Pfungwa Dzekutanga dzeZvinhu Zvisina Kurongeka
Muzviverengero uye dzidziso yeprobability, ma random variables ndeimwe yepfungwa dzakakosha, ichivhara mukaha uripo pakati pezviitiko zvisina tsarukano uye ongororo yemasvomhu inoyerwa. Kuburikidza nema random variables, tinogona "kushandura" mhedzisiro yekuedza kusina tsarukano—iyo pakutanga inosanganisira zviitiko kana mapoka—kuita nhamba dzinogona kugadziriswa: kuverenga mikana yadzo, kuipfupikisa neavhareji, kuyera kupararira kwadzo, uye kutodziita modelling uchishandisa kugoverwa kwakananga. Chinyorwa chino chinokurukura pfungwa huru dzema random variables, mhando dzadzo, uye pfungwa huru dzakadai se probability function, cumulative distribution function, expected value, uye variance.
1. Chii chinonzi variable isina kurongeka?
Muchidimbu, chisaririra chisina kurongeka (random variable) ibasa rinoongorora mhedzisiro yega yega kubva panzvimbo yemuenzaniso kuenda kunhamba chaiyo. Nzvimbo yemuenzaniso iunganidzwa wemhedzisiro yese inogona kuitika kubva mukuyedza kusina kurongeka.
Semuenzaniso, ngatitii tamonera die rine mativi matanhatu. Nzvimbo yemuenzaniso i {1, 2, 3, 4, 5, 6}. Tinogona kutsanangura random variable \(X\) se "nhamba inoonekwa padie." Ipapo \(X\) inogona kuva nenhamba kubva pa1 kusvika pa6, ine mukana wakaenzana kana die iri rakarurama.
Mumwe muenzaniso: tinotenderedza mari mbiri. Nzvimbo yemuenzaniso i {HH, HT, TH, TT}. Kana tikatsanangura variable isingachinjiki \(Y\) se "nhamba yemisoro (H) inoonekwa", saka:
– HH → \(Y = 2\)
– HT → \(Y = 1\)
– TH → \(Y = 1\)
– TT → \(Y = 0\)
Pano tinoona kuti ma "random variables" haafanire "kuratidza" mhedzisiro yekutanga zvakananga; inzira yekupa nhamba dzemhedzisiro dzisina kurongeka zvichienderana nezvinodiwa neongororo.
2. Mhando dzezvinhu zvisingatarisirwi: zvakasiyana uye zvinoenderera mberi
Kazhinji, ma "random variables" akakamurwa kuita mhando mbiri huru:
a) Zvimiro zvisina kurongeka zvakasiyana
Chinosiyana-siyana chisina kurongeka ichinhu chisina kurongeka chine kukosha kunogona kuverengwa chimwe nechimwe (chinoverengwa), kazhinji muchimiro chenhamba dzese kana seti yakasiyana yemitengo chaiyo.
Muenzaniso:
– Huwandu hwevana mumhuri (0, 1, 2, 3, …)
- Nhamba yemota dzinopfuura nepa toll post muminiti imwe chete
- Huwandu hwezvinhu zvisina kunaka kubva kuzvigadzirwa gumi zvakaongororwa
Kune ma "discrete random variables", mukana wekuti kukosha kwega kwega kunogona kuratidzwa zvakananga muchimiro che "probability mass function".
b) Zvimiro zvisingachinjiki zvinoenderera mberi
Continuous random variable imhando yerandom variable inogona kutora ma values pa continuous interval pamutsetse wenhamba chaiyo (uncountable), semuenzaniso ma values ese ari pakati pe0 ne1, kana ese ma positive real values.
Muenzaniso:
– Kureba kwemunhu
- Nguva yekumirira kwevatengi pakaunda
- Tembiricha yemhepo panguva yakatarwa
Kune continuous random variable, mukana wekuti chero nguva ipi zvayo uve zero. Nokudaro, mukana uyu unoverengerwa pamusoro pemhando dzakasiyana-siyana (semuenzaniso, pakati pemaminitsi gumi nemaviri negumi nemaviri), uchishandisa basa rekuti probability density .
3. Mabasa emukana: PMF nePDF
Pfungwa inotevera inokosha ndeyekuti mukana "unonamirwa" sei nemutengo wechinhu chisingawanzoonekwi.
a) Basa reKuwanda kweMhinduro (PMF)
Kune discrete random variable \(X\), PMF inotsanangurwa se:
\[
p(x) = P(X = x)
\]
nechirongwa che:
1. \(p(x) \ge 0\) yezvose \(x\)
2. \(\sum_x p(x) = 1\)
Muenzaniso uri nyore: dhayisi rakanaka
\[
P(X=k)=\frac{1}{6}, \quad k=1,2,3,4,5,6
\]
b) Basa reKuwanda kweMukana (PDF)
Kune inoenderera mberi isina kurongeka variable \(X\), tinoshandisa PDF \(f(x)\) kuitira kuti mukana uripo pa interval \([a,b]\) uve:
\[
P(a \le X \le b) = \int_a^bf(x)\,dx
\]
nechirongwa che:
1. \(f(x) \ge 0\)
2. \(\int_{-\infty}^{\infty} f(x)\,dx = 1\)
Zvakakosha kusimbisa kuti kune continuous random variable, \(P(X=x)=0\) kune yega yega kukosha kwe \(x\). Mikana inogara iine zvazvinoreva pakukurukura ma ranges.
4. Basa rekugovera rinounganidzwa (CDF)
Zvingave zvakasiyana kana kuti zvinoenderera mberi, zvisingatarisirwi zvinogona kutsanangurwa ne cumulative distribution function (CDF), iyo inotsanangurwa se:
\[
F(x) = P(X \le x)
\]
CDF ine zvinhu zvakakosha zvakawanda:
– Kukosha kwe \(F(x)\) kunogara kuri pakati pe0 ne1
– \(F(x)\) haideredzi (haiduredzi)
– \(\lim_{x\to -\infty}F(x)=0\) uye \(\lim_{x\to\infty}F(x)=1\)
Kune ma "variables" akasiyana, CDF ine chimiro che "staircase" (inokwira panzvimbo dzakati wandei). Kune ma "continuous variables" anoenderera mberi, CDF inowanzo nyorova uye ndiyo chinhu chakakosha muPDF:
\[
F(x)=\int_{-\infty}^{x} f(t)\,dt
\]
5. Kuyera kwetsika yepakati: kukosha kunotarisirwa (kutarisira)
Kana tangoziva kugoverwa kwemukana, tinowanzoda kupfupisa shanduko isina kurongeka nenhamba imwe chete inomiririra "avhareji yayo yenguva refu." Iyi ndiyo kukosha kunotarisirwa kana kuti kutarisira.
a) Tarisiro dzakasiyana-siyana
Kana \(X\) isina kurongeka:
\[
E[X] = \sum_x x\,p(x)
\]
b) Kutarisira kwezvinoshanduka-shanduka zvinoramba zvichienderera mberi
Kana \(X\) iri kuramba ichienderera mberi:
\[
E[X] = \int_{-\infty}^{\infty} x\,f(x)\,dx
\]
Kutarisira hakusi nguva dzose kwakafanana ne "kukosha kunowanzoitika" (maitiro), uye hakusi nguva dzose kukosha kunogona kuitika, asi kunobatsira zvikuru pakuita sarudzo, kufanotaura, uye kuongorora njodzi.
Muenzaniso wekushandisa: Mubhizinesi, zvinotarisirwa zvinogona kushandiswa kuverenga avhareji yepurofiti inotarisirwa yehurongwa, tichifunga nezvezviitiko zvakasiyana-siyana uye mikana yazvo.
6. Zviyero zvekupararira: kusiyana uye kutsauka kwakajairika
Zvimiro zviviri zvisina kurongeka zvinogona kuva netarisiro imwe chete asi mazinga akasiyana ekusava nechokwadi. Nokudaro, tinoda zviyero zvekupararira, kureva kusiyana uye kutsauka kwakajairwa.
Kusiyana kwe \(X\) kunotsanangurwa se:
\[
Var(X)=E[(XE[X])^2]
\]
Kutsauka kwakajairika ndiko mudzi wechikwere wemusiyano:
\[
\sigma = \sqrt{Var(X)}
\]
Mafomura anoshanda anowanzoshandiswa:
\[
Var(X) = E[X^2] – (E[X])^2
\]
Kana musiyano wacho wakura, kupararira kwe \(X\) kukosha kubva paavhareji kunowedzera, zvinoreva kusava nechokwadi kwakanyanya.
7. Kugoverwa kwezvingangoitika kunowanzoshandiswa
Mukuita, ma "random variables" akawanda anotevera mamwe mapatani ekugovera. Mamwe ma "provisions" anozivikanwa ndeaya:
– Bernoulli: mhedzisiro miviri (kubudirira/kukundikana), semuenzaniso chokwadi-nhema, mupenyu-afa.
– Binomial: huwandu hwevakabudirira kubva mukuyedzwa kweBernoulli, semuenzaniso huwandu hwevadzidzi vakapedza kudzidza kubva kuvanhu makumi maviri.
– Poisson: huwandu hwezviitiko munguva/nzvimbo, semuenzaniso huwandu hwemafoni anouya paminiti.
- Yakafanana inoenderera mberi: zvese zvinodiwa panguva imwe neimwe zvingangoitika zvakaenzana.
– Zvakajairika (Gaussian): zviitiko zvakawanda zvechisikigo uye zvemagariro zvinosvika pakupararira uku, zvakaita sekukanganisa kwekukwirira kana kuyera.
Kusarudza kugoverwa kwakakodzera kunobatsira kuti kutevedzera nekuongorora zvive zvakarurama.
8. Sei zvinhu zvisingatarisirwi zvichikosha?
Zvimiro zvisina kurongeka ndizvo hwaro hwe:
- Nhamba dzekufungidzira: kufungidzira huwandu hwevanhu zvichibva pamienzaniso
- Kuongororwa kwefungidziro: kusarudza kana chikumbiro chinotsigirwa nedata
- Kudzidza kwemuchina: kutevedzera kusava nechokwadi uye mukana wekufanotaura
- Kutarisira njodzi: kuyera mukana wekurasikirwa uye mamiriro ezvinhu akaipisisa
- Uinjiniya nesainzi: kugadzirisa zviratidzo, kuvimbika kwehurongwa, dzidziso yekuisa mumutsara
Nezvinhu zvisingatarisirwi, tine mutauro wemasvomhu wekutaura nezvekusava nechokwadi nenzira yakarongeka.
Mhedziso
Chinoshanduka-shanduka chinoreva pfungwa huru mudzidziso yekufungidzira inoongorora mhedzisiro yekuedza kwakasarudzika kune nhamba. Zvinoshanduka-shanduka zvinogona kunge zvakasiyana kana kuti zvichienderera mberi, uye chimwe nechimwe chine nzira yakasiyana yekuratidza mikana kuburikidza nePMF kana PDF. Uyezve, CDF inopa nzira yakajairika yekuona kuunganidzwa kwezvinogona kuitika. Mukupfupisa kugoverwa, tarisiro inoshandiswa sechiyero chetsika yepakati uye kusiyana/kutsauka kwakajairwa sechiyero chekupararira. Kunzwisisa pfungwa idzi dzekutanga kuchaita kuti kudzidza misoro yepamusoro senge kugoverwa kwezvinogona kuitika, kufungidzira kwenhamba, kudzoka, uye kutevedzera njodzi uye ongororo yedata yemazuva ano.
Kana muchida, ndinogonawo kuwedzera mibvunzo yemuenzaniso nehurukuro dzavo (dzakasiyana uye dzinoenderera mberi) kuti pfungwa yezvinyorwa zvisina kurongeka ive nyore kunzwisisa.
