Darajar da ake tsammani ta Rarraba Binomial
Rarraba binomial yana ɗaya daga cikin rarrabawar yiwuwar da aka fi samu a cikin ƙididdiga da yuwuwar. Wannan rarrabawa yana bayyana adadin nasarorin da aka samu a cikin jerin gwaje-gwajen binary masu zaman kansu (gwaji tare da sakamako biyu kawai: nasara ko gazawa). Don fahimtar rarraba binomial mafi kyau, yana da mahimmanci a fahimci manufar ƙimar da ake tsammani, wanda ke bayyana matsakaicin ƙimar gwaji da aka maimaita a cikin dogon lokaci. Wannan labarin zai sake duba manufar ƙimar da ake tsammani a cikin mahallin rarraba binomial.
Ma'anar Rarraba Binomial
Rarrabawar binomial tana tasowa ne a cikin mahallin da muke yin gwaje-gwaje iri ɗaya da masu zaman kansu, kuma kowace gwaji tana da sakamako biyu daban-daban, waɗanda galibi ake kira "nasara" da "kasawa." Misali, jujjuya tsabar kuɗi (kawuna ko wutsiya), amsa tambayar jarrabawa (gaskiya ko ƙarya), ko gwajin likita (wanda aka warkar ko wanda ba a warkar ba).
An bayyana rarrabawar binomial ta hanyar sigogi biyu:
– n , adadin gwaje-gwaje.
- p , yuwuwar samun nasara a kowace gwaji.
A takaice dai, idan X wani canji ne na bazuwar da ke wakiltar adadin nasarorin da aka samu a gwaje-gwajen n, to X ya biyo bayan rarraba binomial tare da sigogi n da p, waɗanda aka nuna a matsayin X ~ Binomial(n, p).
Aikin Yiwuwa
Aikin yiwuwar rarrabawar binomial kamar haka:
\[ P(X = k) = \binom{n}{k} p^k (1-p)^{nk} \]
Ina:
– \( \binom{n}{k} \) shine ma'aunin binomial, wanda aka ƙididdige shi azaman \( \frac{n!}{k!(nk)!} \).
– \( k \) shine adadin nasarorin da ake so.
– \( n \) shine adadin gwaje-gwajen.
– \( p \) shine yuwuwar samun nasara a kowace gwaji.
– \( (1-p) \) shine yuwuwar gazawa a kowace gwaji.
Darajar da ake tsammani
Ƙimar da ake tsammani ko ma'anar rarrabawar yiwuwar yana ɗaya daga cikin mahimman ma'auni na tsakiyar wuri. Ga rarrabawar binomial, ƙimar da ake tsammani na canjin bazuwar X wanda ke bin Binomial(n, p) shine:
\[ E(X) = np \]
Shaidar Darajar da ake tsammani
Domin fahimtar dalilin da yasa ake tsammanin ƙimar rarraba binomial shine np, zamu iya amfani da kadarar layi na ƙimar da ake tsammani kuma mu ga yadda masu canjin binary zasu haɗu da gudummawarsu.
Bari mu ayyana \(X \) a matsayin adadin nasarorin da aka samu a gwaje-gwajen binary na n. Musamman ma, bari \(X_i \) ya zama canjin da bazuwar da ke nuna sakamakon gwajin i-th, tare da \(X_i = 1 \) idan gwajin i-th ya yi nasara, da kuma \(X_i = 0 \) idan ya gaza. Sannan, za mu iya rubuta \(X \) kamar haka:
\[ X = X_1 + X_2 + \ldot + X_n \]
Tunda kowanne \( X_i \) mai canjin binary ne tare da yuwuwar samun nasara p, ƙimar da ake tsammani ta \( X_i \) ita ce:
\[ E(X_i) = 1 \cdot p + 0 \cdot (1-p) = p \]
Za mu iya amfani da haƙƙin layi na ƙimar da ake tsammani don ƙimar da ake tsammani na X:
\[ E(X) = E(X_1 + X_2 + \ldot + X_n) \]
\[ E(X) = E(X_1) + E(X_2) + \ldot + E(X_n) \]
\[ E(X) = p + p + \ldots + p \]
\[ E(X) = np \]
Wannan yana nuna cewa ƙimar da ake tsammani na rarraba binomial shine np.
Misali Mai Misali
Ka yi la'akari da yanayin da muke juya tsabar kuɗi sau 10. Bari mu tambayi menene ƙimar da ake tsammani na adadin juyawar da za ta haifar da kai.
A wannan yanayin:
– n = 10 (adadin jifan tsabar kuɗi)
– p = 0.5 (yiwuwar samun kawunan, tunda tsabar kudin gaskiya ne)
Don haka, ƙimar da ake tsammani ita ce:
\[ E(X) = np = 10 sau 0.5 = 5 \]
Wannan yana nufin idan muka juya tsabar kuɗi sau 10 akai-akai a cikin dogon lokaci, a matsakaici za mu sami kawunanmu sau 5.
Bambancin da Daidaitaccen Bambancin
Baya ga ƙimar da ake tsammani, yana da mahimmanci a fahimci bambancin da karkacewar daidaito na rarraba binomial.
Ga rarrabawar binomial, bambancin \( \sigma^2 \) da karkacewar misali \( \sigma \) sune kamar haka:
\[ \sigma^2 = np(1-p) \]
\[ \sigma = \sqrt{np(1-p)} \]
Bambancin yana auna nisan da aka bazu daga ƙimar da ake tsammani. Bambancin da aka saba dashi shine tushen murabba'in bambancin kuma yana auna yaduwar bayanan, amma a cikin raka'a ɗaya da bayanan asali.
Kammalawa
Rarraba binomial wani muhimmin ra'ayi ne a cikin kididdiga da yuwuwar samu, wanda ake yawan samu a aikace-aikace daban-daban na duniya, tun daga kasuwanci zuwa kimiyyar zamantakewa da ilmin halitta. Darajar da ake tsammani na rarraba binomial, wanda aka ƙididdige a matsayin np, yana ba da muhimmiyar fahimta game da matsakaicin adadin nasarorin da aka samu a cikin jerin gwaje-gwajen binary. Ta hanyar fahimtar ra'ayoyin ƙimar da ake tsammani, bambancin, da karkacewar daidaito, za mu iya samun fahimtar halayen rarraba binomial da kuma yadda yake bayyana wasu abubuwan da ke faruwa a rayuwar yau da kullun.
Wannan ilimin yana da matuƙar amfani ba kawai a nazarin bayanai da ƙididdiga ba, har ma a wajen yanke shawara wanda ke buƙatar kimanta yiwuwar da rashin tabbas. Fahimtar ƙimar da ake tsammani da kuma rarrabawar binomial na iya taimaka mana mu yi hasashen da ya fi daidai da kuma yanke shawara mai zurfi.
