8) = P(X = 8) + P(X = 9) + P(X = 10) + P(X = 11) + P(X = 12) \]
Verenga chimwe nechimwe:
– Kune \( P(X = 8) \):
\[ P(X = 8) = {12 \sarudza 8} (0.5)^8 (0.5)^4 \]
\[ {12 \sarudza 8} = 495 \]
\[ (0.5)^{12} = 0.0002441406 \]
\[ P(X = 8) = 495 \cdot 0.0002441406 = 0.1208496 \]
– Kune \( P(X = 9) \):
\[ P(X = 9) = {12 \sarudza 9} (0.5)^9 (0.5)^3 \]
\[ {12 \sarudza 9} = 220 \]
\[ P(X = 9) = 220 \cdot 0.0002441406 = 0.05371094 \]
– Kune \( P(X = 10) \):
\[ P(X = 10) = {12 \sarudza 10} (0.5)^{10} (0.5)^2 \]
\[ {12 \sarudza 10} = 66 \]
\[ P(X = 10) = 66 \cdot 0.0002441406 = 0.01611328 \]
– Kune \( P(X = 11) \):
\[ P(X = 11) = {12 \sarudza 11} (0.5)^{11} (0.5)^1 \]
\[ {12 \sarudza 11} = 12 \]
\[ P(X = 11) = 12 \cdot 0.0002441406 = 0.002929688 \]
– Kune \( P(X = 12) \):
\[ P(X = 12) = {12 \sarudza 12} (0.5)^{12} \]
\[ {12 \sarudza 12} = 1 \]
\[ P(X = 12) = 1 \cdot 0.0002441406 = 0.0002441406 \]
Danho rechitatu: Wedzera mikana yese.
\[ P(X \geq 8) = 0.1208496 + 0.05371094 + 0.01611328 + 0.002929688 + 0.0002441406 \inenge 0.1938477 \]
Saka, mukana wekuti kubudirira kunosvika kasere mumiedzo gumi nemiviri ungangoita 0.1938477 kana 19.38%.
Mhedziso
Kugoverwa kwebinomial ipfungwa huru muhuwandu hwezviverengero iyo inokosha mukushandiswa kwakawanda. Nekunzwisisa maitiro ekuverenga mikana yezviitiko zvakasiyana-siyana zvekugoverwa kwebinomial, sezvakaratidzwa mumienzaniso iri pamusoro, tinogona kushandisa pfungwa iyi mumamiriro ezvinhu chaiwo. Ichi chiitiko chinosimbisawo kunzwisisa kwedu kuti maumbirwo emhedzisiro anoshanda sei mumamiriro ezvinhu akajeka uye akarongeka.
