Inani Elilindelekile Lokusatshalaliswa Kwe-Binomial
Ukusatshalaliswa kwe-binomial kungenye yezindlela ezivame ukuhlangatshezwana ngayo ukusatshalaliswa kwamathuba ahlukene kuzibalo kanye namathuba. Lokhu kusatshalaliswa kuchaza inani lempumelelo ochungechungeni lwezilingo ezizimele ze-binary (izilingo ezinemiphumela emibili kuphela: impumelelo noma ukwehluleka). Ukuze uqonde kangcono ukusatshalaliswa kwe-binomial, kubalulekile ukuqonda umqondo wenani elilindelekile, elichaza inani elimaphakathi lesivivinyo esiphindaphindwayo esikhathini eside. Lesi sihloko sizobuyekeza umqondo wenani elilindelekile kumongo wokusatshalaliswa kwe-binomial.
Incazelo Yokusatshalaliswa Kwe-Binomial
Ukusatshalaliswa kwe-binomial kuvela ezimweni lapho senza khona izivivinyo eziningana ezifanayo nezizimele, futhi isivivinyo ngasinye sinemiphumela emibili ehlukene, evame ukubizwa ngokuthi “impumelelo” kanye “nokwehluleka.” Isibonelo, ukuphonsa uhlamvu lwemali (amakhanda noma imisila), ukuphendula umbuzo wokuhlolwa (okuyiqiniso noma okungamanga), noma isivivinyo sezokwelapha (eselaphekile noma engelaphekile).
Ukusatshalaliswa kwe-binomial kuchazwa ngamapharamitha amabili:
– n , inani lezilingo.
– p, amathuba okuphumelela esivivinyweni ngasinye.
Empeleni, uma u-X enguguquguquko olungahleliwe olumelela inani lempumelelo kuzivivinyo zika-n, khona-ke u-X ulandela ukusatshalaliswa kwe-binomial ngamapharamitha u-n no-p, abizwa ngokuthi u-X ~ u-Binomial(n, p).
Umsebenzi Wamathuba
Umsebenzi wamathuba wokusatshalaliswa kwe-binomial ungokulandelayo:
\[ P(X = k) = \binom{n}{k} p^k (1-p)^{nk} \]
Kuphi:
– \( \binom{n}{k} \) yi-binomial coefficient, ebalwa njenge \( \frac{n!}{k!(nk)!} \).
– \( k \) yinani elifiswayo lempumelelo.
– \( n \) inani lezilingo.
– \( p \) amathuba okuphumelela esivivinyweni ngasinye.
– \( (1-p) \) amathuba okwehluleka esivivinyweni ngasinye.
Inani Elilindelekile
Inani elilindelekile noma isilinganiso sokusatshalaliswa kwamathuba kungenye yezindlela ezibaluleke kakhulu zendawo ephakathi. Ngokusatshalaliswa kwe-binomial, inani elilindelekile lokuguquguquka okungahleliwe u-X okulandela i-Binomial(n, p) ngu:
\[ E(X) = np \]
Ubufakazi Benani Elilindelekile
Ukuze siqonde ukuthi kungani inani elilindelwe lokusatshalaliswa kwe-binomial lingu-np, singasebenzisa i-linearity property yenani elilindelwe futhi sibone ukuthi iziguquguquko ze-binary zihlangana kanjani nomnikelo wazo.
Ake sichaze \( X \) njengenani lempumelelo kuzivivinyo ze-n binary. Ngokuqondile, ake \( X_i \) kube yi-variable engahleliwe echaza umphumela wesivivinyo se-i-th, kanye \( X_i = 1 \) uma isivivinyo se-i-th siyimpumelelo, kanye \( X_i = 0 \) uma kuyiphutha. Ngemuva kwalokho, singabhala \( X \) njengo:
\[ X = X_1 + X_2 + \ldots + X_n \]
Njengoba i-\( X_i \) ngayinye iyi-variable enama-performance okuphumelela u-p, inani elilindelekile le-\( X_i \) lithi:
\[ E(X_i) = 1 \cdot p + 0 \cdot (1-p) = p \]
Singasebenzisa impahla yokulingana yenani elilindelekile yenani elilindelekile lika-X:
\[ E(X) = E(X_1 + X_2 + \ldots + X_n) \]
\[ E(X) = E(X_1) + E(X_2) + \ldots + E(X_n) \]
\[ E(X) = p + p + \ldots + p \]
\[ E(X) = np \]
Lokhu kubonisa ukuthi inani elilindelekile lokusatshalaliswa kwe-binomial lingu-np.
Isibonelo Esichazayo
Cabanga ngendaba lapho sijika khona uhlamvu lwemali olufanele izikhathi eziyi-10. Ake sibuze ukuthi inani elilindelekile lenani lamajika aphumela emakhanda liyoba yini.
Esimweni esinjalo:
– n = 10 (inani lokuphonswa kohlamvu lwemali)
– p = 0.5 (amathuba okuthola amakhanda, njengoba uhlamvu lwemali lulungile)
Ngakho-ke, inani elilindelekile yileli:
\[ E(X) = np = 10 \izikhathi 0.5 = 5 \]
Lokhu kusho ukuthi uma sijika uhlamvu lwemali izikhathi eziyi-10 ngokuphindaphindiwe esikhathini eside, ngokwesilinganiso sizoba namakhanda izikhathi ezi-5.
Ukwehluka kanye nokuphambuka okujwayelekile
Ngaphezu kwenani elilindelekile, kubalulekile futhi ukuqonda ukuhlukahluka kanye nokuphambuka okujwayelekile kokusatshalaliswa kwe-binomial.
Ngokusatshalaliswa kwe-binomial, ukuhluka \( \sigma^2 \) kanye nokuphambuka okujwayelekile \( \sigma \) kungokulandelayo:
\[ \sigma^2 = np(1-p) \]
\[ \sigma = \sqrt{np(1-p)} \]
Ukwehluka kulinganisa ukuthi idatha isakazeke kangakanani kusukela enanini elilindelekile. Ukuphambuka okujwayelekile kuyimpande yesikwele yokwehluka futhi kulinganisa nokusabalala kwedatha, kodwa ngamayunithi afanayo nedatha yokuqala.
Isiphetho
Ukusatshalaliswa kwe-binomial kuwumqondo oyisisekelo kuzibalo kanye namathuba, okuvame ukutholakala kuzinhlelo zokusebenza ezahlukahlukene zomhlaba wangempela, kusukela ebhizinisini kuya kwisayensi yezenhlalo kanye ne-biology. Inani elilindelekile lokusatshalaliswa kwe-binomial, elibalwa njenge-np, linikeza ukuqonda okubalulekile ngenani elimaphakathi lempumelelo ochungechungeni lwezilingo ze-binary. Ngokuqonda imiqondo yenani elilindelekile, ukuhlukahluka, kanye nokuphambuka okujwayelekile, singathola ukuqonda okucacile kwezici zokusatshalaliswa kwe-binomial nokuthi kuchaza kanjani izimo ezithile empilweni yansuku zonke.
Lolu lwazi luwusizo kakhulu hhayi nje ekuhlaziyweni kwedatha kanye nezibalo, kodwa futhi nasekuthathweni kwezinqumo ezidinga ukuhlola amathuba kanye nokungaqiniseki. Ukuqonda inani elilindelekile kanye nokusatshalaliswa kwe-binomial kungasisiza senze izibikezelo ezinembe kakhudlwana futhi senze izinqumo ezinolwazi oluthe xaxa.
