Imibuzo Eyisibonelo Exoxa Ngenani Elilindelekile Lokusatshalaliswa Kwe-Binomial
Ukusatshalaliswa kwe-binomial kuwukusatshalaliswa okuhlukile okusetshenziswa kaningi kwizibalo ukuchaza amathuba enani elithile lempumelelo ezivivinyweni eziningana ezenziwa ngokuzimela. Lokhu kusatshalaliswa kuwusizo kakhulu emikhakheni ehlukahlukene, njengezomnotho, i-biology, kanye nesayensi yezenhlalo. Umqondo owodwa obalulekile okufanele uwuqonde ekusatshalalisweni kwe-binomial inani elilindelekile. Lesi sihloko sizoxoxa ngomqondo wenani elilindelekile ekusatshalalisweni kwe-binomial ngezinkinga eziningana zezibonelo kanye nengxoxo yazo.
Incazelo Yokusatshalaliswa Kwe-Binomial
Ukusatshalaliswa kwe-binomial kuchaza inani lempumelelo ezivivinyweni ze-\( n \) ezinemiphumela emibili engaba khona: impumelelo noma ukwehluleka. Lokhu kusatshalaliswa kubonakala ngamapharamitha amabili ayinhloko:
– \( n \): inani lezilingo
– \( p \): amathuba okuphumelela esivivinyweni esisodwa
Lokhu kusatshalaliswa kuvame ukubizwa ngokuthi i-B(n, p). Umsebenzi we-probability mass (PMF) wokusabalalisa kwe-binomial uthi:
\[ P(X = k) = \binom{n}{k} p^k (1-p)^{nk} \]
lapho \( \binom{n}{k} \) kuyi-binomial coefficient, ebalwa kanje:
\[ \binom{n}{k} = \frac{n!}{k!(nk)!} \]
Inani Elilindelekile Ekusabalalisweni Kwe-Binomial
Inani elilindelekile lokusatshalaliswa kwe-binomial yinani elimaphakathi lempumelelo ekuhlolweni kwe-\( n \), futhi lakhiwe kanje:
\[ E(X) = n \izikhathi p \]
Imibuzo Eyisibonelo Nengxoxo
Isibonelo Umbuzo 1
Umbuzo:
Ake sithi umcwaningi wenza ucwaningo lokutshala izithombo eziyi-10, ngayinye inethuba lokukhula elingu-0.7. Lingakanani inani lezithombo ezilindelekile ezizokhula?
Ingxoxo:
Kuyaziwa:
– \( n = 10 \)
– \( p = 0.7 \)
Inani elilindelekile, \( E(X) \), libalwa kanje:
\[ E(X) = n \izikhathi p \]
\[ E(X) = 10 \izikhathi 0.7 \]
\[ E(X) = 7 \]
Ngakho-ke, inani elilindelekile lenani lembewu ekhula yimbewu ezi-7.
Isibonelo Umbuzo 2
Umbuzo:
Esivivinyweni, amathuba okuthi umfundi aphendule yonke imibuzo ngendlela efanele angu-0.8. Uma kunemibuzo eyi-15 esivivinyweni, yiliphi inani elilindelekile lezimpendulo ezifanele?
Ingxoxo:
Kuyaziwa:
– \( n = 15 \)
– \( p = 0.8 \)
Inani elilindelekile, \( E(X) \), libalwa kanje:
\[ E(X) = n \izikhathi p \]
\[ E(X) = 15 \izikhathi 0.8 \]
\[ E(X) = 12 \]
Ngakho-ke, inani elilindelekile lenani lezimpendulo ezifanele yimibuzo eyi-12.
Isibonelo Umbuzo 3
Umbuzo:
Inkampani yokunyathelisa ikhiqiza amaphepha anethuba lokuba namaphutha angu-0.02. Ngosuku olulodwa lokusebenza, ifektri ikhiqiza amaphepha angu-500. Lingakanani inani elilindelekile lamaphepha anephutha ngosuku olulodwa?
Ingxoxo:
Kuyaziwa:
– \( n = 500 \)
– \( p = 0.02 \)
Inani elilindelekile, \( E(X) \), libalwa kanje:
\[ E(X) = n \izikhathi p \]
\[ E(X) = 500 \izikhathi 0.02 \]
\[ E(X) = 10 \]
Ngakho-ke, inani elilindelekile lenani lamaphepha anephutha ngosuku olulodwa amashidi ayi-10.
Ukwanda Kwemibono Ekuqondeni
1. Ukwehluka kanye nokuphambuka okujwayelekile:
Ngaphezu kwenani elilindelekile, kubalulekile futhi ukuqonda ukuhlukahluka kanye nokuphambuka okujwayelekile ekusabalalisweni kwe-binomial. Ukuhlukahluka kokusabalaliswa kwe-binomial kwakhiwe kanje:
\[ \text{Var}(X) = n \times p \times (1 – p) \]
Ukuphambuka okujwayelekile kuyimpande yesikwele yomehluko:
\[ \umbhalo{SD}(X) = \sqrt{n \izikhathi p \izikhathi (1 – p)} \]
2. Isicelo Ezivivinyweni Zezibalo:
Ezivivinyweni noma ezivivinyweni zemfundo, amaphuzu alindelekile angasetshenziswa ukukala isilinganiso esilindelekile samaphuzu omfundi noma iqembu labafundi, okusiza ekuhlaziyweni kwezifundo zemfundo kanye nokuhlolwa kokusebenza kahle kokufundisa.
3. Izifundo Zezimo Eziphathelene Nezifo:
Isibonelo, ocwaningweni lokudluliswa kwesifo, amathuba okuthi isiguli silulame angalinganiswa kusetshenziswa ukusatshalaliswa kwe-binomial. Ukwazi inani elilindelekile kuvumela ochwepheshe bezempilo ukuthi bahlele izinsiza zezokwelapha ezidingekayo ngokusekelwe enanini elibikezelwe leziguli ezilulame.
Isiphetho
Ukusatshalaliswa kwe-binomial kuyithuluzi elibalulekile kwizibalo elisiza ukuchaza amathuba okuphumelela ochungechungeni lwezilingo. Inani elilindelekile ekusatshalalisweni kwe-binomial lingumqondo oyinhloko ochaza inani elimaphakathi lempumelelo elindelekile. Ngezibonelo okuxoxwe ngazo, singabona ukuthi inani elilindelekile libalwa futhi lisetshenziswa kanjani ezimweni ezahlukene. Ukuqonda okuqinile kwalomqondo kwenza abacwaningi kanye nochwepheshe bakwazi ukwenza izinhlelo ezingcono futhi benze izinqumo ezinolwazi oluthe xaxa ngokusekelwe kudatha engenzeka.
Ukusatshalaliswa kwe-binomial akubalulekile nje kuphela ku-probability theory kanye nezibalo kodwa futhi kubaluleke kakhulu ezinhlotsheni ezahlukene zokusebenza ezisebenzayo. Ngakho-ke, ukufunda lokhu kusatshalaliswa kanye nomqondo wenani elilindelekile kunikeza isisekelo esiqinile ekuhlaziyweni kwedatha nasekuthathweni kwezinqumo.
