Cov Ntsiab Lus Tseem Ceeb ntawm Random Variables
Hauv kev suav lej thiab kev xav txog qhov muaj feem yuav tshwm sim, cov hloov pauv random yog ib qho ntawm cov tswv yim tseem ceeb tshaj plaws, txuas qhov sib txawv ntawm cov xwm txheej random thiab kev ntsuas lej. Los ntawm cov hloov pauv random, peb tuaj yeem "txhais" cov txiaj ntsig ntawm kev sim random - uas thaum xub thawj muaj cov xwm txheej lossis pawg - mus rau hauv cov lej uas tuaj yeem ua tiav: xam lawv qhov muaj feem yuav tshwm sim, sau lawv nrog qhov nruab nrab, ntsuas lawv qhov kev tawg, thiab txawm tias ua qauv rau lawv siv cov kev faib tawm tshwj xeeb. Tsab xov xwm no tham txog cov tswv yim yooj yim ntawm cov hloov pauv random, lawv cov hom, thiab cov tswv yim tseem ceeb xws li qhov muaj feem yuav tshwm sim, kev faib tawm cumulative, tus nqi xav tau, thiab variance.
1. Ib qho kev hloov pauv random yog dab tsi?
Hauv cov lus yooj yim, ib qho random variable yog ib qho function uas map txhua qhov tshwm sim los ntawm ib qho sample space mus rau ib qho real number. Lub sample space yog cov sau ntawm txhua qhov tshwm sim uas ua tau ntawm ib qho kev sim random.
Piv txwv li, xav tias peb dov lub tuag rau sab. Qhov chaw piv txwv yog {1, 2, 3, 4, 5, 6}. Peb tuaj yeem txhais qhov hloov pauv random \(X\) ua "tus lej uas tshwm sim ntawm lub tuag." Tom qab ntawd \(X\) tuaj yeem muaj tus nqi ntawm 1 txog 6, nrog rau qhov muaj feem sib npaug yog tias lub tuag ncaj ncees.
Lwm qhov piv txwv: peb tig ob lub npib. Qhov chaw piv txwv yog {HH, HT, TH, TT}. Yog tias peb txhais qhov hloov pauv random \(Y\) ua "tus lej ntawm cov taub hau (H) uas tshwm sim", ces:
– HH → \(Y = 2\)
– HT → \(Y = 1\)
– TH → \(Y = 1\)
– TT → \(Y = 0\)
Ntawm no peb pom tias cov hloov pauv random tsis tas yuav "cuam tshuam" qhov tshwm sim thawj zaug ncaj qha; lawv yog ib txoj hauv kev los muab cov nqi lej rau cov txiaj ntsig random raws li qhov xav tau ntawm kev tshuaj xyuas.
2. Hom ntawm cov hloov pauv random: discrete thiab tas mus li
Feem ntau, cov hloov pauv random raug muab faib ua ob hom tseem ceeb:
a) Cov hloov pauv random sib cais
Ib qho kev hloov pauv random discrete yog ib qho kev hloov pauv random uas nws cov nqi tuaj yeem suav ib qho los ntawm ib qho (suav tau), feem ntau yog nyob rau hauv daim ntawv ntawm cov lej lossis ib pawg cais ntawm cov nqi tshwj xeeb.
Piv txwv:
– Tus naj npawb ntawm cov menyuam hauv tsev neeg (0, 1, 2, 3, …)
- Tus naj npawb ntawm cov tsheb hla dhau tus nqi them hauv 1 feeb
- Tus naj npawb ntawm cov khoom tsis zoo los ntawm 10 yam khoom uas tau tshuaj xyuas
Rau cov hloov pauv random discrete, qhov probability ntawm txhua tus nqi tuaj yeem qhia ncaj qha hauv daim ntawv ntawm probability mass function.
b) Cov hloov pauv random tas mus li
Ib qho hloov pauv random tas mus li yog ib qho hloov pauv random uas tuaj yeem coj cov nqi ntawm lub sijhawm tas mus li ntawm kab lej tiag tiag (tsis suav), piv txwv li txhua tus nqi ntawm 0 thiab 1, lossis txhua tus nqi tiag tiag zoo.
Piv txwv:
- Qhov siab ntawm ib tug neeg
- Lub sijhawm tos ntawm cov neeg siv khoom ntawm lub txee
- Qhov kub ntawm huab cua ntawm ib teev twg
Rau ib qho kev hloov pauv random tas mus li, qhov muaj feem yuav tshwm sim ntawm txhua qhov chaw yog xoom. Yog li ntawd, qhov muaj feem yuav tshwm sim tau suav dhau ntawm ntau yam nqi (piv txwv li, ntawm 10 thiab 12 feeb), siv qhov muaj feem yuav tshwm sim.
3. Cov haujlwm ntawm qhov yuav tshwm sim: PMF thiab PDF
Lub tswv yim tseem ceeb tom ntej yog qhov muaj feem cuam tshuam li cas "txuas" rau tus nqi ntawm qhov hloov pauv random.
a) Qhov Feem Pua ntawm Qhov Yuav Muaj Tau (PMF)
Rau ib qho discrete random variable \(X\), PMF yog txhais li:
\[
p(x) = P(X = x)
\]
nrog rau kev muab ntawm:
1. \(p(x) \ge 0\) rau txhua \(x\)
2. \(\sum_x p(x) = 1\)
Piv txwv yooj yim: cov dice ncaj ncees
\[
P(X=k)=\frac{1}{6}, \quad k=1,2,3,4,5,6
\]
b) Kev Ua Haujlwm Ceev ntawm Qhov Yuav Muaj Peev Xwm (PDF)
Rau ib qho kev hloov pauv random tas mus li \(X\), peb siv PDF \(f(x)\) kom qhov feem pua ntawm qhov sib nrug \([a,b]\) yog:
\[
P(a \le X \le b) = \int_a^bf(x)\,dx
\]
nrog rau kev muab ntawm:
1. \(f(x) \geq 0\)
2. \(\int_{-\infty}^{\infty} f(x)\,dx = 1\)
Nws tsim nyog hais ntxiv: rau qhov hloov pauv random tas mus li, \(P(X=x)=0\) rau txhua tus nqi ntawm \(x\). Qhov muaj feem yuav yeej ib txwm muaj txiaj ntsig thaum tham txog ntau yam.
4. Kev faib tawm ntawm cov khoom sib sau ua ke (CDF)
Txawm yog discrete lossis continuous, random variables tuaj yeem piav qhia los ntawm cumulative distribution function (CDF), uas yog txhais tias yog:
\[
F(x) = P(X \le x)
\]
CDF muaj ntau yam khoom tseem ceeb:
- Tus nqi ntawm \(F(x)\) yeej ib txwm nyob nruab nrab ntawm 0 thiab 1
– \(F(x)\) tsis txo qis (tsis txo qis)
– \(\lim_{x\to\infty}F(x)=0\) thiab \(\lim_{x\to\infty}F(x)=1\)
Rau cov hloov pauv sib cais, CDF yog "ntaiv" zoo li (nce ntawm qee qhov chaw). Rau cov hloov pauv tas mus li, CDF feem ntau yog du thiab yog qhov tseem ceeb ntawm PDF:
\[
F(x)=\int_{-\infty}^{x} f(t)\,dt
\]
5. Kev ntsuas ntawm qhov sib txawv ntawm nruab nrab: tus nqi xav tau (kev cia siab)
Thaum peb paub qhov kev faib tawm ntawm qhov muaj feem yuav tshwm sim, peb feem ntau xav sau cov lej sib txawv nrog ib tus lej uas sawv cev rau nws "tus nqi nruab nrab mus ntev." Qhov no yog tus nqi xav tau lossis kev cia siab.
a) Kev cia siab sib txawv
Yog tias \(X\) yog discrete:
\[
E[X] = \sum_x x\,p(x)
\]
b) Kev cia siab ntawm cov hloov pauv tas mus li
Yog tias \(X\) txuas ntxiv mus:
\[
E[X] = \int_{-\infty}^{\infty} x\,f(x)\,dx
\]
Kev cia siab tsis yog ib txwm zoo ib yam li "tus nqi feem ntau tshwm sim" (hom), thiab tsis yog tus nqi uas yuav tshwm sim tiag tiag, tab sis nws muaj txiaj ntsig zoo rau kev txiav txim siab, kev kwv yees, thiab kev tshuaj xyuas kev pheej hmoo.
Piv txwv ntawm daim ntawv thov: Hauv kev lag luam, kev cia siab tuaj yeem siv los xam qhov nruab nrab ntawm cov txiaj ntsig ntawm ib lub tswv yim, xav txog ntau yam xwm txheej thiab lawv qhov tshwm sim.
6. Kev ntsuas ntawm kev tawg: kev hloov pauv thiab kev hloov pauv tus qauv
Ob qho kev hloov pauv random tuaj yeem muaj tib qho kev cia siab tab sis qib sib txawv ntawm kev tsis paub tseeb. Yog li ntawd, peb xav tau kev ntsuas ntawm kev tawg, uas yog variance thiab standard deviation.
Qhov sib txawv ntawm \(X\) yog txhais li:
\[
Var(X)=E[(XE[X])^2]
\]
Tus qauv sib txawv yog lub hauv paus plaub fab ntawm qhov sib txawv:
\[
\sigma = \sqrt{Var(X)}
\]
Cov qauv siv tau uas feem ntau siv:
\[
Var(X) = E[X^2] – (E[X])^2
\]
Qhov sib txawv ntau dua, qhov sib kis ntawm cov nqi \(X\) los ntawm qhov nruab nrab ntau dua, uas txhais tau tias qhov tsis paub tseeb ntau dua.
7. Kev faib tawm qhov muaj feem yuav siv ntau zaus
Hauv kev xyaum, ntau cov hloov pauv random ua raws li qee cov qauv kev faib tawm. Qee qhov kev faib tawm nrov yog:
– Bernoulli: ob qho txiaj ntsig (kev vam meej/kev ua tsis tiav), piv txwv li muaj tseeb-cuav, muaj sia-tuag.
– Binomial: tus naj npawb ntawm kev vam meej los ntawm \(n\) Bernoulli kev sim, piv txwv li tus naj npawb ntawm cov tub ntxhais kawm tiav los ntawm 20 tus neeg.
– Poisson: tus naj npawb ntawm cov xwm txheej hauv lub sijhawm/qhov chaw, piv txwv li tus naj npawb ntawm cov xov tooj hu tuaj hauv ib feeb.
- Kev sib txuas ntxiv mus tas li: txhua tus nqi hauv lub sijhawm sib txawv yog qhov yuav tshwm sim sib npaug.
- Ib txwm (Gaussian): ntau yam xwm txheej ntuj tsim thiab kev sib raug zoo hauv zej zog los txog rau qhov kev faib tawm no, xws li qhov siab lossis qhov yuam kev ntsuas.
Kev xaiv qhov kev faib tawm kom raug pab ua qauv thiab kev tshuaj xyuas kom raug dua.
8. Vim li cas cov hloov pauv random thiaj tseem ceeb?
Cov hloov pauv random yog lub hauv paus rau:
- Cov ntaub ntawv txheeb cais: kwv yees cov pej xeem raws li cov qauv
– Kev sim hypothesis: txiav txim siab seb qhov kev thov puas raug txhawb nqa los ntawm cov ntaub ntawv
- Kev kawm tshuab: qauv kev tsis paub meej thiab kev kwv yees qhov yuav tshwm sim
– Kev tswj hwm kev pheej hmoo: ntsuas qhov yuav tshwm sim ntawm kev poob thiab cov xwm txheej hnyav
- Kev tsim kho thiab kev tshawb fawb: kev ua cov cim qhia, kev ntseeg tau ntawm lub kaw lus, kev tshawb fawb txog kab ke
Nrog cov hloov pauv random, peb muaj cov lus lej los tham txog qhov tsis paub meej.
Xaus
Ib qho kev hloov pauv random yog lub tswv yim tseem ceeb hauv kev tshawb fawb txog qhov muaj feem uas qhia cov txiaj ntsig ntawm kev sim random rau cov nqi lej. Cov kev hloov pauv random tuaj yeem sib cais lossis txuas ntxiv mus, thiab txhua tus muaj txoj hauv kev sib txawv los sawv cev rau qhov muaj feem los ntawm PMF lossis PDF. Ntxiv mus, CDF muab ib txoj hauv kev sib koom ua ke los saib qhov sib sau ua ke ntawm qhov muaj feem. Txhawm rau sau luv luv txog kev faib tawm, qhov kev cia siab yog siv los ua qhov ntsuas ntawm qhov sib txawv nruab nrab thiab qhov sib txawv / tus qauv sib txawv ua qhov ntsuas ntawm kev faib tawm. Kev nkag siab txog cov ntsiab lus yooj yim no yuav pab txhawb kev kawm cov ncauj lus siab dua xws li kev faib tawm qhov muaj feem, kev kwv yees txog kev suav lej, kev rov qab, thiab kev ua qauv kev pheej hmoo thiab kev tshuaj xyuas cov ntaub ntawv niaj hnub.
Yog tias koj xav tau, kuv kuj tuaj yeem ntxiv cov lus nug piv txwv thiab lawv cov kev sib tham (discrete thiab continuous) kom ua rau lub tswv yim ntawm cov hloov pauv random yooj yim dua rau kev nkag siab.
