Kukosha Kunotarisirwa Kwekugoverwa KwemaBinomial
Kugoverwa kwebinomial ndeimwe yemigovero inowanzo wanikwa yehuwandu hwezviitiko muhuwandu hwezviverengero uye mukana. Kugoverwa uku kunotsanangura huwandu hwekubudirira mumiedzo yebinary yakazvimirira (miedzo ine mhedzisiro miviri chete: kubudirira kana kukundikana). Kuti unzwisise zviri nani kugoverwa kwebinomial, zvakakosha kunzwisisa pfungwa yemutengo unotarisirwa, iyo inotsanangura avhareji yemutengo wemuedzo unodzokororwa kwenguva refu. Chinyorwa chino chichaongorora pfungwa yemutengo unotarisirwa muchimiro chekugoverwa kwebinomial.
Tsanangudzo yeBinomial Distribution
Kugoverwa kwebinomial kunoitika mumamiriro ezvinhu apo tinoita miedzo yakawanda yakafanana uye yakazvimirira, uye bvunzo yega yega ine mhedzisiro mbiri dzakasiyana, dzinowanzonzi "kubudirira" uye "kukundikana." Semuenzaniso, kutenderedza mari (misoro kana miswe), kupindura mubvunzo webvunzo (chokwadi kana nhema), kana bvunzo yekurapa (yakarapwa kana isina kupora).
Kugoverwa kwebinomial kunotsanangurwa nematanho maviri:
– n , nhamba yemiedzo.
– p, mukana wekubudirira pamuedzo wega wega.
Zvichitaurwa zviri nyore, kana X iri shanduko isina kurongeka inomiririra huwandu hwebudiriro mumiedzo ye n, saka X inotevera kugoverwa kwe binomial ne parameters n na p, inoratidzwa se X ~ Binomial(n, p).
Basa reKugona
Basa rekuti mukana wekugoverwa kwebinomial unoshanda seinotevera:
\[ P(X = k) = \bhinom{n}{k} p^k (1-p)^{nk} \]
di mana:
– \( \binom{n}{k} \) ndiyo binomial coefficient, iyo inoverengerwa se \( \frac{n!}{k!(nk)!} \).
– \( k \) ndiyo nhamba inodiwa yekubudirira.
– \( n \) ndiyo nhamba yemiedzo.
– \( p \) ndiyo mukana wekubudirira pakuedza kwega kwega.
– \( (1-p) \) ndiyo mukana wekukundikana mukuyedza kwega kwega.
Kukosha Kunotarisirwa
Kukosha kunotarisirwa kana kuti avhareji yekugoverwa kweprobability ndeimwe yezviyero zvakakosha zvenzvimbo yepakati. Kune kugoverwa kwebinomial, kukosha kunotarisirwa kwe random variable X inotevera Binomial(n, p) ndeiyi:
\[ E(X) = np \]
Humbowo hweKukosha Kunotarisirwa
Kuti tinzwisise kuti sei kukosha kunotarisirwa kwekugoverwa kwebinomial kuri np, tinogona kushandisa linearity property yemutengo unotarisirwa uye toona kuti binary variables inowedzera sei mupiro wavo.
Ngatitsanangure \( X \) senhamba yekubudirira mu n binary trials. Zvakanyanya, ngatiti \( X_i \) ive variable isina kurongeka inoratidza mhedzisiro ye i-th trial, ne \( X_i = 1 \) kana i-th trial iri kubudirira, uye \( X_i = 0 \) kana iri kutadza. Zvadaro, tinogona kunyora \( X \) se:
\[ X = X_1 + X_2 + \ldots + X_n \]
Sezvo imwe neimwe \( X_i \) iri binary variable ine mukana wekubudirira p, kukosha kunotarisirwa kwe \( X_i \) ndekwekuti:
\[ E(X_i) = 1 \cdot p + 0 \cdot (1-p) = p \]
Tinogona kushandisa pfuma yemutsara wemutengo unotarisirwa wemutengo unotarisirwa weX:
\[ E(X) = E(X_1 + X_2 + \ldots + X_n) \]
\[ E(X) = E(X_1) + E(X_2) + \ldots + E(X_n) \]
\[ E(X) = p + p + \ldots + p \]
\[ E(X) = np \]
Izvi zvinoratidza kuti kukosha kunotarisirwa kwekugoverwa kwebinomial i np.
Muenzaniso Unoratidza
Funga nezvenyaya yekuti tinokanda mari yakakwana kagumi. Ngatibvunzei kuti huwandu hwema "flips" anoburitsa misoro hunotarisirwa hungave hwakaita sei.
Muchiitiko ichi:
– n = 10 (nhamba yekukanda mari)
– p = 0.5 (mukana wekuwana misoro, sezvo mari yacho yakarurama)
Saka, kukosha kunotarisirwa ndekwekuti:
\[ E(X) = np = 10 \kawa 0.5 = 5 \]
Izvi zvinoreva kuti kana tikatenderedza mari kagumi kakawanda kwenguva refu, paavhareji tichawana misoro kashanu.
Kusiyana uye Kutsauka Kwakajairika
Pamusoro pekukosha kunotarisirwa, zvakakoshawo kunzwisisa musiyano uye kutsauka kwakajairwa kwekugoverwa kwebinomial.
Pakugoverwa kwebinomial, variance \( \sigma^2 \) uye standard deviation \( \sigma \) ndeiyi inotevera:
\[ \sigma^2 = np(1-p) \]
\[ \sigma = \sqrt{np(1-p)} \]
Kusiyana kunoyera kuti data rakapararira sei kubva pahuwandu hunotarisirwa. Kutsauka kwakajairwa ndiko mudzi wepakati wekuchinjana uye kunoyerawo kupararira kwedata, asi muzvikamu zvakafanana nedata rekutanga.
Mhedziso
Kugoverwa kwebinomial ipfungwa huru muhuwandu hwezviverengero uye mukana, inowanzoonekwa muzvishandiso zvakasiyana-siyana zvenyika chaiyo, kubva kubhizinesi kusvika kuzvesainzi yemagariro evanhu uye biology. Kukosha kunotarisirwa kwekugoverwa kwebinomial, kwakaverengerwa se np, kunopa nzwisiso yakakosha pamusoro pehuwandu hwekubudirira mumiedzo yebinary. Nekunzwisisa pfungwa dzekukosha kunotarisirwa, kusiyana, uye kutsauka kwakajairwa, tinogona kuwana kunzwisisa kwakajeka kwehunhu hwekugoverwa kwebinomial uye kuti kunotsanangura sei zvimwe zviitiko muhupenyu hwezuva nezuva.
Ruzivo urwu runobatsira zvikuru kwete chete mukuongorora data uye nhamba, asiwo mukuita sarudzo dzinoda kuongorora mukana uye kusava nechokwadi. Kunzwisisa kukosha kunotarisirwa uye kugoverwa kwebinomial kunogona kutibatsira kuita fungidziro dzakarurama uye kuita sarudzo dzine ruzivo rwakadzama.
