Ka'idoji na Asali na Canje-canjen Bazuwar
A cikin kididdiga da ka'idar yiwuwa, masu canjin bazuwar suna ɗaya daga cikin mahimman ra'ayoyi, suna cike gibin da ke tsakanin abubuwan da suka faru bazuwar da kuma nazarin lissafi mai aunawa. Ta hanyar masu canjin bazuwar, za mu iya "fassara" sakamakon gwaji bazuwar - wanda da farko ya ƙunshi abubuwan da suka faru ko rukuni - zuwa lambobi waɗanda za a iya sarrafa su: ƙididdige yiwuwarsu, taƙaita su da matsakaici, auna yaduwarsu, har ma da yin kwaikwayon su ta amfani da takamaiman rarrabawa. Wannan labarin ya tattauna mahimman ra'ayoyi na masu canjin bazuwar, nau'ikansu, da mahimman ra'ayoyi kamar aikin yuwuwar, aikin rarrabawa mai tarin yawa, ƙimar da ake tsammani, da bambancin.
1. Menene canjin da bazuwar yake nufi?
A taƙaice, canjin bazuwar aiki ne da ke nuna sakamakon kowane sakamako daga sararin samfurin zuwa lamba ta gaske. Sararin samfurin tarin duk sakamakon gwaji bazuwar ne.
Misali, a ce mun mirgina dizal mai gefe shida. Samfurin sararin shine {1, 2, 3, 4, 5, 6}. Za mu iya ayyana canjin bazuwar \(X\) a matsayin "lambar da ta bayyana akan dizal." Sannan \(X\) na iya samun ƙima daga 1 zuwa 6, tare da daidaiton yuwuwar idan dizal ɗin yayi daidai.
Wani misali kuma: muna juya tsabar kuɗi guda biyu. Samfurin sararin shine {HH, HT, TH, TT}. Idan muka ayyana canjin bazuwar \(Y\) a matsayin "adadin kawunan (H) da suka bayyana", to:
– HH → \(Y = 2\)
– HT → \(Y = 1\)
– TH → \(Y = 1\)
– TT → \(Y = 0\)
A nan mun ga cewa masu canji bazuwar ba dole ba ne su "nuna" sakamakon asali kai tsaye; hanya ce ta sanya ƙimar lambobi ga sakamakon bazuwar bisa ga buƙatun nazarin.
2. Nau'ikan masu canji bazuwar: rabe-raben da kuma ci gaba
Gabaɗaya, an raba masu canjin bazuwar zuwa manyan nau'i biyu:
a) Masu canjin bazuwar daban-daban
Maɓallin bazuwar da ba a iya ƙirgawa ba shi ne maɓallin bazuwar da za a iya ƙirgawa ɗaya bayan ɗaya (wanda za a iya ƙirgawa), yawanci a cikin nau'in lambobi ko wani saitin takamaiman ƙima daban.
Misali:
– Adadin yara a cikin iyali (0, 1, 2, 3, …)
– Adadin motocin da ke wucewa ta wurin biyan kuɗi a cikin minti 1
– Adadin kayayyakin da suka lalace daga samfura 10 da aka duba
Ga masu canji daban-daban na bazuwar, ana iya bayyana yiwuwar kowace ƙima kai tsaye a cikin nau'in aikin taro na yuwuwar.
b) Masu canjin da bazuwar da ke ci gaba da canzawa
Canjin bazuwar da ke ci gaba da canzawa wani canjin bazuwar ne wanda zai iya ɗaukar ƙima a kan tazara mai ci gaba akan layin lambobi na gaske (wanda ba a iya ƙirgawa ba), misali duk ƙima tsakanin 0 da 1, ko duk ƙima na gaske masu kyau.
Misali:
– Tsawon mutum
- Lokacin jiran abokin ciniki a kan tebur
– Zafin iska a wani lokaci
Ga mai canzawa mai ci gaba, yuwuwar da ake samu a kowace ma'ana ba ta da tabbas. Saboda haka, ana ƙididdige yiwuwar a kan kewayon ƙima (misali, tsakanin mintuna 10 da 12), ta amfani da aikin yawan yuwuwar yiwuwa.
3. Ayyukan Yiwuwa: PMF da PDF
Muhimmin ra'ayi na gaba shine yadda ake "haɗa" yiwuwar da ƙimar wani canji bazuwar.
a) Aikin Yiwuwar Mass (PMF)
Ga wani mai canzawa mai canzawa \(X\), ana bayyana PMF a matsayin:
\[
p(x) = P(X = x)
\]
tare da tanadin:
1. \(p(x) \ge 0\) ga duka \(x\)
2. \(\sum_x p(x) = 1\)
Misali mai sauƙi: fensir mai laushi
\[
P(X=k)=\frac{1}{6}, \quad k=1,2,3,4,5,6
\]
b) Aikin Yawan Yiwuwa (PDF)
Don ci gaba da canjin bazuwar \(X\), muna amfani da PDF \(f(x)\) don haka yuwuwar da ke kan tazara \([a,b]\) ita ce:
\[
P(a \le X \le b) = \int_a^bf(x)\,dx
\]
tare da tanadin:
1. \(f(x) \ge 0\)
2. \(\int_{-\infty}^{\infty} f(x)\,dx = 1\)
Yana da kyau a jaddada: ga mai canzawa mai ci gaba, \(P(X=x)=0\) ga kowace ƙimar \(x\). Yiwuwar koyaushe tana da ma'ana lokacin da ake tattauna jeri.
4. Aikin rarrabawa na tarin bayanai (CDF)
Ko dai mai rarrabawa ne ko mai ci gaba, za a iya bayyana masu canji bazuwar ta hanyar aikin rarrabawa mai tarin yawa (CDF), wanda aka bayyana a matsayin:
\[
F(x) = P(X \ le x)
\]
CDF yana da wasu muhimman halaye:
– Ƙimar \(F(x)\) koyaushe tana tsakanin 0 da 1
– \(F(x)\) baya raguwa (ba ya raguwa)
– \(\lim_{x\to -\infty}F(x)=0\) da kuma \(\lim_{x\to\infty}F(x)=1\)
Ga masu canji daban-daban, CDF yana da siffar "matakala" (yana tashi a wasu wurare). Ga masu canji masu ci gaba, CDF gabaɗaya yana da santsi kuma shine ainihin abin da ke cikin PDF:
\[
F(x)=\int_{-\infty}^{x} f(t)\,dt
\]
5. Ma'aunin yanayin tsakiya: ƙimar da ake tsammani (tsammani)
Da zarar mun san rabon yiwuwar, sau da yawa muna son taƙaita canjin bazuwar da lamba ɗaya da ke wakiltar "matsakaicin ƙimarsa na dogon lokaci." Wannan shine ƙimar da ake tsammani ko tsammani.
a) Tsammani masu canzawa daban-daban
Idan \(X\) ba a raba shi ba:
\[
E[X] = \sum_x x\,p(x)
\]
b) Tsammanin masu canzawa masu ci gaba
Idan \(X\) yana ci gaba:
\[
E[X] = \int_{-\infty}^{\infty} x\,f(x)\,dx
\]
Tsammani ba koyaushe yake daidai da "ƙimar da ta fi faruwa akai-akai" (yanayi) ba, kuma ba koyaushe shine ƙimar da ake iya faruwa da gaske ba, amma yana da matukar amfani don yanke shawara, hasashe, da kuma nazarin haɗari.
Misalin aikace-aikace: A cikin kasuwanci, ana iya amfani da tsammanin don ƙididdige matsakaicin ribar da ake tsammani na dabarun, la'akari da yanayi daban-daban da yuwuwar su.
6. Ma'aunin yaɗuwa: bambancin ra'ayi da karkacewar da aka saba
Masu canji guda biyu na bazata na iya samun tsammani iri ɗaya amma matakan rashin tabbas daban-daban. Saboda haka, muna buƙatar ma'aunin watsawa, wato bambanci da karkacewar da aka saba.
An bayyana bambancin \(X\) kamar haka:
\[
Var(X)=E[(XE[X])^2]
\]
Bambancin da aka saba dashi shine tushen murabba'in bambancin:
\[
\sigma = \sqrt{Var(X)}
\]
Dabaru masu amfani waɗanda galibi ake amfani da su:
\[
Var(X) = E[X^2] – (E[X])^2
\]
Mafi girman bambancin, haka nan yawan yaduwar ƙimar \(X\) daga matsakaicin, wanda ke nufin ƙarin rashin tabbas.
7. Rarraba yiwuwar da ake yawan amfani da su
A aikace, yawancin masu canjin da bazuwar ke biye da wasu tsare-tsaren rarrabawa. Wasu shahararrun rarrabawa sune:
– Bernoulli: sakamako biyu (nasara/rashin nasara), misali gaskiya-ƙarya, da rai-matattu.
– Binomial: adadin nasarorin da aka samu daga gwaje-gwajen Bernoulli, misali adadin ɗaliban da suka kammala karatu daga mutane 20.
– Poisson: adadin abubuwan da suka faru a cikin tazara ta lokaci/sarari, misali adadin kira mai shigowa a minti ɗaya.
- Ci gaba iri ɗaya: duk ƙimar da ke cikin tazara suna da yuwuwar daidai gwargwado.
– Na al'ada (Gaussian): abubuwa da yawa na halitta da zamantakewa suna fuskantar wannan rarrabawa, kamar tsayi ko kuskuren aunawa.
Zaɓar rarrabawa da ta dace yana taimakawa wajen yin ƙira da nazari su zama daidai.
8. Me yasa masu canjin bazuwar suke da mahimmanci?
Masu canjin bazata sune tushen:
– Ƙididdigar ƙididdigewa: kimanta sigogin yawan jama'a bisa ga samfurori
- Gwajin Hasashe: yanke shawara ko da'awar tana da goyon bayan bayanai
- Koyon injina: yin samfurin rashin tabbas da kuma yiwuwar hasashen
- Gudanar da Hadari: auna yiwuwar asara da kuma mummunan yanayi
– Injiniyanci da kimiyya: sarrafa sigina, amincin tsarin, ka'idar layi
Tare da masu canji bazuwar, muna da harshen lissafi don yin magana game da rashin tabbas ta hanyar tsari.
Kammalawa
Ma'aunin bazuwar wani muhimmin ra'ayi ne a cikin ka'idar yiwuwar da ke nuna sakamakon gwaje-gwajen bazuwar zuwa ƙimar lambobi. Ma'aunin bazuwar na iya zama daban ko ci gaba, kuma kowannensu yana da hanyar wakiltar yuwuwar ta hanyar PMF ko PDF. Bugu da ƙari, CDF tana ba da hanya ta gama gari don duba tarin yuwuwar. Don taƙaita rarrabawa, ana amfani da tsammanin a matsayin ma'aunin yanayin tsakiya da bambancin/bambancin daidaito a matsayin ma'aunin watsawa. Fahimtar waɗannan mahimman ra'ayoyi zai sauƙaƙa koyon batutuwa masu ci gaba kamar rarraba yuwuwar, kimanta ƙididdiga, koma baya, da ƙirar haɗari da nazarin bayanai na zamani.
Idan kana so, zan iya ƙara misalai na tambayoyi da tattaunawarsu (marasa bambanci da kuma ci gaba) don sauƙaƙa fahimtar manufar masu canji bazuwar.
